BRIEFS Summary of paiers publishd in this month’s research quarterly, I& EC Process Design and Development RESPONSE SURFACE METHODS I N HETEROGENEOUS KINETIC MODELING
I n process development studies for reacting systems, a reaction model must usually be specified which adequately describes the reaction kinetics; however, process improvement considerations often demand experiments different from those optimal for reaction rate modeling. A compromise can be obtained by examining the empirical representations of the important responses, obtained via response surface methodology. These empirical representations of the response surfaces can directly assist in the specification of a reaction rate model, indicate regions of further experimentation desirable in the modeling program, and provide information necessary for process improvement and preliminary process design and economic considerations. The discussion is illustrated by data on the vapor-phase isomerization of n-pentane in the presence of hydrogen and over a supported metal catalyst.
J . R. Kittrell, Chevron Research Go., Richmond, Calif,, and John Erjavec, University of Wisconsin, Madison, W i s .
IND. END.CHEM.PROCESS DESIGN DEVELOP. 7, 321-327 (1968) GAS-SOLID HEAT TRANSFER IN FLUIDIZED BEDS
The unsteady-state heat transfer between gas and solid particles was studied in fluidized beds, 2 and 4 inches in diameter. A system of transient heating and cooling of glass beads, silica gel, and alumina between 130’ and 288’F. was used. The effect of several variables, particle size (0.004 to 0.2411 inch), bed settled heights (0.8 to 16 inches), particle densities (8.5 to 206 pounds per cu. foot), thermal conductivities (0.013 to 1.8 B.t.u./(hr.)(sq. ft.) (OF./ft.), and air velocities (0.543 to 4.347 feet per second) on the space-averaged heat transfer coefficient, U , was investigated. Using a new approach in interpreting the driving force, a correlation for two ranges of Reynolds numbers, 10 to 60 and 60 to 2200, was developed.
R. 5. Mann and L . C. L. Feng, Department of Chemical Engineering, University of Ottawa, Ottawa, Canada IND.END.CHEM.PROCESS DESIGN DEVELOP. 7,327-335 (1968) PREDICTION OF VAPOR COMPOSITIONS IN VAPOR-LIQUID SYSTEMS
A method of calculating vapor-liquid equilibria in multicomponent mixtures is described. No experimental information of multicomponent mixtures is necessary. The data needed can be obtained from the binaries. Two models were used to calculate the activity coefficients: the Wilson equation and the Black equation. The Black equation has been extended in order to make it possible to extrapolate to other temperatures. Parameters in the equations are calculated by a nonlinear least-squares fit of observed data. The efficiency of the method when using the Black and Wilson equations in predicting vapor-liquid equilibria has been compared. The efficiency in predicting equilibria a t other temperatures, some more than 40’ C. from those where data were obtained, has also been investigated, The Wilson equation has been more accurate in nearly all cases studied.
Ivars Neretnieks, Royal Institute of Technology, Stockholm, Sweden
binary mass transfer equations predicted the ternary efficiencies as well as a more complex method which included ternary interactions. The average absolute deviation between predicted and actual ternary efficiencies was 8 efficiency 70for either prediction method.
David A . Diener and J . A . Gerster, University of Delaware, Newark, Del. 79711
IND.ENG.CHEM.PROCESS DESIGN DEVELOP. 7, 339-345 (1968) FEEDBACK TIME-OPTIMUM PROCESS CONTROLLERS
A study of feedback, time-optimum, switching controllers is reported with emphasis on their practical design characteristics for process control. An overdamped second-order plus dead time model is assumed to represent adequately the plant dynamics for servo-control purposes. Optimum control of the model is described and analytical switching functions are given. Synthesis of the optimum controller and of a useful suboptimal controller is described for a n analog computer. The controllers are tested on plant simulations differing from the model and on a real process involving temperature control.
