(y/x)n = breakage function, dimensionless circulating load, dimensionless throughput of mill, tons per hr make-up feed of system, tons per hr size frequency distribution, micron-' constant appearing in selection function, micron-" minute-' constant appearing in Rosin-Rammler equation, micron-" T,/(T~ r d ) = mixing degree, dimensionless exponent appearing in selection function, breakage function, and Rosin-Rammler equation for size distribution, dimensionless tonnage of finished product, tons per hr cumulative undersize fraction, dimensionless cumulative oversize fraction, dimensionless Kz* = selection function, minute-' flow rate of tailings, tons per hr milling time, minutes holdup of mill, tons dummy variable for particle size, microns x,/x,* = dimensionless cutoff size of classifier, dimensionless cutoff size of classifier, microns (PI'KTI')~'. = reference particle size, niicrons maximum particle size in feed material, microns particle size, microns y/x,* = dimensionless particle size, dimensionless
+
SUBSCRIPTS D = mill discharge E = mill inlet F = new feed
P T
= =
finished product tailings
GREEKLETTERS 6(t)
~(y) 7
Te,7d
ill) $(x)
= = = = = = =
delta function (unit pulse function), dimensionless fractional recovery of classifier, dimensionless W,/B = average retention time of particles, minutes parameters appearing in +(t), minutes TY/F = W / P , minut.es retention time distribution function, minute-' arbitrary function of x
Literafure Cited
Austin, L. G., Klimpel, R . Ii.,Ind. Eng. Chem., 56 ( l l ) ,18 (1964). Broadbent, S. R., Callcott, T. G., Phil. Trans. Roy. SOC.,249, 99 (1956). Chfijo, K., Kagaku Kogaku to Kagaku K i k a i (Annual Report of Japan Soc. Chem. Engrs.), 7, 1 (1949). Gaudin, A. &I., hleloy, T. P., Trans. A I J l E , 223, 40, 43 (1962). Gilvarry, J. J., J. A p p l . Phys., 32, 391, 400 (1961). Gilvarry, J. J., ibid., 33, 3211, 3214 (1962). Harris, C. C., Trans. A I M E , 241, 359 (1968). Herbst, J. A., Fuerstenau, U. W., ibid., 241, 538 (1968). Kelsall, 1). F., Stewart, P. S. B., Reid, K. J., Trans. Inst. d l i n . M e t . , Section C , 77, C120 (1968). Mori, Y., Jimbo, G., Yamazaki, RI., Preprints for Second European Symposium on Comminution, Amsterdam, Sect'ion F, p. 497 (September 1966). lleid, K. J., Chem. Eng. Sci., 20, 953 (1965). Tamura, K., Tanaka, T., Ind. Eng. Chem., Process Des. Develop., 9, 165 (1970). Tanaka, T., Zement-Kalk-Gips, 10, 409 (1957). Tanaka, T., ibid., 11, 92, 298, 396 (1968). RECEIVED for review J ~ l 22, y 1970 ACCEPTED March 30, 1971
Power Consumption in Solid-Liquid Slurries Archie D. McPhee' and Norman 1. Brown2 National Center for Fish Protein Concentrate, National Marine Fisheries Service, National Oceanic and Atmospheric Administration, C. S.Department of Commerce, College P a r k , M d . 20740
The power consumed in fully baffled mixing vessels of various sizes was measured for a unique solid-liquid two-phase system consisting of raw ground fish and isopropyl alcohol. The design criteria strictly adhered to in this investigation was the requirement of 100% mixing throughout with a straight vane impeller without disk. A relationship was obtained for this unique two-phase system from which the power requirement for scale-up can b e calculated. Many commercial mixing devices are presently installed with strong emphasis on power consumption, and only secondary consideration is given to the degree of mixing attained. This article recommends design changes for solid-liquid mixing equipment which, for many physical and chemical processes, may improve product yields and at the same time reduce the mixer power consumption, thereby resulting in a significant increase in operating profits.
Baffled mixing vessels of various sizes were investigated in order to obtain-a general relationship for the calculation of mixing power required for geometrically similar vessels of any diameter. The fluid under consideration concisted of a solidliquid slurry containing two parts by weight of azeotropic isopropyl alcohol and one part by weight of raw, frozen, To whom correspondence should be addressed. Present address, Office of the Foreign Secretary, National Academy of Sciences, Washington, L). C. 20418. 456 Ind. Eng. Chem.
