A Mass
7 The operation ot agitation i s appraised from the standpoint of t w o separate types of interrelated actions: (1) those characteristic of the impeller and stream discharged by it, and (9)those dependent upon the nature of fluid flow required to produce the desired result in the fluid. The first i s the result of impeller performance; the second dictates the agitation requirement. In other words, the action of an impeller produces a definite mass flow; the amount and speed of flow required for a given result are controlled by the physical properties of the fluids, solids, or gases to be handled. Both subjects can be expressed in terms OF mass flow. The three prime performance characteristics of impellers are discharge capacity, discharge velocity, and power required to produce them. Data are given showing the relation of the discharge characteristics and power required for several sizes of propellers. The relations are general and w e l l established. Methods For evduation OF impeller performance and the action OF the discharge stream are proposed. Means For evaluating agitation requirements are discussedi examples are given for operations such as maintaining or producing suspensions, which show h o w an impeller can be selected to produce a desired result once the agitation requirements are determined. A plea is made for establishing firm bases for the measurement of data incident to agitation problems, and for separating the component problems to logical foundations which are susceptible to analysis by the mass flow technique. The photograph i s a view OF the impeller and tank OF the differential dynamometer used to obtain the data on which this report i s based (see page 500).
Velocity Theory for
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MIXING EOUIPMENT COMPANY, INC., ROCHESTER, N. Y . I
HE lack of accurate quantitative terms to describe agitation has complicated tpe use of agitators for many years. There are no published data which permit the calculation of agitation requirements for any particular operation, much less the selection of type and size of agitator required. Analysis of this statement reveals that actually two problems must be solved The first is to secure complete performance data on all agitator impellers, and the second is to compute agitation requirements in terms of impeller performance. The apparent difficulty of arriving a t a method of computjng agitation requirements has led to almost complete neglect of both problems, so that today there is neither a method of computing these requirements nor adequate performance data on various impellers. Much of the work in this field has sought to combine the two problems in a n effort to solve both simultaneously-for example, Hixson and Baum’s work with propellers in liquid-solid systems (2). They showed the performance of propeller agitators i n terms of the results accomplished on a specific application. Work of this kind is useful, but it has one fundamental f a u l e t h a t the measurements of impeller performance are apt to result in misleading generalizations. To overlook the correlation of the results with other similar applications precludes the prediction of performance in any other system; therefore, truly basic data are not being obtained. Likewise, agitation requirements which are found apply only to the particular combination of materials used and are of littlenvaluein calculating the requirements of any other combination.
Present addrese, University of Virginia, Charlottesville, Va
One method of computing agitation requirements is based on using certain ratios of power per unit volume. The success of this method depends on a wide range of experience with the impellers used in order to determine proper ratios for a particular operation. Each different type of impeller requires a different -power ratio for the same operation with the result that the experience of those who specify agitators is confined to only one or two types of impellers. This is one of the most widely used methods of determining agitation requirements and should be adequate proof that the application of agitators is an art, for no field which puts such a premium on experience should be regarded as a science. Nevertheless, because the method is successful within limits, considerable work has been done to develop methods for computing the power requirements of some types of impellers under limited conditions. These studies have added little fundamental data to .the solution of the problems.