P . R . Latour, L . B. Kopjel, and D . R . Coughanowr, Purdue University, West Lafayette, Ind. IND.END.CHEM.PROCESS DESIGN DEVELOP. 7, 345-353 (1968) EXPERIMENTAL ANALYSIS OF FORCED CONVECTION FILM BOILING FROM A FLAT HORIZONTAL PLATE
An experimental study of the effect of velocity on film boiling heat transfer from a flat plate for two liquids, hexane and methanol, has shown that for velocities less than 4 feet per second the NN“ relationship is: NNu = 0.141 ( N ~ , ) O . ’ ~ ( i V , , , ) 0 . 4 8 ( ~ ‘ / p ) 1 . 8 8 . This study verifies the NPrKrelationship in the model of Owens. The experimental N R e effect is much smaller than his model would predict. The large effect of the ratio of the liquid to vapor viscosity is an effect not accounted for in any existing models for forced convection film boiling. A high speed photographic study indicated that Taylor instability analysis was appropriate. The experimental “most dangerous wave length” of 0.76 compared favorably with the theoretical value of 0.69 inch. J . F . Zemaitis, Jr., and R. I . Kermode, Carnegie Institute of Technology, Pittsburgh, Pa. 40506
IKD.ENG.CHEM.PROCESS DESIGN DEVELOP. 7, 354-359 (1968)
CORRELATION OF GRAVITATIONAL FORCE FOR ABSORPTION IN PACKED COLUMNS
Re-examination of gas absorption data in packed columns at different gravitational force levels resolves existing differences and agrees with the correlation of Onda, Sada, and Murase. An end effect correction is proposed to be applied for gravitational forces in excess of those at sea level. A unified correlation is developed and trends with liquid Reynolds number and body force are presented.
IND.ENG.CHEM.PROCESS DESIGN DEVELOP. 7,335-339 (1968)
G . 5. Jackson and J . M . Marchello, University of Maryland, College Park, M d . 20740
POINT EFFICIENCIES IN DISTILLATION OF ACETONEMETHANOL-WATER
IND.ENG.CHEWPROCESS DESIGN DEVELOP. 7, 359-361 (1968)
Point efficiencies measured on a rectangular split-flow sieve tray, 5 by 6 inches, are reported for the acetone-methanol-water system as well as the systems acetone-methanol, acetone-water, and methanol-water. For the composition ranges investigated in all these systems the mass transfer was gas-phase-controlling. From the binary data a correlation developed for the number of gas-phase transfer units, N Q , was used as a basis for the prediction of efficiencies in the ternary system. A prediction method based on 90
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
CALCULATION OF THE TRANSIENT BEHAVIOR OF SOLVENT EXTRACTION PROCESSES
A method for calculating the transient behavior of countercurrent solvent extraction processes permits multiple feed and exit streams to simulate multicontactor cascades. Cocurrent mass transfer efficiencies represent nonequilibrium distribution behavior. Agreement between calculated and experimentally observed
J. T . Lowe, Savannah River Laboratory, E. I . du Pont de Nemours €3 Go., Aiken, S. C. 29801
uses a high mole ratio of water to ethylene oxide in the feed. At the high mole ratios used in commercial hydration reactors, the selectivity obtained in a distillation column reactor can be exceeded by either the plug flow or the perfect mixer reactor. Calculations show that over a wide range of relative volatilities, the performance of a distillation column reactor for a series-parallel reaction does not exceed that of a plug flow reactor using a high mole ratio in the feed.
INDENG.CHEM.PROCESS DESIGN DEVELOP. 7, 362-366 (1968)
T . E. Corrigan and J . H. Miller, Ohio State University, Columbus, Ohio
concentration profiles is good. The usefulness of the additional information obtained from transient calculations which is not available from steady-state calculations is described. This method can be used to optimize startup and shutdown procedures in nuclear fuel processing equipment.
IND.ENG.CHEM. PROCESS DESIGN DEVELOP. 7,383-384 (1968) AMMONIA, A VERSATILE LIQUID EXTRACTION SOLVENT
Liquid ammonia is a versatile solvent for making chemicals, solvents, fuels, and lubricants from a wide variety of petroleum fractions. A continuous extraction pilot plant equipped with a 20stage mixer-settler extractor was used to make 194 extractions with 32 different petroleum fractions. The effects of solvent to feed ratio, reflux ratio, number and distribution of stages, solubility pattern, boiling point range of the feedstocks, and solvent recovery techniques on product yields and purities were investigated.
