Nature and Nedsure of AdhCta
.
w.o&&,
COLUMBIA UNIVERSITY, NEW YQRK, N. Y.
B A n agitation device i s a basic part of a l l unit process machinery. It i s like other unit operations in that i t i s the means by which specific optimum physical and chemical conditions are provided. In such cases it is, therefore, a prime process variable and must be controlled, There i s no truly rational basis for agitation equipment design; a l though the over-all operation appears to b e simple, i t i s invariably found to consist of a number of related factors which are difficult t o evaluate quantitatively. Since the general objectives are so varied and the range of conditions is so extensive and involved, the formulation OF a general, unified, and practical expression tor agitation does not appear possible. The most fruitful attempts to measure agitation have been made by dividing the subject into categories or systems
based on the physical natures OF the materials to be agitated or mixed, such as liquid-liquid, liquid-solid, liquid-gas, solid-gas, solid-solid, and solid-liquid-gas. Various workers have attempted to evaluate and relate the factors required to produce an arbitrary standard result, such as a definite dispersion of a finely divided solid in a liquid (mixing index), the dispersion of an immiscible liquid (standard emulsions) in another liquid, a definite rate of solution of a solid in 4 liquid (intensity of agitation), etc. Several have suggested and illustrated the use of models and dimensional analysis as a general practical method of approach to problems in agitation. The criteria of agitation, the value of previous studies, unsolved problems, suggestions for future studies are discussed, and a bibliography is included.
I T A T I O S is a unit operation of primary importance in the chemical manufacturing industries. An agitation device is a basic part of practically all unihprocessniachinery. It may be defined as the unii; operation by which particles of the components of a mass of materials are put in such space relation to one another that a desired result may be obtained in a minimum time with the least expenditure of energy, Although much attention of an empirical nature has been paid to the engineering and economic aspects of agitation, until recently little study has been made of the basic principles upon which it depends. The most fruitful attempts to measure agitation have been made by dividing t’he subject into categories or systems based on the physical natures of the materials to be agitated or mixed such as liquid-liquid, liquid-solid, liquid-gas, solid-gas, solid-solid, gas-gas, and solid-liquid-gas. These have been studied by various workers who have attempted to evaluate and relate the factors required t’o produce arbitrary standard results such as definite dispersion of a finely divided solid in a liquid (mixing index) or the dispersion of an immiscible liquid (standard emulsions) in another liquid, a definite rate of solution of a solid in a liquid (intensity of agitation) or the dispersion of a finely divided solid in a carrier gas, the rate of reaction in an agitated chemical system, the time required to produce a standard particle dispersion pattern in a solid-solid system, the time required to bleach a n oil by a solid adsorbent, and the net power required to produce arbitrary standards in the various systems. I n connection with some of these studies, several workers have suggested and illustrated the use of models and dimensional analysis as a g e l l e d practical method of approach to problems in agitation. Many of t’hese studies have been made with standard types of agitation devices such as paddles, propellers, turbines, pulsators, tumblers, mills, etc., for determining their performance characterist,ics and relative values as agitating devices. The operations to which agitation is applied have been enumerated by Wood, Whittemore, and Badger (126), Seymour (91), Hixson and Crowell ( 5 4 , Valent,ine and MacLean (low, 103, 105), Olive (82), and Maclean and Lyons (73). Types and structural features of agitating apparatuses have been described by Fischer (37, 38), Tyler (loo),Vollrath (Zla),Killeffer (65), Carpenter (19), Valentine and MacLean (102, lob), Olive (82),Badger and McCabe (S), and Antoni (1). Valentine and MacLean (104) de-
scribed the history of the development of agitating and mixing machinery. The use of air as a means of agitating is described by Kauffman (63). Pierce and Terry (83)shoa-ed how agitation can be used to improve heat transfer coefficients. FLUID MOTION
Early workers in the field of agitation recognized that the principles governing fluid motion are of basic importance in the quantitative study of any fluid agitation system. Although the science of fluid motion has received much attention, unfortunately for agitation it has been developed along three distinct lines: hydrodynamics, a study of the behavior of liquid fluids in motion in which abstract mathematics has been applied t o certain simplifying assumptions as a physical basis; hydraulics, in which experimental data, closely related to the physical facts, have been utilized to determine the equations of motion; and lately, that phase of aerodynamics which considers the forces on solids moving in compressible fluids, particularly air. Application of the eyuarions of hydrodynamics to engineering problems is exceedingly difficult since these equations become complex when dealing with the actual motion of fluids, and particularly when it is no longer possible to ignore factors of viscosity, surface tension, compressibility, boundary shape, and inertia (54). These factors tend toward the production of considerable turbulence which is desirous for purposes of agitation. Application of hydraulics to the same problems has been limited because of the. lack of an adequate unitary theory (4,and only during the past few years have the works of von Karman (62), Bakhmeteff (0, and Prandtl (8.4)led to a n e v method of attack on the problem. The earliest work in fluid motion was done in the seventeenth century by blariotte, Galileo, Torricelli, and Pascal. The term “hydrodynamics” was introduced by Daniel Bernoulli (1700-83) who discovered the theorem still known by his name. I n 1683 Newton proposed in his “Principia” the theory of the resistance to motion of bodies in fluids. D’Alembert (1717-83) also investigated resistance and first developed the equation of continuity for a liquid. Euler (1707-83) and LeGrange (1736-1813) formed the equations of motion for a perfect fluid and developed the mathematical theory. Navier (1785-1836) and Stokes (18191903) developed the equations of motion of a viscous fluid. The latter also worked out the equations for the velocity of falling 488
. 489
June, 1944
,
I
spheres in viscous fluids which are so important in sedimentation processes. Helmholz (1821-94) founded theories of vortex and discontinuous motion. Excellent treatises on hydrodynamics have been written by Lamb (70), Green (44), and Milne-Thomson (77). The distinction between fluid motion in conduits at high and low speeds was made by Boussinesq (9) and others, and the subject was clarified indisputably by the color band experiments of Reynolds (86). A filament of dyed water was introduced a t the center of a horizontal tube in which water was flowing, and the nature of the filament was observed a t varying velocities of flow. At low speeds the filament remained intact, but as the velocity was increased, the filament became thinner and seemed to oscillate slightly until a t high water velocities the filament became entirely disrupted. The disruption was attribubed to a change in the nature of flow of the liquid, and the velocity a t which the change takes place has been determined in various ways by Schiller (89), Bond (8),and Fage and Townend (36). At low speeds the term “laminar” or “streamline” and a t high speeds the term “turbulent” have been applied to describe the motion. The criterion for change from streamline to turbulent motion is measured by the