T
IMPELLER PERFORMANCE
’
499
Most of the work on impeller performance hrts resulted in data on the power necessary to drive the impeller. This is only one of three fundamental characteristics of any impeller performance. The other characteristics are discharge capacity and discharge velocity. Determination of the power required to drive an impeller (without wide application experience) is useful only to ensure that a suEciently large source of power is applied to the machine to drive it at the chosen speed. The power required is,
500
INDUSTRIAL AND ENGINEERTNG CHEMISTRY
in itself, no indication of the effectiveness of the impeller as a n agitator, for one of the primary functions of an agitator impeller is to create turbulence by the movement of liquid, and this does not bear the same relation to the power input for every type of impeller or for the same impeller run at different speeds. As an illustration, Figure 1 is a logarithmic plot of rotational speed against horsepower for three-bladed marine propellers, one 6 inches in diameter with an 8-inch pitch, and the other, 10 inches in diameter with a 10.9-inch pitch. The graph shows that the 6-inch propeller must be run a t 1350 revolutions per minute to draw 1 h.p., whereas the 10-inch propeller will draw 1 h.p. a t 720 r.p.m. Using the figures for the blade area as given by Figure 1, the 6-inch propeller will discharge only 454 gallons per minute while drawing 1 h.p., whereas the 10-inch propeller will discharge 1210 gallons per minute for the same amount of power. I n propeller applications experience has shown that a different result is obtained for the same power input at different rates of Auid flow. The discharge capacity, or mass of fluid discharged, is a fundamental performance characteristic of any agitator impeller.
Figure 1. Log Graph of Speed vs. Horsepower for Three-Bladed Marine-Type Propellers
Figure 1 illustrates another point which may not be generally appreciated. Both of the lines are straight, and therefore the dope of each is the index showing the variation of power with speed. The slope of both of these lines is almost exactly 3; thus the power varies as the cube of the speed. The curves are plotted from data obtained by a dynamometer; of the hundreds of different sizes and types of impellers tested on this machine, the variation from this cube law has been negligibly small. This cube relation is not presented as a new fact, for it has been covered adequately in texts on hydraulics, but only to point out that one accurate power measurement in a liquid a t a known speed is all. that is needed to determine the power requirements for that impeller in the liquid for any speed. Views of the differential type dynamometer used to obtain these data are shown on pages 498 and 499. The tank with which it is serving is 8 feet in diameter and 8 feet deep. The capacity of the dynamometer is approximately 7 h.p.; its speed may be varied from 27 to 1600 r.p.m. Some of the impellers tested are shown in Figure 2 . Simple dynamometers can be built to obtain these results so quickly and accurately that is seems a waste of effort to devote much time to developing empirical formulas for speed-power functions. The importance of the third characteristic, discharge velocity, becomes apparent when the problem of suspending heavy solids in light liquids is considered. A specific example is the suspending of sand in water where the sole object of agitation is to maintain the sand in uniform suspension throughout the vessel. This means that there must be no settling in any part. It may be assumed that there is some minimum velocity which must be maintained in all parts of Lhe vessel to prevent the settling of the sand. Since the velocity of the stream discharged from the impeller decreases with the distance it travels from the impeller, the lowest velocities will be obtained at the greatest distance from the
Vol. 36, No. 6
impeller measured along the path of travel. The assumption, then, is that a certain minimum velocity is required in the most remote parts of the vessel and that the velocities a t the intermediate points will be higher. If the initial or discharge velocity of the impeller is not equal to the required minimum velocity, it is ob\ ious that the impeller cannot maintain the uniform suspension, even though it may displace a large volume in relation to the volume of the vessel. This condition is sometimes encountered with paddle agitators, and in many instances another type of agitator using less power and discharging less volume of liquid has been successfully substituted; the reason is that the velocity of discharge in the latter case was high enough to maintain the necessary minimum in the remote parts of the vessel. As more work is done on agitation problems, other important performance characteristics may be discovered. The ones mentioned are the more obvious and will serve to indicate the futility of attempting to compare the performance of various impellers on the basis of only one of several characteristics which influence their effectiveness as agitators. There is a striking similarity between the performance characteristics of a fan and an agitator impeller; moreover, the conditions under which they are used are similar. For instance, the fan is frequently used in a large room in much the same way an agitator impeller is used in a large vessel. Both of them are employed to secure movement of the medium which surrounds them. Agitator impellers are frequently used in draft tubes similar to the way a ventilating fan is placed in a wall t o exhaust the air from a room. Many years ago