Paul Barton, R. H. McCormick, and M . R. Fenske, Pennsylvania State University, University Park, Pa. 76802
IND.END.CHEM.PROCESS DESIGN. 7, 366-371 (1968) KINETIC STUDY OF NEUTRAL SULFITE PULPING OF ASPEN CHIPS
A delignification rate expression suitable for digester design and control in neutral sulfite pulping of aspen chips has been developed. The rate equation takes into account the effects of temperature, concentration of cooking liquor, and time of reaction on pulping rate in the temperature range of 338' to 385' F. and concentration range of 7.5 to 30 grams of NazSOa per liter expressed as NagO. A molal ratio of 4 parts of NazCOs per 3 parts of NaiS08 was used in the cooking liquors. A rigorous method has been developed to compute the equivalent cooking times based on time-temperature data for a given cook. Data on yields and lignin contents at different concentrations and temperatures on Ross diagrams show the relative rates of removal of lignin and nonlignin fractions. For H-factor computations, tables on relative rates have been prepared in the temperature range of 212" to 400" F. An excellent correlation has been observed between H-factor and yield. Graphs showing H-factor-yield relationships at various concentrations are presented. N. C. S. Chari, Forest Products Division, Owens-Illinois, Im., Toledo, Ohio 43607
IND.ENC.CHEM. PROCESS DESIGN DEVELOP. 7,372-377 (1968)] MECHANISM OF GRAVITY FLOW OF PARTICLES FROM A HOPPER
From measurements of the flow rate and holdup of the outflowing particles from a bin, the existence of a dynamic arch was observed, and a simplified flow model was assumed, according to which every particle coming into the hypothetical dynamic arch falls freely down from its location under gravity. The average flow rate equation could be derived from the average location of a dynamic arch and the material balance in a hopper. Comparison of our equation with several empirical flow rate equations showed a satisfactory agreement, and indicated the validity of our concept for the flow mechanism of particles from a hopper. Furthermore, by observation of pulsating solids discharge, the collapsing frequency of solids was investigated, and consideration of the balance of discoidal block above the hopper outlet led to the flow criterion for the design of gravity-flow hoppers.
Kunio Shinohara, Yasuo Idemitsu, Keishi Gotoh, and Tatsuo Tanaka, Hokkaido University, Sapporo, Japan
IND.ENG.CHEM.PROCESS DESIGN DEVELOP. 7,378-383 (1968) EFFECT OF DISTILLATION ON A CHEMICAL REACTION
The reaction of ethylene oxide and water to form ethylene glycol in a distillation column reactor was simulated on a digital computer to test the hypothesis that 100% selectivities could be obtained in this type of reactor. I t was found that a distillation column does not surpass the performance of a plug flow reactor if the latter
OPERATION OF A SPRAY COLUMN WITH A DENSE PACKING OF DROPS
Operation of a spray column with a dense packing of drops and high holdups is desirable for commercial application of spray columns for extraction and heat transfer. The best shape of column for obtaining and maintaining a stable dense packing of drops is a constant diameter column with a conical extension at the top, in which the upper interface is maintained. The average drop size is in the same range for dense and dispersed packings of drops. The local holdup decreases up the column at high flow rates of either phase when the continuous phase is introduced at the top and center of the column proper. For kerosine drops, 3 to 4 mm. in diameter, the velocity of the kerosine at the coalescence interface is 0.7 cm. per second. The range of superficial velocities, for which a constant holdup dense packing can be obtained, is given.
Ephraim Kehat and Ruth Letan, Chemical Engineering Department, Technion-Israel Institute of Technology, Haqa, Israel
IND.ENG.CHEM. PROCESS DESIGN DEVELOP. 7, 385-389 (1968)
KINETICS OF THE HYDROREMOVAL OF SULFUR, OXYGEN, AND NITROGEN FROM A LOW TEMPERATURE COAL TAR
The results of the batch hydrogenolysis of S, 0, and N compounds of a low temperature tar in the presence of a WSZcatalyst indicated that all the hetero atoms can be completely removed at 500OC. and a pressure of 1500 p.s.i. Cracking, aromatization, and possibly dehydrogenation reactions start a t temperatures above 375OC. The hydrogenation of aromatic hydrocarbons starts at a pressure of 1250 p.s.i. The hydroremoval reactions of S, 0, and N are all first-order with respect to the heterocyclic molecules. Sulfur removal follows a true Arrhenius temperature dependence, but the removal of 0 and N shows slight curvatures in the Arrhenius plot which can be resolved into two parts, each approaching linearity with a break at 400OC. and having different activation energies. The magnitude of energies of activation and enthalpies of activation obtained suggest that chemical reactions but not physical processes are rate-controlling. The surface C-0, and C-N reaction involving the rupture of the C-S, bonds of the heterocyclic molecules appears to be the rate-determining step.