Reynolds number, D v p / ~where , D is the pipe diameter, V the fluid velocity, p the fluid density, and p the fluid viscosity, all in consistent units. The commonly accepted value for the Reynolds number a t the point where the nature of the flow changes is 2100, but there is a transition zone between Reynolds numbers of 2100 and 3100 where the exact nature of flow is dependent on conditions of the fluid channel. The nature of streamline flow is well known. This motion has also been termed “laminar” since it has been found that the motion of the particles throughout any parallel plane, taken in the direction of the motion, is the same, both with regard to direction and magnitude of velocity. The velocity distribution in a close channel with streamline motion has been shown, both mathematically and experimentally, to be parabolic. Recent experiments of Fage and Townend (35) have indicated, however, that laminar motion is not necessarily rectilinear. They observed the motion of colloidal particles in water flowing in a rectangular conduit and found that this motion was sinusoidal; but for laminar motion as distinguished from turbulent, there was no particle motion perpendicular to the over-all stream motion. Turbulency of fluids in motion has received much attention by investigators in recent years. I t has been described by Bakhmeteff (4), Dryden (28), and von Karman (68) as a random motion of the particles constituting the fluid stream. I t is convenient to think of a fluid in turbulent motion as having a mean velocity U in a direction z, with a superimposed random motion resulting from instantaneous deviations from U at any point. The instantaneous deviating velocity has components u, v, and w in the z, y, and z directions, respectively, a t any point. Techniques, particularly the use of a fine hot wire anemometer (as),have been developed for measuring u‘ which is the root mean square averThe percentage turbulence can be deterage u(u’ = -). mined as the ratio of u’to the mean velocity a t the point, U. Turbulence can be measured in terms of the “mixing length” of Prandtl (84). This quantity is the average distance through which an eddy, which is formed by the irregular motion of a turbulent stream, moves before breaking up and losing its identity. Other work so far done to define turbulence is lengthy, and the reader is referred to Bakhmeteff (4), Dryden (28), von Karman (621, and Prandtl (84) for details of the theories involved. The nature of fluid motion adjacent to a solid interface has been studied. Even for highly turbulent main-stream conditions, the works of Stanton and co-workers (96, 96) Dryden and Keuthe (291, Burgers and Zijnen (18), Elias (38),Hansen (47), and Fage and Townend (35),have indicated the existence of a layer adjacent to the interface in laminar motion with a linear velocity gradient, having a zero value a t the interface. With increasing distance from the wall, the nature of the fluid motion changes from
streamline to turbulent, and beyond this transition zone the motion is entirely turbulent. The preceding is of interest in connection with the NernstBrunner theory (79) for the rate of heterogeneous processes. I n the original theory it was assumed that a stagnant layer of fluid existed adjacent to the interface between the two phases! even when agitation in the main stream was taking place. This concept was modified by Van Name and Hill (110) who stated that the fluid adjacent to the interface was in motion, but that in this moving layer, motion perpendicular t o the solid interface becomes small and does not affect materially the rate a t which dissolved substances are transported to and from the surface. Later Fage and Townend confirmed experimentally this observation concerning vertical component of velocity adjacent to a solid surface, and King (66) showed that the Nernst-Brunner theory was still tenable even though the fluid in the vicinity of the interface was not stationary. Simultaneous fluid flow and mass transfer are of interest in the field of agitation since this field is so often concerned with the dissolution of solids and gases by liquids under motion. Theoretical equations have been derived by analogy to heat transfer for the absorption of gases by liquids and the vaporization of liquids when brought .in contact with moving gas streams. Data for mass transfer in gas-liquid systems is fairly complete and verified. With one exception (68); the solid-liquid system, which is of far greater importance in agitation, has yet t o be investigated. A summary of theory and data for gas-liquid systems has been given by Sherwood (92). Dimensional analysis and the use of model experiments, as shown by Hixson and Baum (61,62, 67) is of untold value, both for planning experiments and the utilization of experimental data in the field of fluid flow. The theory and application of dimensional analysis to correlate results for entirely different fluid mediums, such as air and water, in one basic equation are ably illustrated by the work of Stanton and Pannell (97). CRITERIA OF AGITATION
Velocity of Heterogeneous Reactions. The phase of the problem of agitation that has received more attention than any other is agitation efficiency. Because of the complexity of the fluid motion existing in a system undergoing agitation, a direct mathematical attack is precluded and the several methods of a quantitative nature so far proposed make use of an “indicating substance’’ whose distribution throughout the system is a result of a11 the forces acting in the fluid. The effect of agitation on yelocity of heterogeneous reactions, particularly those concerned with dissolution of solids in liquids, has been studied extensively. Fick (36) in 1855 applied the laws of Fourier for heat conduction and Ohm for flow of electricity to the case of diffusion of dissolved substances through liquids. Fick’s law states that the quantity of solute dW which diffuses through area A in time dt, through distance dz, a t right angles to plane of A , is given by:
dW/dt = -DAdC/dx where C is the concentration, and D is a constant called “diffusivity”, usually given in terms of sq. cm. per second. The law holds true for constant temperature only, and is based on the assumption that the only motion involved is due t o molecular agitation. Fick proved the law for diffusion in one direction only, upward against the force of gravity. The law is not concerned with the process of solution but merely with the process of diffusion of a substance in solution. Weber (115) confirmed Fick’s law by an electrolytic method, and also determined the effect of temperature and concentration on the diffusivity constant, D. The kinetic theory for liquids has been only partly developed, and unlike the case of gaseous diffusion, no completely satisfactory equation for diffusivity in liquids can be given. Fairly complete tables are to be found in Volume V of International Critical
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
490
Tables. Arnold (8) gave an empirical equation which can be used in the absence of any direct experimental data. Noyes and Whitney (80) first studied the process by which solid substances dissolved in their own solution. They rotated cylindrical sticks of benzoic acid and lead chloride in initially fresh water a t constant velocities and constant temperature, and analyzed the resulting solutions after 10, 30, and 60 minutes of contact with the solid. They postdated that a film of saturated solution was always present at the surface of the solid, and the rate of solution was dependent on the rate of diffusion of this saturated solution to the main body of the fluid. For finite concentration gradients, Fick’s law becomes:
-DAAC/Ax
dW/dt
The thickness of the layer, Ax, through which the dissolved substance passed, wm stated to be dependent on the intensity of the agitation. If constancy of diffusivity D (which does not vary much, even with comparatively large concentration changes) and area A is assumed, Fick’s law a t constant agitation becomes: d W / d t = k’AC
