BRIEFS Summary of papers published in this month’s research quarterly, I@EC Fundamentals SALT CONCENTRATIONAT PHASE BOUNDARIES IN DESALINATION BY REVERSE OSMOSIS
ANALYSIS OF THE FLUX EQUATIONS OF MULTICOMPONENT DIFFUSION
Water removal from saline solutions by reverse osmosis causes the salt concentration at the phase boundary to be increased over that in the Bulk solution, increasing the effective osmotic pressure. The magnitude of this effect is investigated for both turbulent and laminar flow in round tubes, and for laminar flow in a twodimensional channel. It is concluded that the salt buildup effect may become a serious problem as membranes are developed which make it possible to operate at high water fluxes.
A general scheme of flux equations for multicomponent diffusion is proposed as a consistent generalization of Fick’s law. By the use of matrix theory, the characteristic roots of the matrices of multicomponent diffusion coefficients defined in this uniform manner are shown to be invariant under transformations of coordinate systems. The direct analogy of these characteristic roots to true binary diffusivities is completed by the use of fundamental information from the thermodynamics of irreversible processes to prove these roots always to exist as real, positive numbers. This consistent mode of multicomponent flux representation is applied to the solution of the general Jinearized equations of multicomponent mass transfer. The results not only afford a basis for a uniformity of definition heretofore not available, but also offer a groundwork for the better understanding of multicomponent diffusional phenomena.
T. K. Sherwood, P. L. T. Brian, and R. E. Fisher, Massachusetts Institute of Technology, Cambridge, Mass., and Lawrence Dresner, Oak Ridge National Laboratory, Oak Ridge, Tenn. IND.ENC.CHEM.FUNDAMENTALS 4,113-118 (1965)
MASS TRANSPORT FROM A URANIUM SPHERE TO LIQUID CADMIUM IN HIGHLY TURBULENT FLOW
Mass transfer coefficients from single l / ~ - , a / ~ - , and ‘/a-inchdiameter spheres of solid uranium to liquid cadmium at 500” to 600’ C . were measured in the wide Reynolds number region of 80 to 400,000. The dissolution rate of uranium was diffusioncontrolled over the entire range. The plot of Jn us. Reynolds number exhibited a very sharp peak or maximum at a Reynolds number of 70,000. These values of J D were about 50% lower than those for organics in the low velocity regions. In the high velocity region the values were about the same. Heat-treated uranium spheres with random, fine-grained structures gave J D values several times greater than spheres with nontreated, coarsegrained structures.
E. D . Traylor, Leslie Burris, and C. J . Geankoplis, Argonne National Laboratory, Argonne, Ill., and Ohio State University, Columbus, Ohio IND.ENG.CHEM.FUNDAMENTALS 4, 119-125 (1965) MASS TRANSFER DURING SHORT SURFACE EXPOSURES IN COUNTERCURRENT FLOW
Numerical solutions have been obtained for mass transfer behavior during countercurrent flow of two fluid phases across a short contact interval fixed in space. Two simplified models were considered: Model 1 postulates a simple penetration behavior for both phases, whereas Model 2 postulates a laminar boundary layer behavior for one phase and a simple penetration model for the other. The over-all rate of mass transfer for these models is higher than that predicted by the classical two-film addition of resistances equation, the maximum deviation being +20% for Model 1 and +14% for Model 2.
C. Judson King, University of California, Berkeley, Calif.
IND. ENG.CHEWFUNDAMENTALS 4,125-129 (1965) MASS TRANSFER FROM FREE DROPS
A photographic technique was employed to measure the rate of extraction of methyl Carbitol out of individual drops of a benzenecarbon tetrachloride mixture dispersed in water. Because of the high distribution coefficient favoring the aqueous phase and the large ratio of continuous to dispersed phase, the dispersed phase resistance was the only one of significance. A 14-fold decrease in transfer rate during the first second after formation was attributed to the decay of internal circulation induced by the formation process. Observations indicated that the drops were internally stagnant after several seconds of life. B. Duane Marsh and William J . Heideger, University of Washington, Seattle, Wash. IND.ENG.CHEM.FUNDAMENTALS 4, 129-133 (1965)
E?. T. Cullinan, Jr., Carnegie Institute of Technology, Pittsburgh, Pa. IND.END.CHEM.FUNDAMENTALS 4, 133-139 (1965) APPLICATION OF THE ENSKOG RELATIONSHIPS FOR PREDICTION OF THE TRANSPORT PROPERTIES OF SIMPLE SUBSTANCES
The viscosity ratio, p/p*, and the correlation of Damasius and Thodos for the Enskog modulus, bpx, have been used for argon to establish the effect of temperature and pressure on the quantity, b. The dimensionless parameter, b p c , has been correlated with reduced temperature for constant reduced pressures. The combined contributions of the b p x and bp, correlations permit the calculation of the effect of pressure on viscosity, thermal conductivity, and self-diffusivity, using the Enskog transport property relationships. These relationships have been applied in the dense gaseous and compressed liquid regions and, whenever possible, were compared with experimental values for argon, krypton, nitrogen, and oxygen. For these substances, whose critical compressibility factors are all approximately z , = 0.291, the calculated transport properties were found to be in good agreement with experimental values. These comparisons were also made for carbon dioxide ( z , = 0.275) and found to be reasonably accurate.