Process Des. Develop., Vol. 10,
No. 4, 1971
ground fish. The density difference between the heavier raw fish and the liquid was'initially about 12 lb/ft3 and the fish was ground with a Hobart grinder using a '/l&i. end plate. The solids content in the final slurry was about 15Yo by weight. Apparatus and Procedure
Figure 1 depicts the apparatus used in this experiment. It is an experimental agitator kit supplied by the Bench Scale Equipment Co., Dayton, Ohio. The initial portion of this in-
Figure 1. Experimental apparatus 1. Tachometer 2. Glair vessel and baffler 3. Impeller shaft 4. Motor ond variable speed drive
variations caused noticeable increases in the power required to achieve the criterion of 100% mixing. In addition, the most effective baffle ratio was also determined experimentally, and 10% baffles were found to be more effective for mixing than baffles which were of the vessel diameter. Furthermore, 10% baffles were also found to be equally as effective for mixing as 15% baffles, but they resulted in less power consumed than the wider baffles, so a J / T ratio of 10% was used for the extended study. After the geometry of the system was fixed, a series of experiments was conducted using baffled mixing vessels of 6-, 8-, lo-, 12-, and 15-in. i.d. Each vessel was carefully filled with predetermined amounts of alcohol and fish, and the H/T ratio was kept constant a t unity. Four 10% baffles, mounted vertically in the tank, projecting radially from the wall and located 90’ apart were used. Offset baffles were tried, but the power requirement for the offset baffles showed no improvement over that of equivalent standard baffles, and the degree of mixing deteriorated noticeably. Data were obtained for a series of impeller speeds in which the motor speed was increased from the minimum to the maximum, and then a second set of data was obtained where the motor speed was decreased from the maximum to the minimum. Correlation of Data. By use of the fundamental dimensions of force, mass, length, and time, and the Pi Theorem of Buckiugham (1915), dimensional analysis was carried out on the system, and the following relationship was obtained for systems in which geometric similarity exists between model and protot,ype:
5. Force gauge
vestigation consisted of making decisions which fixed the geometric ratios of the system, in order to maintain geometric similarity throughout the entire investigation-Le., such ratios as the ratio of impeller diameter to tank diameter. To decide on the type of impeller to use, all types of impellers provided were tested under simulated adverse conditions. Of all the impellers tested, only a flat blade turbine without disk did not undergo extreme vibrations when a fibrous solid material was agitated in water. With all other types of impellers tested, “roping” occurred a t the impeller, causing an imbalance which resulted in noisy vibrations. Because this condition could occur in practice, a &vane flat blade turbine without disk was chosen for the extended study. It was also determined experimentally that a DIT ratio of 0.5 was necessary from the viewpoint of better mixing as well as lower power consumption. Smaller D / T ratios would not SUSpend all of the solids in’the liquid to provide a uniform slurry throughout, except a t very high impeller speeds. The criterion of total suspension is very important in all chemical and physical processes of this type. Oldshue (1969) defines five criteria for mixing which can help to interpret the type of work required in a mixing vessel. Average settling velocities for the system considered were less than 6 ft/min and 100% mixing, or complete uniformity, is defined as a slurry having the same composition throughout. To obtain 100% mixing, surface agitation is obviously necessary. The effect of the geometric ratio D/C on the impeller power required to accomplish the criterion of 100% mixing was also determined experimentally for the system under consideration, and it was found that a D/C ratio of 2.0 resulted in a reduction of power. This last ratio proved to be quite sensitive, as small
The term (D/Do) is required to describe the shear stress associated with solid particles circulating through a liquid medium. This group may be thought of as an equivalent shear stress term, where D is a characteristic diameter for the system and D, is a characteristic length. A plot of log P,gc/pNaD3vs. log ND2p/p for the data obtained is presented in Figure 2, which shows that turbulent flow was obtained a t relatively low Reynolds numbers. For the system studied and with the equipment available, it is not possible to obtain data in the laminar zone, but laminar mixing was not of interest in this study. For the turbulent region the exponent a of the impeller Reynolds number is aero and since the density p was kept essentially constant during the course of this investigation, Equation 1 can, for turbulent flow, be rearranged to:
Thereforea log-log plot of P,/NsDKvs. D will give a straight line of slope b. Such a plot for the data obtained is presented in Figure 3, from which we obtain the value b = 2/3. For process design, one generally fixes the impeller diameter D ; therefore, the two variables Po and N are free to vary, but not independently. One must be able to predict the impeller speed required to obtain 100gro mixing for a particular impeller diameter. The recommended scale-up relationship relating impeller speed to impeller diameter for process design for the particular two-phase system studied is:
N I / N z = DI/DI Ind. Eng. Chem. Proserr Der. Develop., Vd. 10, No. 4, 1971
(3) 457
- 4
14.-
Transition range b -
Turbulent range
D= 7- V2" D: 5 "
D: 2-1/2"
X
0 0
Increosing
impeller speed
+X
Decreosing impeller speed
w*
1.