Since the concentration at the interface, assuming that equilibrium has been established between the solid and solvent, is saturation concentration C,, and if the concentration in the main body of the liquid is represented by C, the concentration change is: AC =
c,
-c
Substituting, d W / d t = k‘(G,
- C)
which can also be written as d C / d t = k(C8
- C)
which in the integrated form becomes kt = h(C,
- C,) - ln(C, - C)
Noyes and Whitney verified this final equation by proving experimentally the constancy of k. Since the amount of solid dissolved w&s considered negligible, no correction on constant k was made for reduction in surface as dissolution proceeded. Bruner and Tolloczko (15 ) extended the Noyes-Whitney equation to include more varied and more soluble substances. They showed that constant IC was dependent on the exposed area, structure of exposed surface, stirring rate, temperature, and arrangement of apparatus. They also introduced the idea of the finite layer thickness, y, and the concept that k i5 determined by the quotient, D / y . Drucker (27) also investigated the Noyes-Whitney equation and found that the rate at which solids dissolved was inversely proportional to the volume of the solvent at constant agitation. Nernst (79) and Brunner (14) investigated the Noyes-Whitney equation for cases of simple solution, neutralization, action of acids on metaIs, and electrolytic actions. Nernst included all heterogeneous reactions in his generalization that, at every boundary between two phases, equilibrium is established with practically infinite velocity, and that the rate a t which all heterogeneous reactions proceed, in the absence of secondary influencing effects, is governed by the diffusion of the active substances to or from the interfacial surface. Brunner (14) also calculated the supposed film thickness existent when various substances dissolved in their own solutions and found this thickness to be in the order of magnitude, 0.03 mm. Senter (QO), Spear (94),Bodenstein and Fink (79, Jablczynski (69), Meyer (75), Roth (88), Delaham (%), Van Name and coworkers (107-1103, Hevesy (@), Collenberg @2), Heller (48), Hixson and Crowell (64), and King and Braverman (67) found experimental evidence to confirm the Nernst-Brunner theory. There are several types of reaction where the Nernst-Brunner theory does not hold. As originally presented, the theory was
Vol. 36, No. 6
found to be nonapplicable to those reactions where secondary influences existed. These secondary influencing effects were investigated by Drucker (27) and enumerated by Van Kame and Hill (111) as: formation of an insoluble coating a t the interface, two-stage reactions, passivity, and the evolution of a gas a t the interface as a result of chemical reaction. But heterogeneoue reactions have been investigated (15, 111) where, even in the absence of any of these secondary influences, the theory did not apparently hold. These reactions were characterized by a comparatively high temperature coefficient for the velocity and nonsensibility to agitation in direct contrast to those where the Nernst-Brunner theory held. Brunner (16) introduced the idea. that the reaction velocity a t the interface was not necessaiily infinite, in order to explain those reactions where the theory did not hold. Van Name. and Hill (111) segregated all heterogeneous reactions into three classes: (I) The chemical reaction proceeds much faster than the rate of diffusion, and the observed reaction rate will be determined by the rate of diffusion; (11) the chemicai reaction is much slower than the rate of diffusion and is, therefore, controlling; and (111) chemical reaction rate and diffusion rate are both of the same order of magnitude, and the observed rate is a function of both. These investigators also showed that, where the chemical reaction itself was first order, the observed reaction rate of type 111 reactions would also be first order, notwithstanding the influence of the diffusion reaction. The theory as presented by Van Name and Hill was extended and verified by King and Braverman (67) and King (66). Van Arsdale (106) and Davis and Crandall ($4) found experimental data on absorption of gases by liquids to confirm the theory Lewis and Whitman (71) applied what is, in essence, the Van Name and Hill theory to the absorption of gases in Iiquids. Marc (74), Ericson-Auren and Palmaer (33), and Wilderman (1$3) found the Kernst-Brunner theory untenable. Wilderman, in particular, attacked the entire theory, and in a severe analysib of the experimental results of Brunuer ( 1 4 , threw considerable doubt on the diffusion concept of heterogeneous reactions. The most striking evidence he presented in opposition to the NernstBrunner theory 7%as that the rate of solution in the same solvent of different faces of the same crystal of gypsum varied from face to face. Earlier Wilderman (192) presented an equation analagous to that of Noyes and Whitney, but he explained the theory of solution on the basis of osmotic solution pressures. I n conclusion, the law, as stipulated in its original form by Xoyes and Whitney, has been substantiated by experiment, regardless oi the mechanism by which the process takes place (64). The use of the dissolution .or reaction velocity constant of B heterogeneous reaction as a criterion of agitation was first suggested by Murphree (78), who derived a rate equation in terms of the linear dimensions of the crystals involved. Later, Heller (48) proposed that the rate of solution of various metals in corrosive liquids undergoing agitation could be used to establish what he termed a “normal agitation”; by comparison with normal agitation, the effect of agitation on a given process in any apparatus could be determined. Gunness and Baker (45) pointed out the shortcomings of this method, and declared that it is not applicable to commercial agitation since the rate of flow next to the metallic interface, and not turbulence in the main body, is the controlling factor on the reaction rate constant. That the method has other shortcomings is indicated by Roetheli and coworkers (39, 87) on the rorrmion rates of steel in oxygenated water, kept in motion by a stirrer. They found that speed of stirring had a pronounced effect on type of corrosion product formed and, hence, on rate of corrosion. At low speeds a granular coating of magnetic iron oxide was formed, which offered little resistance t o oxygen diffusion; but at high speeds a protective coating of gelatinous ferric hydroxide was formed, which offered considerable resistance to oxygen diffusion. Because of the opposing tendency of the two processes, the corrosion rate actually went through a maximum value a t n certain agitator speed
491
June, 1944 Hixson and Crowell (64),who used the rate of solution of crystals and tablets as a criterion of agitation, included the effect of a variable surface in their determination of the dissolution constant. They derived a general expression for the reaction velocity in terms of area and concentration. The derivation was based on the assumptions that agitation was the same against all parts of the surface, that no breaking up of particles occurs, that the surface is proportional to the two thirds power of the weight, and that the agitation is sufficiently intense to assure constancy of the concentration throughout the main bulk of the fluid. The expression was termed the “cube root law”:
dW/dt V(dW/dt)
or
-KgA(C, - C ) -KW’/a(W,
(1)
- Wo+ W )
(2)
Integration of Equation 2 leads t o the exact expression for the cube root law, which is cumbersome. Special cases were considered: A. When W, = Wo,substitution in Equation 2 leads to
V(dW/dt)
-KWS’a
(3)
which integrates to give (4)
B. When the concentration change is negligible, (C, constant and Equation 1 becomes dW/dt
5
- C) is
-KaW’l/3
(5)
which, on integration, yields
C. When the surface is constant, Equation 1 becomes
dW/dt = K,(Ca but since ( W/V)
- C)
(7)
+ Co = C, dW = VdC and Equation 7 becomes dC/dt = &(C,
- c)
(8)
which is the same as the Noyes-Whitney equation.