David A. Lennert and George Thodos, Northwestern University, Evanston, Ill.
IND.END.CHEM.FUNDAMENTALS 4, 139-141 (1965) GENERALIZED RECIPROCITY RELATION BETWEEN RATES AND AFFINITIES OF SIMULTANEOUS CHEMICAL REACTIONS
The general reciprocity relation, a v l / b A n = ?)vt/bA1, between the rates and affinities of simultaneous over-all reactions is derived on the basis of the De Donder form of chemical thermodynamics. The Onsager symmetry relation, 112 = 1 2 1 , between mutual influence coefficients is a limiting case of this relation for the neighborhood of equilibrium where rates and affinities are linear functions of one another. Some kinetic implications of the general reciprocity relation are examined in the case of the “triangular N 3 P +.M . reaction” M
Pierre Van Rysselberghe, Stanford University, Stanford, Calif. IND.END.CHEM.FUNDAMENTALS 4, 142-145 (1965)
MAGNETIC EFFECTS IN CERTAIN SYSTEMS OF CHEMICAL ENGINEERING INTEREST
The chemical engineer may be called upon to design a process which must operate in a region of strong magnetic fields. In particular, this problem arises in the design of blankets for a hightemperature plasma reaction system which must be contained
(Continued on page 9 9 ) VOL. 5 7
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Turbine agitators: technology replaces black art Chemical engineers tend to regard fluid agitation with a mixture of awe, frustration and disgust. Awe when they have been led to believe that this unit operation is too mysterious to handle with confidence, frustration when they try t o correlate published literature t o develop a unified discipline, and disgust over the endless exceptions to almost every rule so far developed. The temptation is t o pretend that analysis is not worth the time because the argument goes precision exceeds accuracy by many orders of magnitude. There are some troublesome areas in agitator design and application which are often subject t o misinterpretation. We offer here a few clear statements about turbine agitators which we believe will be helpful. (If any of them agitate you in any way, let us know. We'll be happy to discuss them further with you.)
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IMPELLERS The impeller is the heart of the agitator. Consider it as your process does - i n terms of the performance characteristics rather than the physical form. The three predominant flow patterns produced are Tangential flow, Radial flow, and Axial flow. Tangential flow occurs in the laminar range where essentially no radial discharge takes place. Although any impeller might perform in this way under certain conditions, this area is best served by impellers in the paddle family. Radial flow impellers were at one time the major type used for processes through the entire Reynolds range of flow. Technological developments in recent years indicate more efficient mixing in most applications with axial flow impellers. Radial flow designs are now desirable only in such applications as gas dispersion and continuous staged systems. The majority of mixing applications - blending, solids suspension, heat transfer, dissolving, and wetting - are best approached first as Axial Flow Turbine agitator problems. The reason is that pitched blade 'turbines require on the order of onehalf the HP of straight blade turbines for most applications. Through major emphasis by Chemineer over the past decade on the greater efficiency and greater effectiveness of the Axial flow impeller it has now become a standard in the industry.
SHAFT DESMN Shaft design is by far the major mechanical design problem encountered in agitation equipment. Shaft deflection and critical frequency are the controlling variables here, not bear. ing life or toque requirements. Once horsepower and speed are established. shafts must be designed to operate well below the critical frequency and with minimum deflection.