I_pL.t
-+--+--
Impeller Power Number vs. Impeller Reynolds Number
Iu 3
:I
ILv3
IMPELLER REYNOLD NUMBER
Figure 2. Turbulent flow obtained at relatively low Reynolds numbers
7 I slope
I
I
Impeller Diameter( f
Figure 3. Log-log plot of P , / N 3 D 5 v s . D 458
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 4, 1971
= 2/3
4.0
MOOlFlED MIXING WESSEL (Not to swlr )
Discussion
The D I T ratio of 0.5 used in this investigation is higher than that recommended in most of the literature. Furthermore, DIG‘ ratios slightly greater than, and slightly less than 2.0 resulted in ail increase in power. Investigations initially carried out a t a D I T ratio of 1/3 resulted in much higher impeller speeds and a greater power requirement t o attain the criterion of ‘ ‘ l O O ~ o mixing”. Inspection of the data of Randolph and Iskander (1968) reveals that a D/C ratio of 1.0 and a D/T ratio of 4/15 were used in their studies of a liquid-solid two-phase system, and their resulting imlieller speeds were 650 rpni and 850 rpm. -1fter making some asamnptions about the geometry of the system used iii their iiivestigatioii, it was determined experimentally that 100% mixing could be achieved for their system ut aii impeller speed of only 175 rpm for a 15-in. diameter vessel if a D I T ratio of 112 and a D / C ratio of 2.0 were used. Therefore the lion-er required by Randolph and Iskander in their in2 to 3’:;~ times greater than necessary. o critical that in some chemical reactioii systems it will be advisable t o install a false flat bottom in conical bottom tanks, as shown in Figure 4. The improved mixing obtained a t lower power could increase product yields significantly in such systems, resulting in higher profit’swith a low payout time. The power coiisumed in fully baffled mixi measured for a uiiique solid-liquid two-phase wide raiige of impeller speeds for different vessel diaiiieters, maintaining the concept of geometric similarity throughout. The following relationship was obtained:
(4) where a = 0 for the turbulent region. For scale-up of geometrically similar systems in the turbulent region, the following relationship is recommended : h-1/1V2 = D2/Di (5) For the systems studied, Equation 5 can be reduced to:
N = 2.08 ft(sec)-’.l/D (6) Equations 4 and 6 are the recommended equations for the design of geometrically similar mixing vessels for the unique system studied in this investigation and for mixing in the turbulent region. The power requirement so calculated is the power required at the impeller to perform the work of mixing, and does not include such factors as friction losses in gear boxes. A flat blade turbine without disk was chosen in preference to other types of impellers because “roping” occurred with a fibrous solid causing an imbalance which resulted in noisy vibrations for all types of impellers tested except the flat blade turbine. The impeller Reynolds number a t turbulent flow was significantly reduced with the system geometry used, and the exponent which expresses the effect of impeller diameter on power number is larger than expected. However, the criterion of 100% mixing is very severe, and the additional requirement of a flat blade turbine also increased the power requirement. Offset baffles cannot be used because the degree of mixing deteriorates too much. The fluid density used in the Reynolds number calculations was the true density obtained from pycnometer measurements. The slurries studied in this investigation were pseudoplastic in behavior; therefore, the viscosity used in these same calculations was the limiting viscosity at high rates of
Figure 4. False flat bottom counteracts critical D / C ratio
shear obtained with a laboratory viscometer specially designed for this purpose by the senior author (patent to be applied for). Nomenclature
c =
height of iinpeller above tank bottom, ft D = impeller diameter, ft D , = reference impeller diameter, 1 ft functional relationship 9 = acceleration due to gravity, 32.17 ft/sec2 gc = conversion factor, 32.17 ft-lb,jlbfjsec2 H = liquid depth, ft J = baffle width, it k‘= proportioiiality const’ant I , = impeller blade length, ft A- = impeller speed, rps and r i m nb = nuniber of baffles, dimensionless n , = number of impeller blades, dimensionless Po = power, ft-lbf/sec s = impeller pitch, ft T = tank diameter, ft Ti7= impeller blade width, ft
s =
Greek Symbols p
=
p
=
liquid density, lb,/ft3 liquid viscosity, lb,/ft sec
References
Bates, 11. L., Fondy, P. L., Corpstein, I:. It., Znd. Eng. Chem. Process Des. DeveloD.. 2. 310 11963). Bessel, E. S., Everett: H . J., Heise, 11. C., Rushton, J. H., Chem. Eng. Progr., 43, 649 (1947). Buckingham, E., Trans. Amer. Sac. M e c h . Eng., 37, 263 (1915). Fondy, P. I,., Bates, N. L., AIChE J . , 9, 338 (1963). Hixson, A. W., Tenney, A. H., Trans. AZChE, 31, 113 (1935). Lyons, Emerson J., Chem. Eng. Progr., 44, 341 (1958). Alack, 1). E., Uhl, V. W.,Chem. Eng., 54, 119 (1947). Miller, S. A , , AIann, C. A., Trans. AIChE, 40, 709 (1944). Oldshus, J. Y., Znd. Eng. Chem., 61(9), 79 (1969). Randolph, A. D., Iskander, RI. It.! “Experimental Study of Concentration ProEles and Withdrawal Rates of Solids in a Well Mixed Vessel, Symposium on Mixing, Sixty-First Annual Meeting A.I.Ch.E., 1968. Rushton. J. H.. Costich. E. W.. Everett, H. J., Chem. Ena. Progr., 46, 467 (19,50). Schwartzberg, H., Ph.D. Thesis, Xew York University, 196.5. White. A. NcL.. Brenner. E.. Phillias. G. A.. hlorrison. RI. S.. Trans. Amer. inst. Cheni. Ehg., 30,’570 (1934). RECEIVED for review September 23, 1970 ACCEPTEDMarch 15, 1971 Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 4, 1971
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