C = concentration at time t C, = saturation concentration
K
= cube root law constant
Kz, Kg, K I , & = constants A = surface area of crystals a t time t t = time V = volume of solution W = weight of crystals a t time t W, = initial weight of crystals W, = weight of crystals required to saturate volume V The law was verified for different solvents and solids and also for various types of agitation. The idea of standard agitation was introduced, and the effects of variations from the standard a@tation were measured i n terms of the dissolution constant. Hixson and Baum (61)developed a simplified form of the cube root law which gives comparable results with the exact expression. Hixson and Wilkens (67) further verified the cube root law. They measured the rate of solution of benzoic acid tablets in solvents of varying viscosity and density, both with free rotational and impeded rotational agitation with four baffles. They suggested a design method by which large agitating systems could be determined by experiments with small laboratory apparatus. I n this method a plot of the two dimensionless quantities, D 2 N / V against KD/Ced (where D is stirrer length, N is stirrer speed, V is kinematic viscosity, K is cube root law constant, C, is solubility, and d is diffusivity) for the small apparatus can be used t o design larger installations. These investigators also compared agitating system designs by the cube root law constant.
Wilhelm, Conklin, and Sauer (124) integrated, without limitations, the diffusion rate equation involved when particles of sodium chloride are freely suspended and dissolved i n water. The graphical solution of the integral gave a series of curves which can be used to predict the rate of solution of salt where values of diffusion rate constant k are known for particular agitation conditions and, of more importance, for the determination of k and its correlation with agitation conditions. Watson (114) measured the rate of solution of sodium carbonate decahydrate crystals when packed into a bed and subjected to the dissolving action of a stream of water. H e found that the rate of solution varied as the 0.7power of the weight of the crystals, a n excellent check on the value that Hixson and Crowell assumed to be a/*. The effect of agitator speed on the. velocity of heterogeneous reactions has been the subject of much investigation. Investigators have usually reported the reaction velocity constant as a function of agitator speed:
K = a(N)B with values for p ranging from zero to one (6,I S , 14,15, 40,48, 68,60,67, 69, 74, 75, 79, 107, 123). Where the reaction taking place is diffusion-controlled, it has been observed that the value of exponent @ is one or approximately equal to one (66, 67, 110). This is in strict accordance with the Nernst-Brunner theory, since it is to be expected that the thickness of the film adjacent to the interface will be nearly proportional to the stirrer speed, and hence the reaction velocity constant (K = D / y ) will be directly proportional t o the speed. On the other hand, for reactions controlled by the rate of the interfacial reaction, values for @ have been found t o approach zero (16, 67). Where both the diffusion and reaction rates are controlling, the process falls under category I11 as given by Van Name and Hill ( i l l ) ;it is usually found that, if a sufficiently wide range of stirrer speeds is taken, fl will vary from zero t o one. Reid and co-workers (68,76) studied the effect of stirring on the rates of several different reactions and found three types into which these reactions could be classified: (1) The reaction rate is a linear function of the stirrer speed; ( 2 ) the reaction is a linear function of the stirrer speed only after a certain speed is attained; and (3) the reaction rate is independent of the stirrer speed. They used a baffled vessel and a Witt (186)disk type of stirrer rotating between 3000 and 13,000 r.p.m. The exact shape of the stirrer was determined by the system under consideration, whether gas-liquid or solid-liquid. Results for type 1 reaction could be expressed by K=a+bN where a and b are constants. I n this category they found the following reactions to fall: ethylation of benzene in the presence of aluminum chloride, oxidation of sodium arsenite by oxygen, oxidation of nitrotoluene by alkaline potassium permanganate, and reduction of nitrobenzene by iron in presence of 0.1 N hydrochloric acid. Results for type 2 reactions can be expressed by K = b‘(N
- N’)
(for straight portion of curve only) where b’is a constant and
N‘ is the rate of stirring a t which the linear part of the curve intersects the axis if prolonged. To this class of reactions belong the hydrogenation pf cottonseed oil in the presence of a nickel catalyst and the solution of iron in 0.1 N sulfuric acid. The solution of magnesium in hydrochloric acid in the presence of potassium nitrate as a depolarizer, as studied by King and Braverman (67) falls in this category. To class 3 belong the saponification of ethyl benzoate and the hydrolysis of benzyl chloride. Marc (74) also found certain salts whose rate of crystallization was independent of stirrer speed. Watson (12.4) passed water through a bed of sodium carbonate decahydrate crystals and measured the effect of mass velocity on
492
INDUSTRIAL AND ENGINEERING CHEMISTRY
the rate of solution. A curve was obtained which showed that, as the velocity was increased, the rate of solution approached an asymptotic maximum. A similar observation was made by Van Arsdale (106) on the absorption of gases by liquids. The effect of other variables on the dissolution constant has been determined. Jablcsynski and Jablonski (60) measured the effect of the presence of organic solvents on the velocity of an inorganic heterogeneous reaction. Heller ( 4 8 )measured the effect of temperature, a t constant stirring, on the rate of solution of tin and cadmium in ferric chloride solutions. Wilderman (123) measured the temperature coefficient of the velocity with which bemoic acid dissolved in water under constant agitation. Van Name and Edgar (109) measured the temperature coefficient of the velocity of the reaction between certain metals and dissolved iodine, and found this coefficient to be 1.3. Van Name and Hill (110) measured the effect of dissolved alcohol and sugar on the rate with which cadmium dissolved in an iodine solution in the presence of potassium iodide. The results indicated that the rate of solution a t constant stirrer speed varied indirectly as the viscosity of the solution. Later Van Name (107) measured the temperature coefficient of reaction velocity when cadmium was dissolved in a solution of iodine in the presence of potassium iodide. H e showed that the main effect of temperature was due t o the change in solution viscosity. The coefficient K / e T was shown to be constant a t any temperature in the range studied (0" to 65' C,), where K is the reaction velocity constant, 8 is the fluidity of the solvent, and T is the absolute temperature. King and Braverman also found a similar viscosity effect (6'7). Finally, Hixson and Wilkens (67) measured the effects of vessel size, viscosity, and agitation speed on the rate with which tablets of benzoic acid dissolved in water and various oils. Solid-Liquid Suspension Method. White and co-workers (118121) and Hixson and Tenney (56) tested several agitators by their ability to suspend particles of sand in fluid mediums. White and
-
-