NE= (1.681) (d) (10F(fm) (fa)
-4
LZ
where fa
(L)
(W
=F L+S
Nc =critical shaft w e e d h = bearing span factor L =shaft length S =spacing of bearings W =impeller Weight d =shaft diameter frn =material modulus filCtDl
The general equation for first critical speed shows that as the bearing span goes down the critical speed goes up, where all other variables remain the same. At the same time, shorter bearing spans decrease the danger of shaft deflection. Mounting requirements and bearing loads go up under such arrangements but mounting rigidity and bearing design and selection on Chemineer agitators far exceed the imposed loads of the installation. Since the shaft diameter is determined by critical frequency design requirements, it always exceeds by a healthy margin the torque requirements of the iob. The old nonsense that bearinn _w .a n cannot be less than
onesixth the shaft extension had meaning only in earlier days before high capacity bearings were available.
SPEED CHANOES Much has been said about the advantages of changing gears quickly in turbine agitators. This option is available with varying degrees of ease from most turbine agitator manufacturers-but it is often a misleading benefit. To illustrate, consider the Power Number formula in its simplified form. p
.c Np=
INDUSTRIAL A N D ENGINEERING CHEMISTRY
w
factor
p =fluid density .. = immller speed Y
-impeller
diameter
Since the power requirement changes in propoltion t o N3, a small change in speed has a major effect on power; for example, one AGMA speed increase nearly doubles the power requirement. You're immediately faced with the need to change the motor. A second danger in changing gears lies in the gearbox itself. A modest change in speed, accompanied by the appropriate motor change, may exceed the gearbox rating limits and cause major damage. Although you can compensate for this danger by changing impellers, you probably will have t o redesign the shaft also. All in all, these changes are a major and usually uneconomic revision in the design of your agitator. What this discussion means is just this: (a) A quick change option is likely to give you nothing but a , false sense of security. (b) If you anticipate the need for flexibility, incorporate a Veri. able' speed or belt drive feature. Either will allow Small increments of change in speed without danger or redesign.
DESlOW EVALUATIOW Here are a couple of things to keep in mind. A mixing job is not just a horsepower job. If you have enough HP and enough time, any irregular lump on the end of a shaft will do a mixing job. But. you can't buy wisely on the basis of horsepower. You can't even buy effectively on the basis of torque. True optimization in your fluid process requires correlation of impeller style, impeller diameter, speed, and system geometry. When this is done you may find, for example, that a 5'HP. 84 rpm agitator will be more effective and less costly than a 7% HP, 100 rpm design. Secondly, value analysis applies perfectly to fluid agita. tion. At Chemineer we learned long ago that there aren't any quick "hardware" answers t o fluid agitation problems. Every process problem yields to a rigorous application of a unified technology which results in the optimum balance of performance with economy. We do not dally with the technology of fluid agitation. This is why we say that every recommendation of Chemineer agitators starts from a process analysis and that you get more efficient, more effective fluid agitation as a result. The comprehensive performance warranty which Chemineer provides is your guarantee not only that the equip ment is well made but also that it will perform the process function for which you bought it. Chemineer, Inc., P. 0. Box 1000, Dayton, Ohio, 45401.
Cinli NI. 36 om Raden' S t n i c l Carl
98
$43
=pow*, -conversion
BRIEFS magnetically. Although many magnetic effects are so small as to be negligible, careful design practice requires that anticipated effects be evaluated and not merely assumed to be small. This article presents analyses by which several magnetic effects may be evaluated. Magnetically induced shifts of equilibria, magnetic effects on rate processes, and the possible effects of magnetic field gradients are considered, and the magnitudes of some effects are computed.
and McNelly-were fair. The correlations of Levy, JakobLinke, Insinger-Bliss, Miyauchi-Yagi, and Nishikawa were poor.
Frederick W. Camp and Ernest F. Johnson, Princeton University, Princeton, N . J .
The effective thermal conductivity, k,, of water was measured in a vertical annulus in which the inner cylinder was heated and the outer cylinder and bottom plate were rotated. Qualitative analysis of the flow patterns due to natural convection and rotation indicates that the effect of rotation is to counteract the flow caused by natural convection. The experimental results verify this and establish the rotation speed, no*, at which the heat transfer rate becomes a minimum. At this speed k, approaches the true thermal conductivity of the fluid, and the heat transfer characteristics are poorer than with no rotation. Above no*, the effective thermal conductivity increases.