fan manufacturers realized the necessity of knowing the number of cubic feet per minute the fans would displace, the velocity at which the air was moved, the effect of back pressure on the discharge capacities, as well as the power consumptions under the various conditions. So important is this information that they have set up codes which tell not only how the measurements are to be made but also the instruments which must be used so that all results obtained will be strictly comparable. Many fan manufacturers have also helped to establish ventilation standards as well as to determine the losses which will be experienced in leading the air through various shapes of ducts. By hnalogy, it is essential that similar information be developed for the manufacturers and users of agitator equipment. Determining the discharge capacity, the discharge velocity, and power consumption, together with the variation of these factors with variation of viscosity and density of various fluids, will require time and effort. The problems involved, however, should not be considered insurmountable for already much of the power data is known; some information is also available on t,he discharge capacities and velocities. The experiences of fan manufacturers should be a guide to indicate the steps to be taken and the instruments to be developed. The fan manufacturers found that laboratory tests were essential. There 1s no reason to assume that paper and pencil will be adequate substitutes for a laboratory in the investigation of agitator performance. It has already been pointed out that the power requirements can best be determined by a few actual measurements with a dynamometer. The discharge capacities and velocities should be susceptible to measurement by means of Pitot tubes. Perhaps the tubes will not be conventional in design, as indicated by some investigations. However, some means of actual measurement is preferable to theoretical calculations. The theoretical methods for calculating these factors would be different for different types of impellers, and the results would always be open to question until verified experimentally; actual measurements made with a standard instrument under prescribed conditions would be less open to question. The necessity of using standard test conditions becomes apparent when we realize how impossible it is, for example, to place a Pitot tube at the exact point of discharge of an impeller. Further, the velocity of the discharge stream decreases as it travels away from the impeller; thus it is necessary to standardize
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on a flow measurement made at some standard distance in order t o determine' the discharge capacity of a n impeller. In addition, a stream discharging from a n impeller induces flow in the surrounding liquid, and it is impossible to tell whether the velocities measured are due to the liquid passing through the impeller or to the flow induced in the surrounding liquid. With the measurements taken under k e d conditions, the induced flow will always be in the same ratio to that actually pumped by the impeller. The establishment of standard test codes such as those proposed here requires the cooperation of all concerned. It has been demonstrated that different procedures might yield contradictory results because they would not be strictly comparable, and this would cause more confusion in the field than now exists. It is not intended that all who are concerned with agitation problems should help set up the codes. This is primarily the problem of equipment manufacturers, bot there is no doubt that they will welcome the aid of others who may be in a position to contribute to the work. This proposal is not intended to cover t h e complete details of equipment, instruments, and procedures which should be followed to obtain all of the desired information. However, there is a lack of data on impeller performance; such data are readily obtainable and will be of most use when determined as nearly as possible on the same basis. It
B. C.
is important to remember that even with complete data on the three basic impeller characteristics-power, discharge capacity, discharge velocity-it is still essential to know agitator requirements as a function of the particular process application. However, even if the agitation requirements are not known, agitator performance data will be of great use as a basis for intelligent application of the impeller to many operations. It is common practice, as previously mentioned, to select " impellers on a power per unit volume basis. Successful iesults are obtained if there has been sufficient previous experience with the particular impeller, and for liquids of similar physical properties. This method amounts to using only one of the performance characteristics and is so limited in application that, even if the power consumption of a new style of impeller were known, it would not be possible to use the new impeller interchangeably with those on which experience had been gained. On the other hand, if all three performance characteristics were known, it should be possible to substitute one impeller for another in almost any operation with which the user had experience. Data allowing this to be accomplished would be extremely useful, for i t would then be possible to determine whether two different types of agitators could perform the same operation. Furthermore, i t would make possible the substitution of one impeller for another, as might be dictated by mechanical limitations.
A . Marine propeller
Fan-tvpe turbine Shrouded radial-flow turbine
Figure 2.