co-workers (121) measured the distribution in water of approximately 65-mesh sand in a tank fitted with a paddle agitator, rotating a t constant speed. They found that the sand concentration reached a maximum within 2 minutes after the paddle was started, an indication of the surprising efficiency of paddle agitators as previously confirmed by Badger and co-workers (126) and Hill (60). They also found some classification of the sand with this apparatus, leading to the suggested use of the simple paddle agitator as a classifier (120). Later these investigators (129) extended their work to include variation in particle size and the relative amounts of sand and water. They suggested the use of the agitation speed, a t which the maximum saturation of sand in water occurs, as a measure of agitation intensity. Finally, White and Sumerford (118) tested simple paddle agitators of various size in cylindrical vessels to determine the optimum dimensions by the sand suspension method. To summarize: If R feet represents the tank diameter, best suspension will occur if the agitator blade is made R feet long and R feet wide, and the clearance between paddle and tank bottom is l/g R feet. These data, coupled with power requirements for simple paddle agitators (116, 117, 118) can be used to determine optimum dcsign. The importance of proper sampling in the suspension method was pointed out by Gunness and Baker ( 4 5 ) . The use of sample tubes inserted in the side of a tank, as used by White and coworkers, has several objectionable features. .Hixson and Tenney (66) used a better technique in which a simple plunger device was inserted to the desired position in the tank, and a sample was withdrawn almost instantaneously. These investigators measured the sand-suspending characteristics of a 45" propeller in a cylindrical tank, with both water and sucrose solutions as the suspending fluid. The ratio of sand to liquid was kept constant in all experiments a t 1 part of sand to 10 parts of liquid by weight. The mixing index proposed and used by Hixson and Tenney differs from that of White and co-workers; the latter suggested the
Vol. 36, No. 6
agitator speed required to reach the limiting value of sand concentration as the index, whereas the former proposed the ratio of average sand or liquid concentration a t several points in the apparatus to the concentration that would exist if even distribution occurred. The method suggested by Hixson and Tenney seems to be more universally applicable since it is adaptable to the measurement of agitation intensity a t any stirrer speed (45). Liquid-Liquid Systems. The time necessary to mix two miscible liquids completely or the ability of a stirrer to emulsify two nonmiscible liquids have been suggested as criteria of agitation. Wood, Whitternore, and Badger (126) tested a paddle stirrer by measuring the time necessary to mix a salt solution completely with water. The salt solution was introduced into the bottom center of a stirred tank of water, and the distribution of salt was measured by electrical conductivity cells, located at several points throughout the tank. The time necessary for the conductivity cells to come to a constant reading was used as an indication of solution turbulence. Hixson and Cervi (20) measured intensity of agitation with a propeller type agitator. They varied propeller speed and measured the effect of this variation on the degree of distribution of kerosene in water. Dodd (26) measured the degree of mixing of two miscible liquids by noting the time necessary for the striations due to density differences to disappear. He used the systems gasolinecarbon disulfide and brine-water to form the striae, which he made discernible by projection on a screen with a strong beam of light. A reciprocating stirrer was used, and the disappearance of the striae occurred abruptly, determinable within 3% of the total mixing time. Bissell (6) showed an adaptation of the striae method to measure the lines of flow in an agitated vessel. Kambara, Oyamada, and Matsui (61) tested various types of agitating impellers by noting their ability to cause suspensions of oil and water. The oil was dyed, and the degree of suspension in water was determined by the intensity of color as measured by a photoelectric cell. Of the wide variety of impeller shapes tested, propellers were found to be most efficient. Brothman (11) gave a method for selecting emulsifying equipment by small laboratory tests, based on their ability to produce emulsions. Other Methods. Various methods for testing agitators have been proposed and used. Hill (50) cited a case where adsorption of a dye on a solid, u-hich was kept in suspension in a liquid by a simple paddle agitator, was used to test agitation efficiency. Esselen and Hildebrand (34) applied ultraslow-motion photography t o a study of agitators. Hixson and Luedeke (55) suggested total power to the driving element, less the power consumed by wall friction, as a criterion of mixing efficiency. MacLean and Lyons (73) proposed pumping capacity in conjunction with entrainment as a measure of mixing efficiency. Bissell (6) suggested the bleaching action of fuller's earth on vegetable oils to determine mixer efficiency, since this action is dependent on the degree of agitation. Time to bleach to a certain color could be used as a reference point. Gunness and Baker (45) surveyed and criticized previously proposed criteria of agitation and suggested a method for determining the optimum operating conditions for agitating devices already in use. The mixing of gases prior t o reaction was studied by Chilton and Genereaux (21) by a visual method. They caused two streams of different gases to come together by means of T- and Y-connections, and measured efficiency of mixing by noting the distance downstream where complete mixing occurred. With T-connections and a velocity of the added stream 2.5 times that of the main stream, best mixing took place. POWER REQUIREMENTS Q F SYSTEMS UNDERGOING AGITATION
It is important to realize that power input to a system undergoing agitation may not be a criterion of the efficiency or degree
June, 1944
493
of the agitation (106). Thus, with fairly viscous fluids, a large amount of power may be consumed in producing a vigorous local action in the vicinity of the mixing element, with little or no action outside this region. This would satisfy but one of the requisites for good mixing (105), since a suitable motion of the entire body of material must be maintained. There is a growing tendency in chemical industry t o standardize types of equipment used. Unfortunately standardization of mixing equipment is difficult since a large variety of agitating machinery is available to perform any one particular task. As a result of lack of standards of efficiency and power requirements, it is almost always impossible to pick the best mixer for a specific task. Of all the types of equipment, those in most common use are paddle agitators, the propeller, and the turbine mixer. With the exception of the paddle type agitator, little work has been done on the power requirements of systems undergoing agitation. I n addition t o the above three important types, several other types of apparatus are encountered in industry but usually under strenuous conditions where high viscosity .exists or it is desirous to produce a grinding action simultaneously with a mixing action. Other exceptional cases do exist but not with the frequency of the above two. Power requirements for equipment such as kneaders, dough mills, pug mills, pebble mills, rotary pan mills, emulsifiers, colloid mills, tumbling barrels, and others, are usually impossible to determine without running a load test on the actual equipment. This is a disadvantage as far as design work is concerned, but because of the complexity of the subject, it seems that no other solution will be immediately