IND. ENC.CHEM.FUNDAMENTALS 4,145-150 (1965) KINETICS OF FAST INTERFACIAL REACTIONS IN LAMINAR TUBE FLOW
Previously published mathematical analyses of kinetic studies in a tubular reactor for both heterogeneous and homogeneous reactions usually begin with an exact solution to the flow-diffusion-reaction equations for pipe flow. Considerable difficulty is encountered in applying these to actual experimental data and their usefulness for detailed analyses of the kinetics and mechanisms of such reactions is limited. An asymptotic solution for small contact times circumvents these difficulties and permits calculation of the surface reaction kinetics from experimental conversion data obtained in a suitable tubular reactor. Since transition between “reaction control” and “diffusion control” in such systems occurs at contact times of the order of several seconds (as shown in this paper), very short reactors are required and the experimentalanalytical difficulties are correspondingly severe. However, this mathematical technique proves particularly pertinent to this problem. Supporting results are presented for two systems.
Chatten Cowherd, Jr., and H . E. Hoelscher, The Johns Hopkins University, Baltimore, M d .
D . E. Drayer, Cryogenic Engineering Laboratory, National Bureau of Standards, Boulder, Cola.
IND.ENC. CHEM.FUNDAMENTALS 4, 167-171 (1965) HEAT TRANSFER IN A ROTATING ANNULUS
D . K. Petree, W. L. Dunkley, and J. M . Smith, University of California, Davis, Calif.
IND.ENG.CHEM.FUNDAMENTALS 4, 171-1 76 (1965) WALL EFFECT IN COUETTE FLOW OF NON-NEWTONIAN SUSPENSIONS
Suspensions of fibrous particles in Couette flow exhibited yield stresses, apparent wall slip, and time-dependent behavior. The behavior exhibited cannot be observed in a narrow-gap viscometer, and data from such instruments should be used with caution.
S. R. Morrison and J. C . Harper, University of California, Davis, Calif. IND.ENC.CHEM.FUNDAMENTALS 4,176-181 (1965)
IND.ENG.CHEM.FUNDAMENTALS 4, 150-154 (1965) SOME NONLINEAR VlSCOELASTlC MODELS WITH INCLUSION OF RESULTS FROM MOLECULAR THEORY DYNAMICS OF HEAT REMOVAL FROM AN AGITATED TANK
The dynamics of heat removal from an agitated tank was investigated for the case where the flow rate of coolant through a coil is varied. The measured frequency response of batch temperature in a 2-foot tank was compared with that predicted from four lumped-parameter models and one distributed model. A mixing delay of several seconds was calculated by comparing the data with the response predicted by the distributed model.
Thomas W. Weber, State University of New York, Buffalo, N. Y. and Peter Harriott, Cornell University, Zthaca, N . Y. IND.ENG.CHEM.FUNDAMENTALS 4,155-161 (1965) STRUCTURE OF COMPLEX CHEMICAL REACTION SYSTEMS
A mathematical theory of complex chemical reaction systems of polynomial order (mass action kinetics) is formulated in terms of multilinear (or tensor) algebra. The chonical forms are discussed. The “porcupine theorem” which asserts the existence of straight-line reaction paths in homogeneous open subsystems is demonstrated.
James Wei, Princeton University and Socony Mobil Oil Co., Znc., Princeton, N. J .
IND.ENG.CHEM.FUNDAMENTALS 4,161-167 (1965) NUCLEATE BOILING OF HYDROGEN. A COMPARISON BETWEEN EXPERIMENTAL AND PREDICTED DATA
Heat flux values predicted by 11 nucleate boiling correlations were compared with experimental data for boiling liquid hydrogen. Three correlation$-Forster-Zuber, Forster-Grief, and Cryder-Gilliland-predict heat flux values in good agreement with the experimental data at temperature differences in the neighborhood of 1 ’ F. Three other correlations-Hughmark, Gilmour,
A nonlinear generalization of the general linear viscoelastic model is proposed, which seems capable of giving a reasonable description of non-Newtonian viscosity and normal stresses. It also provides a description of the ‘‘analogies” between the material functions describing steady flow behavior and small amplitude oscillatory motions. By appropriate choice of constants in the model it can be made to give close agreement with the rather successful complex-viscosity expressions obtained by molecular-model calculations. The model thus constructed contains only two constants. The published data on complex viscosity, shear viscosity, and normal stresses of a 5% solution of polyisobutvlene in Decalin can be successfully correlated using only two adjustable parameters.
Thomas W. Spriggs and R. Byron Bird, Unioersity of Wisconsin, Madison, Wis.