Impellers Tested on the Differential Dynamometer
D . Curved-blade radial-flow turbine E. Disponer-blade radial-flow turbine
INDUSTRIAL AND ENGINEERING CHEMISTRY
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To some extent it would also indicate the practical limit of application for the various types of impellers. For example, the discharge capacities of some impellers drop rapidly with an increase of viscosity because small passages become blocked with very viscous material. With an appreciation of the value of such data, those who have had considerable experience in the field of agitation should be in better position to see approximate correlations between performance data and agitation requirements. As an example, the probable correlation between the rate of settling of solids in suspensidn and the discharge velocity might be cited. Only a few tests on different types of impellers would be needed to show that the impeller would have to produce a certain minimum velocity of discharge before it could possibly accomplish the suspension, regardless of its discharge capacity. AGITATION REQUIREMENTS
If a method can be found to compute the agitator requirements from the physical characteristics of the materials involved and the dimensions of the vessel, then performance data become a, necessity. At first glance the problem of arriving a t a method of computing the agitation requirements seems almost insurmountable. However, by attacking some of the more obvious operations, the method of solution for the others may become more apparent. The operation of suspending sand in water seems to be based on a few fundamental principles of physics and hydraulics and should be a good starting point. It is known thht solids in suspension have a certain rate of settling, depending on their size and specific gravity and on the viscosity and specific gravity of the liquid in which they are suspended. The settling velocity of these particles can be computed from Stokes’ law or can be measured in the laboatory. Knowing this velocity it would be possible to compute the velocity of a stream in the opposite direction which would prevent settling. This velocity is the minimum that would be required to suspend the solids and would, therefore, be the minimum velocity that could exist in any part of the vessel if the suspension is to be maintained uniformly. It has been pointed out that the velocity of the discharge stream of an impeller decreases, the farther it is from the point of discharge, and the limiting requirement is that the minimum velocity for suspension be maintained in the most remote parts of the vessel as measured along the path of the discharge stream. If it is assumed that this operation is being performed in a vertical cylindrical vessel with a vertically mounted agitator whose impeller revolves about the axis of the vessel, the conditions which can exist may be visualized more readily. The agitator might be such that it maintained the necessary minimum velocity on the bottom of the vessel; but because of a lack of sufficient discharge capacity or of high enough discharge velocity, it might not maintain this minimum in the upper part of the vessel. The concentration of solids would then be less in the upper parts of the vessel as a result of settling in these low-velocity zones. Such an impeller might be an axial flow type such ae a propeller, of such size and capacity in proportion to the size of the vessel that it “short-circuited” the upper part of the vessel. The same condition can also be established with radial-flow impellers; two-bladed paddle agitators frequently give just such results. Some physical and fluid dynamics data and theory already exist which can be made use of in estimating agitation requirements. The characteristics of flow of a stream discharged from an impeller depends on both the initial mass and velocity of the stream. The distance through which the stream can travel is actually proportional to the kinetic energy ( K E ) of the stream at the point of discharge ( I ) ? since
:.