available. Where standard pieces of chemical equipment other than mixers are used to provide agitation, the power requirements are usually easy to determine. Thus, in using a n outside circulating centrifugal pump in connection with a tank, the horsepower is easily calculated from a knowledge of the pumping capacity and the pressure drop. Similarly, in using a packed tower t o mix gases, extensive data are available for calculating pressure drop through packed towers that can easily be used to determine power requirements. I n general, however, it is inadvisable t o secure mixing by the above methods. For example, Valentine and MacLean state that the power required by a turbine mixer is approximately one thirtieth that required by an outside circulating pump delivering the same volume of liquid (106). Some work has been done on the power requirements of bodies rotating in various fluids, but mainly in connection with turbines and centrifugal pumps. Thomson (99) was probably the first to study the friction of rotating disks. He measured the torque produced on the containing vessel by rotating a smooth disk in water. The data he obtained fit the equation
P
=
3.4
x
10-10n*~5
(9)
between limits of nD from 192 to 518. Thomson stated that the friction was only slightly affected by the clearance between the rotating disk and the containing vessel, a n observation unconfirmed by later investigators. The data of Froude and Coulomb, cited by Unwin (101), indicated a difference between frictional resistance a t low and high speeds. Coulomb found by oscillating thin disks, suspended by fine wires in fluids, that the frictional resistance varied directly with the velocity, was sensitive to viscosity, and was nearly independent of the surface roughness. Because of the nature of the apparatus, the speeds were necessarily slow. Froude's results with towed boards at higher velocities indicated that the frictional resistance varied nearly as the square of the velocity and was sensitive to the nature of the surface roughness. Froude also found that the critical speed at which the nature of the fluid motion changed varied as the surface roughness. These results are in close agreement with those indicated for fluid flow i n round pipes by the Fanning friction equation d F = 2f( Va/g) ( d N / D ) (113). For speeds below the critical (streamline or laminar flow) this equation indicates that the friction varies directly as the
velocity and viscosity, whereas for speeds above the critical (turbulent flow), the equation indicates that the friction varies as approximately the 1.8 power of the velocity. The exact value of this exponent varies with the degree of surface roughness. I n a study of the effect of surface roughness on the friction of rotating disks, Unwin (101) rotated disks varying from 10 to 20 inches in diameter in both water and sugar solutions, using a n adaptation of Froude's dynamometer t o measure torque. The surface roughness was varied from turned and polished brass to sand and gravel cemented on metal. For any given disk of specified roughness, the data fitted the equation I
p
=
kn3D4.85
(10)
with the exact exponent of n varying for different surfaces. Unwin also found a difference in fluid friction at high and low speeds. Thus, at high speeds an increase in chamber size caused a n increase in friction, but at low speeds the resistance decreased considerably as the size of the chamber was increased. When the chamber was roughened, the resistance of the rotating disk increased considerably, in some cases almost as much as when the disk was roughened. Unwin found that a n increase in viscosity caused a n increase in resistance to rotation, but no mathematical relation was given. Odell (81) also confirmed a critical speed for rotating disks. He rotated cardboard disks varying from 15 to 47 inches in diameter in air and measured the power consumed. At low speeds the power absorbed varied as the square of the peripheral velocity and at higher speeds, as the 3.5 power. Odell showed that the product nD2 had the same value at the critical speed for all disks. This product is analagous to the Reynolds number ( D v p / ~or nD2p/p), since in Odell's work the density and viscosity of the fluid were always constant. For speeds above the critical the data are correlated by a n equation of the type:
P = kv3D2 (11) Zahm (1a7) investigated the friction of air on long plates subject to a rectilinear air stream while suspended in a closed channel. The flow in all experiments wax turbulent. Stodola (98) integrated Zahm's equation for the case of a rotating disk and obtained the equation: P = 3 32 x 1 0 - ~ ~ 2 ~ 2 . 8 5 (12) Zahm also stipulated the friction loss was approximately linear with respect t o fluid density-a phenomenon confirmed dimensionally by Rayleigh (86) and experimentally by Stodola (98). Gibson and Ryan (43) investigated the effect of clearance and surface on power losses of rotating disks. They found that the smaller the clearance between the rotating disk and the casing, the lower the power consumed due t o windage. A 12-inchdiameter, rough, cast-iron disk rotating at 2000 r.p.m. in a rough casing with */B-inch clearance required 40% more power to overcome windage losses than did a painted disk and casing under similar conditions. The effect of clearance was more pronounced on the power consumption of smooth disks than for rough disks. Addition of vanes to the rotating disks increased windage losses enormously. Thus, when four radial vanes, I/, inch deep, were added to a 12-inch-diameter brass disk, the power consumed was increased 350%. These experiments show the importance of surface and clearance on power consumption of rotating bodies. Odell (81),Kerr (64), Buckingham ( I T ) , and Stodola (98) have investigated the windage losses of unenclosed disks and turbine wheels. I n general, they found that this loss is composed of two parts: the friction of the vanes and the friction of the disk on which vanes are mounted. Although there is a fairly definite critical value of Reynolds number for smooth rotating disks, for ordinary turbine wheels consisting of vanes and disk there is no sharply defined critical speed. Buckingham's equation for windage losses of rotating turbine wheels is probably most comprehensive: p = apO.QnZ.QD4.8 P 0.1 113)
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INDUSTRIAL AND ENGINEERING CHEMISTRY
494
This equation can be arranged in terms of a Reynolds number:
P = 5600 av3D2 (:UP)
The exponents of Equation 14 check those given by Spannhake (93) and Hanocq (46), with the exception of that on the Reynolds number which these investigators found to be 0.2. Integration of the Fanning friction equation for flow in circular pipes for the case of a rotating disk yields the equation P = 2.94 X 10-10~2,S~4.Sp0.8y0.2 (15) which can be rearranged to
in which the exponent on the Reynolds number checks that of Spannhake. Validity of Equation 16 is substantiated by the data of Ode11 (81), who found that 0.305 horsepower was required to rotate a 47-inch-diameter cardboard disk a t 740 r.p.m. in air a t room temperature, whereas Equation 15 indicates a consumption of 0.29 horsepower under the given conditions. The effect of adding blades on the power consumed by a rotating disk was indicated by Buckingham in terms of coefficient a. He expressed CY as a function of the blade length and the turbine diameter, a = A 1/D (17)
+
The total expression for the power consumed by a rotating turbine wheel can then be given by