IND.ENC.CHEM.FUNDAMENTALS 4,182-186 (1965) STRESS-RELAXING SOLIDS.
RECOIL PHENOMENA
One of the most obvious manifestations of elasticity in a fluid is springiness or recoil. Current studies on viscoelastic fluids do not utilize this phenomenon, so a theoretical and experimental investigation of recoil in pipes was undertaken. The theory yields equations for the time course, amount, and distribution of recoil by using a quasi-linear, hereditary model proposed earlier. Some features of these equations are in agreement with data taken on various solutions; others are not. Sources of discrepancies are probably nonlinear effects, since they become most pronounced at higher velocities. The principal value of the recoil experiment is to serve as a check on other experimental methods for determining the properties of viscoelastic fluids.
N . N. Kapoor, J. W. Kalb, E. A . Brumm, and A . G. Fredrickson, University of Minnesota, Minneapolis, Minn.
IND.END.CHEM.FUNDAMENTALS 4, 186-194 (1965) (Continued on page 707) VOL. 5 7
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Polyester resins prepared with NPG are attractively low in color for one reason: the inherent heat resistance of neopentyl glycol allows you to use higher than ordinary esterification temperatures-and correspondingly shorter cook times-thus preventing the formation of dark colored substances usually developed in long cooks. Additionally, the stability of the symmetrical neopentyl skeleton carries over to fiber glass laminates in excellent resistance to light, heat, weathering and chemical attrition. We have a 60 page soft cover book that goes into detail on the properties of NPG, its reactions, derivatives,
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BRIEFS REAPPRAISAL OF THE CONCEPT OF SETTLING IN COMPRESSION. SETTLING BEHAVIOR AND CONCENTRATION PROFILES FOR INITIALLY CONCENTRATED CALCIUM CARBONATE SLURRIES
I n settling of C a C 0 3slurries initially in compression (>145 grams per liter), the rate of elimination of fluid was not always proportional to the amount that could be eliminated up to infinite time. For slurries with initial concentrations from 10 to 190 grams per liter, the descent of the slurry-supernate interface followed the Deerr-Roberts-Yoshioka equation only after a marked concentration gradient had reached the interface. The solids profiles were fairly close to those predicted from Kynch's theory for rigid spheres, I but the concentration at the bottom reached its maximum gradually rather than instantaneously. Mechanical stress, transmitted by particle-particle contact, is believed to account for this gradual increase as well as for the strongly curved final solids profiles and the increase in mean final concentration with increasing weight of solids per unit area. A review of early literature revealed that clear explanations of true compression have been ignored in favor of unwarranted deductions from an empirical equation.
Elmer M . Tory and Paul T. Shannon, Purdue University, Lafayette, Ind.
IND.ENG.CHEM.FUNDAMENTALS 4,194-204 (1965)
BIAS IN PARTICLE-SIZE ANALYSES BY THE COUNT METHOD
Particle-size analyses by the count method of samples taken from populations which follow distributions that are infinite in extente.g., the log-normal distribution-indicate the existence of maximum size-classes by the usual methods of calculation. The resulting forms of the particle-size distributions based on these samples, as a consequence, are biased. This bias was investigated and a simple correction was introduced which yielded unbiased estimates of the particle-size distributions based on number, area, and volume.
J. E. Gwyn, E. J . Crosby, and W . R. Marshall, Jr., University of Wisconsin, Madison, Wis. IND.ENG.CHEM.FUNDAMENTALS 4, 204-208 (1965)
COMMUNICATION. EXPERIMENTAL NATURAL CONVECTION HEAT TRANSFER FROM WIRES TO THE NONEQUlLlBRlUM CHEMICALLY REACTING SYSTEM: N0z-NO42
Natural convection heat transfer measurements to the nonequilibrium gaseous N02-NO-02 system are presented. Electribally heated horizontal platinum wires from 1 to 10 mils in diameter were immersed in the gas maintained at bulk temperatures from 240' to 600' C. Temperature driving forces of the order of hundreds of degrees are required to cause the increase in heat transfer owing to energy of reaction. The problem of correlating heat transfer rates in nonequilibrium reacting systems is presented.
G. R. Bop# and D . M . Mason, Chemical Engineering Department, Stanford University, Stanford, Calif. IND.ENO.CHEM.FUNDAMENTALS 4, 222-224 (1965)
COMMUNICATION. BUBBLES
RISING VELOCITY OF A SWARM OF SPHERICAL
An expression is proposed relating the velocity of rise of a swarm of spherical bubbles to the velocity of a single bubble. The analysis, based on a cellular spherical model, is restricted to the range, of high but subcritical Reynolds numbers.