K E = MV2/2g K E = qpV2/2g K E = ApV3/2g
Vol. 36, No. 6
whereM = mass V = velocity q = volume per unit time p = density A = cross-sectional area of stream g = gravitational constant Equation 1 indicates that the properties of the stream which permit it to overcome the resistance of the surrounding liquid are the quantity and velocity a t the point of discharge. The equation does not show how much kinetic energy the stream should have to result in a certain velocity at a given distance from the point of discharge, but it does give the relation of the factors which govern its ability to travel any distance. The amount of kinetic energy required is determined by the resistance to flow of the surrounding liquid. Contrary to usual belief, this resistance is affected only slightly by changes in the viscosity of‘ the liquid, since almost all of the flow in agitation problems is turbulent, and viscosity has little effect in turbulent flow. The usual form for the resistance to turbulent flon is ( 1 ):
R = CAP where C = a coefficient A = frontal area V = velocity q = quantity per unit time = AV Since R is a force which must be overcome, it is a measure of the power that must be exerted to maintain flow. Dividing R by p, R / q = C V , an indication that the resistance per unit volume depends on the velocity and that if the quantity is increased by increasing the frontal area A , the power per unit volume of the stream will be unaffected. Therefore, of two impellers having the same capacity of discharge and the same shape, the one with the lowftst velocity of discharge should re. quire the least power. Referring to the graph, Figure 1, the discharge volume of the 6-inch propeller at 2000 r.p.m. is the same as that of the 10-inch propeller at 402 r.p.m., but the velocity of discharge of the 6-inch propeller is 1333 feet per minute and it requires 3.65 h.p., whereas the velocity of discharge of the 10inch is 365 feet per minute and it draws only 0.17 h.p. It should be understood that this statement applies only to geometrically similar impellers. However, it is proof of the fallacy of using small impellers with high discharge velocities to secure higb capacities. The justification for neglecting the viscosity effect on the resistance to turbulent flow can be found in literature on hydraulics or agitation studies. The most recent report, by Stoops and Lovell (S), shows that the power required by propellertype agitators varies as the 0.2 power of the viscosity. As the degree of turbulence decreases, however, the effect of visqosity becomes greater. The classic work of Osbourne Reynolds showed that the exponent of the viscosity term was 1.0 when perfect streamline flow existed in pipes and that the exponent of the speed factor was reduced a corresponding amount. Our experiments show evidence that this is also the case in agitated vessels. It then becomes a problem to determine the degree of turbulence which will be produced in a given system by an agitator SO that the effects of viscosity and speed on the power requirements can be determined. In pipe flow work Reynolds found that the product of pipe diameter, density, and velocity divided by viscosity yielded a value which was a crit+on of the type of flow obtained, and from it the value of the exponents could be found. Attempts to derive a similar relation for agitation systems have not been entirely successful and require further experimentation. The hypothesis upon which the present paper is based assumes that agitation requirements can be defined by establishing certain minimum velocities which must exist in the various parts of the vessel; these velocities are based on the operation to be performed and the physical characteristics of the
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liquids. The most important link in this chain, and the one about which least is known, is the rate of dissipation of the kinetic energy with respect to the resistance offered by liquids of various physical characteristics. It should be possible to derive simple and usable relations from experiments to permit prediction of stream flow characteristics in a variety of liquids. The problem of maintaining a suspension of sand in water was used to illustrate a method of analyzing the problem. The same method can be applied to the problem of suspending the sand after it has settled. I n this case it would be necessary to consider the inertia of the particles as the factor determining the minimum velocity required in the remote parts of the vessel, because a higher velocity is required to pick up the particles than to maintain them in suspension. I n all other respects the problem should be the same as the first and should lend itself to the same analysis.. I n both the preceding illustrative operations the sole object of the agitation was to accomplish and maintain a uniform suspension. Any excess of agitation would not improve the suspension and would thus be useless. This is not the case, however, when the solids to be suspended are to be dissolved in the liquid; here agitation is also used to promote the rate of dissolution. I n this case the minimum velocity which will maintain a uniform suspension is the minimum agitation requirement; the rate of dissolution of the solids will depend on the solution rate constants of the materials involved and the velocity at which the particles are moved in relation to the liquid surrounding them. It is justifiable, therefore, to assume that any degree of agitation above the minimum required to obtain the suspension will show an increase in the rate of solution. This increase in the rate of dissolution with the higher amount of agithtion will be a function of but not necessarily directly proportional to the agitation. The problem then becomes one of economics to determine the proper amount of agitation for a given system. This method of analysis may also be applied