Paddle Agitators. As far back as the sixteenth century Agricola used paddles for agitation which were strikingly similar to those of today (104). Notwithstanding the antiquity of this type, only in recent years has any systematic method for studying its characteristics been introduced. Wood, Whittemore, and Badger (116)were the first to attempt to systematize the problem. They determined the power consumption and mixing efficiency of a paddle stirrer with beveled edges by measuring the net power delivered by the driving motor and the length of time necessary to mix brine solutions with water. Difficulties were encountered in attempting to measure power consumption by this method, since the recording meters fluctuated widely. I n general, these investigators found that, above a certain speed (about 30 r.p.m. for their apparatus), the power requirements increased considerably but without a corresponding increase in mixing efficiency. White and co-workers (117) measured the power requirements of simple paddle agitators by determining the torque required to drive the vertical shaft. These investigators varied fluid properties, paddle size, paddle location, speed, tank diameter, and fluid depth. They correlated the data by plotting the modified Reynolds number, L z N p l p , against PIL3N3pDt1.1Wo.3H0.6. A curve was obtained similar to the usual friction factor plot, with a break separating the conditions of streamline and turbulent flow (113). The exact location of the transition point is not sharply defined but occurs a t a Reynolds number, L2Np/p, of about 50. The equation that fitted all data best was given by p = O.OOO129L2.7ZPO.14N2.86p0.86Dt1.1WO. 3HO.6 (19)
The exponents on this equation for p , p , and N are in close agreement with those of Buckingham for power requirements of rotating disks. Although not for the same type of agitating device, data of Wood, Whittemore, and Badger (126) and of Hixson and Wilkens (67) are included in Equation 19 fairly well. Propeller Type Agitators.
As early as 1681 Robert Hooke pro-
posed the idea of making a screw similar in construction to a windmill to work in water (51). From then on, the development
of propeller use was confined mainly to the propulsion of marine vessels. Only during the present century has the propeller been put to any extensive use as a n agitating device. The design of propellers, from the viewpoint of marine propulsion, has been well investigated by Froude (4l),Dyson ( S I ) , Durand (SO), Curtis and Hawkins (as), and others. Few investigators have, however, measured the power requirements of propellers used as an agitating device, and no comprehensive formula similar to that for paddle agitators can be given. Wood, Whittemore, and Badger (126) used a semipaddle propeller type stirrer in their experiments. Notwithstanding the fact that one face of this paddle was planed off to a 45" angle, the data for power consumption are fairly well correlat,ed by the equation of White and co-workers (117). Hixson and Wilkens (57)measured the power absorbed by fourbladed propellers having a pitch of 45" and rotated jn cylindrical vessels. All the apparatus were dimensionally similar in respect to the size and position of the propeller, liquid depth, and container diameter. Measurements were made both with and without baffles. Power consumption was given in terms of watts per 100 gallons, plotted against r.p.m., for water. I n general, they found that the power consumption per unit volume increased r a p idly as the size of the system was increased and as the free rotational flow was modified with baffles. Although no factor for surface roughness is included in the equation of White and co-workers (ll?),the data of Hixson and Wilkens (57) for systems coated with varnish is fitted by the equation for paddle agitators fairly well, although this equation is for a system with far greater surface roughness. Valentine and MacLean (105) give a nomographic chart for determining power requirements of propeller agitators in terms of pitch, diameter, and speed. This chart can be used only for the roughest estimates, since it does not include the important variables, density and viscosity. Hixson and Tenney (56) measured power consumption of a 45" propeller in liquids varying in viscosity from 0.9 to 44 centipoises. The curves of power against speed are similar to those of Hixson and Wilkens (67). Hixson and Tenney (66) found that the power consumption varied as the 2.8 power of the stirrer speed, a result well in agreement with that of White and co-workers (117) who found a n exponent of 2.86 for paddle agitators. Hixson and Luedeke (66) studied wall friction in liquid agitation systems. They obtained data for friction drag in systems using 45O, 60", and 90' (paddle) stirrers, all arranged with geometrically similar dimensions as previously used by Hixson and Wilkens (67). Power going to wall friction was expressed by
p =
+
kp0.1Qcc0.211a2.1eDf3.60(D 4H)
sin e
(20)
The value of constant k was not determinable, since it is necessary to know the liquid velocity immediately adjacent to the stagnant film at the vessel wall, a figure that was not determined by these investigators. The authors showed the existence of two distinct modes of flow in a system undergoing agitation-laminar and turbulent. Equation 20 is for turbulent flow only, which occurs at'Reynolds numbers, pND2/p, greater than 3 X 106. It is interesting to note t h a t the exponents of p , p , and N agree with those of Equation 5. Turbine Type Agitators. Although the turbine type of machine has been known since the Renaissance, its application as a mixing device has taken place only during the past fifty years. As a mixing device, it is used in a similar manner like a centrifugal pump to impart a pumping motion to the fluid. Essentially it consists of a rotating impeller, similar in construction to that of a centrifugal pump, and a deflecting ring. Actually the impeller can be used without the deflecting ring, but tests (57) have shown that, above certain speeds, the use of the deflecting ring is advantageous because of the better agitation secured. The motion produced by the turbine type mixer is similar t o that of the pro-
495
June, 1944 peller; whereas the latter pumps fluid parallel to the driving shaft, the former imparts a motion radial to the driving shaft. Until recently the only information available on the power consumption of turbine agitators was the limited data of Valentine and MacLean (106)and that found in manufacturers’ catalogs. I n 1942 Hixson and Baum (69), using a turntable dynamometer, measured the torque requirements of turbine agitators under a variety of conditions. They used a modification of the method of Hixson and Luedeke (90)and of Buche (16) for the correlation of their experimental results which was based on an integration of Newton’s law expressing the friction drag resulting from relative motion between a body and a fluid in contact. An equation for the power requirements of a series of dimensionally similar agitators was derived. A standard turbine agitator design waa established, and the effects of single variations on power requirements were detymined. They developed an empirical method for predicting the power requirements of other agitators which differ in more than one manner from the standard design. The comparisons they made between predicted and measured horsepower for plant-size equipment compared favorably. These are the accomplishments on the theoretical side toward the determination and evaluatipn of the basic factors of agitation. Substantial progress has been made toward establishing this unit operation on a quantitative basis. Much remains to be done. NOMENCLATURE FOR POWER REQUIREMENTS
A , B = constants D = disk diameter or turbine diameter a t root of blades, ft. = pipe diameter, ft.