Giuseppe Marrucci, Istituto di Chimica Industrials) University of Naples Naples, Italy
IND.END,CHEM.FUNDAMENTALS 4,224-225 (1965)
COMMUNICATION. ANNULAR FLOW
THE SECOND NORMAL-STRESS DIFFERENCE IN
Standard techniques are used to produce, unlike in prior work, a rigorous expression for the second normal + stress difference, P22 - P33, of a viscoelastic fluid in axial annular flow. The expression is treated by an iterative method, which, when applied to existing data, yields Pzz Pa3 as a unique function of shear rate.
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C. R . Shertzer, University of Delaware, Newark, Del.
IND.ENG.CHEM.FUNDAMENTALS 4,225-226 (1965) THERMODYNAMICS OF SOLID CARBON DIOXIDE SOLUBILITY IN LIQUID SOLVENTS AT LOW TEMPERATURES
A molecular thermodynamic method is described for correlating solubility data for solid carbon dioxide at temperatures below its triple point, 216.56"K. The activity coefficient for carbon dioxide in liquid solution is referred to pure subcooled liquid carbon dioxide at the temperature of the solution and is related to pertinent intermolecular forces by a generalization of Scatchard's equation. The significant quadrupole moment of carbon dioxide as well as acid-base complex formation with unsaturated hydrocarbons is taken into account. The correlation should be useful for predicting solid carbon dioxide solubility in various solvents as well as in solvent mixtures. A . L. Myers and J . M . Prausnitr, National Bureau of Standards, Boulder, Colo., and Universitj of California, Berkeley, Calif.
COMMUNICATION.
A NORMALIZATION FOR THE THIELE MODULUS
Following Petersen's asymptotic method for a general case it is shown how the Thiele modulus, A, can be normalized so that the effectiveness factor, E, is asymptotically A-' €or large A. This provides a universal criterion for definite diffusion limitation and brings together the effectiveness factor curves for other parameters of the system. The method is illustrated by reference to the recent work of Wakao and Smith. A parametric representation of E ( A ) is given in terms of indefinite integrals.
Rutherford A r k , University of Minnesota, Minneapolis, Minn.
IND. ENG.CHEM.FUNDAMENTALS 4, 227-229 (1965)
IND.ENG.CHEM.FUNDAMENTALS 4, 209-212 (1965)
GREEN'S FUNCTIONS AND OPTIMAL SYSTEMS. DIRECTION IN DECISION SPACE
THE GRADIENT
The gradient direction in the space of decision vectors for multistage optimization problems is obtained under general constraints through the solution of variational equations by means of Green's functions. This direction is then used for a steepest ascent solution of the optimization problem.
M . M . Denn and Rutherford Aris, University of Minnesota, Minneapolis, Minn.
IND.ENG.CHEM.FUNDAMENTALS 4, 21 3-222 (1965 )
COMMUNICATION. THE MCCABE-THIELE DIAGRAM FOR A FEED WHICH IS PARTLY VAPOR
A break should be introduced into the McCabe-Thiele construction for the case in which a n equilibrium mixture of liquid and vapor enters a distillation column in such a way that the vapor portion of the feed mingles with the vapor from the plate below the feed entry but does not come in contact with the liquid on that plate. S. D . Cavers, University of British Columbia, Vancouver, B. C., Canada
IND.ENC.CHEM.FUNDAMENTALS 4,229-230 (1965) (Continued on page 703) VOL. 5 7
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INDUSTRIAL A N D ENGINEERING CHEMISTRY
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BRIEFS COMMUNICATION. CONVERGENCE OF A METHOD OF SUCCESSIVE APPROXIMATIONS IN THE THEORY OF OPTIMAL PROCESSES
The convergence of a method of successive approximations used recently for the optimization of discrete and continuous systems is examined. Convergence is assured within some neighborhood of the maximum for nonsingular continuous systems, but attempts to use the algorithm for discrete systems could lead to divergence from starting policies arbitrarily close to the maximum.
Morton M . Denn, University of Delaware, Newark, Del.