to liquid systems in which no solids are involved. These usually fall into two general categories-namely, blending of miscible liquids and blending of immiscible liquids. A special case of the latter is the forming of emulsions. Usually these operations asre performed to promote a chemical reaction, although there are instances where it is done to obtain a mechanical combination of materials. Regardless of the purpose of the agitation, the minimum agitation requirements can be derived from the fundamental factors of the system involved. The problem of blending miscible liquids is somewhat similar to the dissolving of solids in suspension. The difference between the two lies in the fact that with the purely liquid system the minimum velocity requirement is very low; in fact it may be zero if sufficient time is allowed for one liquid to diffuse through the other. Thus it closely resembles the state that exists when the solids to be dissolved are in uniform suspension. The requirements for an impeller to produce the minimum velocity could be calculated in much the same manner as previously described, and it might be possible to predict the time required to complete the blending if the rate of diffusion of one liquid in the other were known. This type of information would have to be determined by the individuals interested in the particular system before an economically sound selection of an agitator could be made. The blending of immiscible liquids may be analyzed in the same manner as the suspension problem, for in this case particlea of one liquid must be uniformly dispersed in the other and the velocity required in the vessel to accomplish this should be related to the rate a t which the liquids will separate. Here again the primary criterion must be derived from the physical nature of the materials involved before they can be translated into minimum performance requirements of an agitator. There is one
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factor in this problem, however, which has not been mentionednamely, the ability of the impeller to break the liquids into small particles before such particles can be dispersed by the stream This is particularly important in forming emulsions or dispersing a gas in a liquid. It is a mechanical action of the agitator impeller on the liquids; it contributes nothing to the motion of the liquids and, therefore, was not included as a performance characteristic of the impeller. This factor is an important consideration for certain operations, and some way should be found to evaluate i t so that impellers can be selected for this particular duty in much the same manner as that proposed for the other operations. No data are available that point to a fundamental basis for investigating this shearing characteristic, but we know that it can be accomplished by high discharge velocities or by special shapes of impellers which break up the liquid particle8 mechanically as they make contact. Experimentation may show that there is a correlation between the impact of the discharge stream and this disintegrating capacity. Probably the degree of disintegration that can be accomplished, due either to the discharge velocity or $0 the mechanical work done on the liquid by the impeller, will depend on such factors as the surface tension of the dispersed phase of the liquid, the specific gravity, and the viscosities involved. All impellers are capable of mechanical disintegration of particle size to some extent, and it would be desirable to evaluate their abilities to permit an intelligent selection of impellers for this purpose. It is conceivable that this factor may be related to the hydraulic efficiency of the impellers. To be more specific, it may be that impellers having low discharge capacity per unit of power input may have a high disintegrating capacity; conversely, those with a high discharge capacity per unit of power input may have a low disintegrating capacity. This is indicated by the way power is used that is supplied to an impeller; only part of the power imparts motion to the liquid, and the remainder is absorbed in skin friction. “Skin friction” is a rather general term intended to cover the friction or drag effect acting against the passage of a body through a liquid, due to the liquid moving relative to the surfaces of the body. This motion of the liquid over the surfaces causes eddy currents to be set up along the whole stream, and these eddy currents may cause much of the mechanical reduction of the liquid particle size. CONCLUSION
The selection and application of liquid agitators is a t present an art rather than a science. Considerable work has been done a t various times in attempts to place agitation on a scientific basis. Unfortunately much of the work done has been on specific applications of the agitators and has therefore contributed very little knowledge of the fundamental principles of the problem; it has been shown that the basic data available to do so is far from complete. The presentation of this theory has bken made in an attempt to stimulate thought along fundamental fluiddynamics lines and to coordinate the efforts of investigators in this field so that results obtained will ultimately yield solutions to the problems of defining the agitator requirements and selecting the agitators which meet these requirements. The mass velocity theory seems to offer a logical method of attack. The authors are well aware of the fact that this presentation of the theory does not cover all phases of the subject. However, it does serve to point out the fundamental principles involved and the data which must be obtained before agitation can be considered a science. LITERATURE U T E D
(1) Gibson, A. H., “Hydraulics and Its Applioations”, 4th MI.,
1930. (2) Hixson, A.
W.,and Baum, 8. J., IND. ENQ.CHEM.,34, 120
(1942). (3) Stoops, C. E.,and Lovell, C . L., Ibid., 35, 845 (1943).