2
$ g
H k 1
L
n
N
N,
p
P
v
v
W
z
cy
@ p
e cc
= tank diameter, ft. = Fanning friction factor, no dimensions = friction loss, ft.-lb./fb. = acceleration due to gravity = 32.2 ft./sec./sec. = height of liquid in tank, ft. = constant = turbine bucker length, ft. = paddle le th, ft.
= stirrer or%sk speed, revolutions/min. = stirrer or disk s ed, revolutions/sec. = length of pipe, = pressure drop, lb./per sq. ft. = horsepower = 550 ft.-lb./sec. = peri heral velocity, ft./sec. = fiuitvelooity, ft./sec. = paddle width, ft. = stirrer pitch angle, degrees = factor in Buckin ham equation = exponent in Bucfingham equation = fluid density, lb./cu. ft. = (1.13 x 12) in Hixson-Luedeke equation = viscosity, lb./sec./ft.
-
SUGGESTIONS FOR FUTURE WORK
What should be the direction and nature of future research on this important unit operation? A few suggestions are ventured which should yield results of practical value: 1. Determination of the effects of design variations on the value of dissolution constant K. The method would be similar to that employed in measuring the power input to agitators of various designs. Establish the rate of dissolution in the standard turbine as the basis, and then measure the dissolution rates over a wide range of Reynolds numbers in apparatus which differ in one way from the standard design. Then make a correctioD chart for the design variations and estimate the dissolution constants for commerical equipment from this chart, and compare them with actual results. 2. Economic balance in agitator design. Select a specific problem-say the production of a certain amount of a 10% sodium chloride solution-and determine the optimum design and size of the agitating equipment to perform this task. The factor to be considered would be amount of power and its cost; the rate of solution (along with time required to discharge and charge the materials to the agitator) would determine the size of equipment. Then the problem would be to see whether a battery of mixers or
a sin le mixer would be best, to evaluate the optimum rotational speef, and to determine the most efficient design. I n all cases selection should be made on the basis of lowest cost or maximum profit per unit of product. 3. The ability of an agitator to suspend solids. Determine the minimum conditions of agitation to keep in suspension solid particles of various shapes, sizes, and weights. These experiments can best be performed in a glass container. Vary the fluid properties and the agitator speed over as wide a range as possible. The results can probably be correlated by a modified form of the Stokes law. A Reynolds number can be defined for agitation systems, and a correlation based on this number seems feasible. The work can be extended to studies in geometrically similar systems to see whether the results in models can be used to predict the results in large-scale equipment. 4. Heat transfer in jacketed and coil-heated agitation vessels. Hixson and Baum suggested that a modified form of the Nusselt type equation be used to correlate heat transfer data in heated agitation vessels. A verificatioh of this would be extremely helpful. This research should be carried out with a series of liquids to give a wide range of fluid properties, and if possible, in78 series of geometrically similar vessels. A steam-heated coil or jacket could be used. 5. Simultaneous mass transfer and chemical reaction. Derive the equations for the case where a material is simultaneously diffusin and undergoing a chemical reaction. Hixson and Baum cf!d this for the c a y of benzoic acid dissolving in a caustic solution; for such a system the rate of reaction depends on the rates of diffusion of the components. This work should be extended to other systems, and the effect of agitation measured on the rate of this type of mass transfer. Problems of this type are of extreme importance industrially-for example, the reduction of or anic oompounds by the suspension of iron filings in the liquicfb agitation, etc. 6. d e power requirements of propeller agitators. No accurate data on the power requirements of propeller agitators has yet been published. It would be necessary to design a dynamometer to measure power input when the drive shaft enters the vessel a t an an le. Otherwise, the procedure would be similar to that used by &xson and Baum. 7. The power requirements for liquids not having true viscosity characteristic#. The power requirements for agitation of slurries, sludges, @astic liquids, pseudo- lastic liquids, and inverted plastic liquids is an important iniustrial problem. The studies to date have been mostly concerned with only truly viscous liquids. It should be extended to studies of other liquids. The same apparatus could easily be used. 8. Mass transfer in liquid-liquid agitation systems. A problem of increasing industrial importance is that concerning the extraction of material from one liquid by a suitable solvent. To increase the rate of extraction, the two liquids are brought into intimate contact by means of an agitator. No data are yet available on the rate of extraction between two liquids, except for continuous extraction in a acked tower. The turbulence produced in such towers is s m d b e c a u s e of the low relative velocities between the two phases due to small differences in density. The resulting transfer rates are also small. Agitators in conjunction with settling chambers give much higher rates. I t would be necessary to derive an equation for this case based on the rate of material transfer to and from the liquid-liquid interface. The efficiency of an agitator for this type of phase combination could be expressed in terms of the interfacial surface area that i t produces. This would also be of some interest in problems of emulsification. Systems which might be most easily studied are rate of distribution of acetic acid between water and benzene, rate of distribution of benzoic acid between water and benzene, etc. The results obtained from such studies, when correlated with those now available, should contribute much toward establishing a rational basis for agitation equipment design and operation. BIBLIOGRAPHY
(1) Antoni, A., G6nie civil, 114,77,109 (1939). (2) Arnold, J. H., J . Am. C h m . Soc., 52,3937 (1930). (3) Badger, W.L., and McCabe, W. L., “Elements of Chemical Engineering”, New York, McGraw-HiU Book Co., 1936. (4) Bakhmeteff, B. A., “Mechanics of Turbulent F~oN”,p. 8 , Princeton Univ. Press, 1936. (6) Bekier, E., and Rodziewica, K., Roczniki Chem., 6 , 869 (1926). (6) Bissell, E. S.,IND.ENQ.CHEM.,30,493 (1938). (7) Bodenstein and Fink, 2.physik. Chem., 60,l (1908). (8) Bond, J. E., Proc. Phys. Boc. (London), 1-43,46 (1931). (9)Boussinesq, J., M e n . amd. eci. Paris, 23, 1-680 (1877).
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