IND.ENG.CHEM.FUNDAMENTALS 4,231-232 (1965)
COMMUNICATION. EXISTENCE OF ASYMPTOTIC SOLUTIONS TO FIXED-BED SEPARATIONS AND EXCHANGE EQUATIONS
IT’S NEVER EASY TO SOLVE A PROCESS PROBLEM.. .
--
IND.ENG.CHEM.FUNDAMENTALS 4,233-236 (1965)
It’s the fat volume of CHEMICAL ENGINEERING CATALOG-bulg ing with the detailed, factual information you need t o work out your solution, All the facts about equipment, materials of construction and specialized services are right there-facts from over 500 supplier companies-up-dated, complete! Next time you tackle a process problem-today-tomorrow-turn first to CEC. You’ll find it a BIG HELP, a reliable partner in your work, the standard basic reference of the industry-since 1916!Keep it on your desk. Work it hard!. , . t o make your work easier.
COMMUNICATION.
REINHOLD PUBLISHING CORPORATION
The existence of asymptotic solutions to the differential equations which describe single-solute fixed-bed separations and exchange processes is proved for nonlinear equilibria under three different sets of conditions: (1) zero longitudinal dispersion and finite mass transfer resistance, (2) equilibrium operation with finite longitudinal dispersion, and (3) finite longitudinal dispersion and finite mass transfer resistance. The use of single-solute asymptotic solutions for describing the behavior of individual species in multicomponent systems is also discussed.
D. 0. Cooney and E. N. Lightfoot, University of Wisconsin, Madison, wis.
DlFFUSlVlTY IN NON-NEWTONIAN LIQUIDS
Measurement of gas absorption rates in laminar liquid jets provides an easy way for determining the diffusivity of the absorbing gas in the liquid. The absorption equation for ideal rodlike jets is a good approximation also for non-Newtonian liquids. The method has been used to measure the diffusivity of COZ and CzH4 in a variety of aqueous non-Newtonian solutions and suspensions. Although the apparent viscosity of these liquids is much higher than that of water, the measured values of diffusivity are of the same order of magnitude as in water, and, in the case of polymer solutions, they appear to increase with increasing polymer concentration.
Gianni Astarita, Istituto di Chimica Industriale, University of Naples, Naples, Italy
IND.ENG.CHEM.FUNDAMENTALS 4, 237 (1965)
8
CHEMICAL ENGINEERING CATALOG
430 PARK AVENUE, NEW YORK, N.Y. 10022
Circle No. 1 on Readers’ Service Card
NO CONTAMINATION
N O CORROSION
Material is forced through flexible tubing-never touches pump. Volume is controlled by size of tubing and speed of pump. Several sizes and types with or without drives and speed controls provide the answer to most problems. PUMP SEVERAL DIFFERENT LIQUIDS SIMULTANEOUSLY:
CORRESPONDENCE
Comments on an article by A. S. Gupta and George Thodos [IND.END.CHEM.FUNDAMENTALS 3, 218 (1964)] on simultaneous mass and heat transfer of gases flowing through packed and distended beds of spheres, and a rebuttal.
Norman Epstein, University of British Columbia, Vancouver, B. C., Canada; George Thodos, Northwestern University, Evanston, Ill.
IND. ENG.CHEM.FUNDAMENTALS 4, 238 (1965)
CORRESPONDENCE
Comments from several different authors on a published paper on the discrete maximum principle.
Some models accommodate up to four tubes at one time - will pump different liquids - different FEED AND MIX: One or more tubes can be feeding material to a mix while a larger tube is recirculating the l i q u i d to produce agitation and thorough mixing. Viscous materials can b e pumped without danger of gumming or plugging. Remove tube and pump is clean. METER ADDITIVES: One or more additives can be pumped to a solution in the exact amount desired by selecting the correct size of tubing and regulating pump speed. A wide range of combinations is available with standard tubing. Sampling can b e accomplished auto.
L. T. Fan, Kansas State University, Manhattan, Kan.; C. S. Wang, California Institute of Technology, Pasadena, Calif.; F. Horn, Impertal College of Science and Technology, London, England; R. Jackson, University of Edinburgh, Edinburgh, Scotland; and M . M . Denn, University of Delaware, Newark, Del.
Capacities from .05 cc to 4.5 GPM. Write for complete information on size, and prices.
IND. ENG.CHEM. FUNDAMENTALS 4, 239-240 (1965)
67 North Main Street 0 Middleport, New York
SIGMAMOTOR, Inc, Circle No. 29 on Readers’ Service Card VOL. 5 7
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