S. A . Qader, W. H. Wiser, and G. R. Hill, University of Utah, Salt Lake City, Utah IND.ENG.C H E ~PROCESS . DESIGN DEVELOP. 7, 390-397 (1968)
HYPERFILTRATION. REDUCTION OF CONCENTRATION POLARIZATION OF DYNAMICALLY FORMED HYPERFILTRATION MEMBRANES BY DETACHED TURBULENCE PROMOTERS
Spiral turbulence promoters positioned away from the membrane surface by small wire runners markedly increase the rejection of 0.01M MgClz by dynamically formed membranes. Salt rejection was increased most at the lowest Reynolds numbers, as expected from other turbulence promoter studies. An unexpected feature of the studies was the observation that the transmission rate through the membrane was increased 10 to 50y0in the turbulence promoter region compared to the values found in the same test with a portion of the tube containing no turbulence promoters.
David G. Thomas and J . S. Watson, Oak Ridge National Laboratory, Oak Ridge, Tenn.
IND.END.CHEM.PROCESS DESIGN DEVELOP. 7,397-401 (1968) VOL. 6 0
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KINETIC STUDIES FOR THE PREPARATION OF ITACONATES BY CONTINUOUS-FLOW AND FIXED-BED METHODS
PERFORMANCE OF A POLYMERIZATION REACTOR IN PERIODIC OPERATION
The esterification of isoamyl alcohol and itaconic acid was studied kinetically in a continuous-flow and fixed-resin bed under various conditions of temperature, acid-alcohol low ratio, and Dowex 50W x 8 resin particle size. Theoretical considerations concerning the rate constants of the two-stage reversible process at steady state and a t equilibrium were discussed, and mathematical equations were derived. Film diffusion increased with increase of contact time or decrease of flow rate; there was no diffusion resistance when the contact time was less than 50 grams per ml. per minute. Pore diffusion was also studied. The resin particles must have high porosity for this type of heterogeneous catalytic reaction.
The performance of a polymerization reactor in periodic operation is analyzed. Analytical estimates of the deviations from the steady-state performance are obtained for both simple addition and combination-termination polymerization mechanisms in the limit of very slow oscillation. The results using reported kinetic data indicate that a marked change in the distribution can be obtained for these polymerization reaction mechanisms.
J . C. Y. Tsao, Ting-Chia Huang, and Hung-Shan Weng, Dejartment of Chemical Engineering, Taiwan Provincial Cheng Kung University, Tainan, Taiwan, Republic of China
IND.ENC.CHEM. PROCESS DESIGN DEVELOP. 7, 401-409 (1968)
DYNAMIC SORPTION BY HYGROSCOPIC SALTS. A COMPARATIVE STUDY
Sorption of water vapor by solid desiccants is one of the methods employed for dehumidification of air. Studies have been made on silica gel, activated alumina, molecular sieves, and salts such as calcium chloride. However, no systematic evaluation of sorption rates of hygroscopic salts such as halides of lithium, cesium, rubidium, and their mixtures has been reported. A method has been devised to compare the sorption rates of hygroscopic salts. Sorption rates of lithium chloride, lithium bromide, cesium fluoride, cesium bromide, and their mixtures have been compared at a constant bulk temperature of 4OoC., with an inlet flow rate of 138.5 cm. per minute and a humidity of 18,000 p.p.m. on a volumetric basis. Lithium chloride and lithium bromide have been studied more intensively. Their sorption rates have been compared a t various humidities, flow rates, and temperatures. For the same surface area, lithium bromide shows a higher sorption rate at all temperatures. A simple empirical model has been developed which can adequately predict the sorption rate data of lithium chloride in the region between anhydrous and saturated solution.
Dharamuir Punwani, D . W . Chi, and D. T. Wasan, Institute of Gas Technology and Illinois Institute of Technology, Chicago, Ill. IND.ENG.CHEM.PROCESS DESIGN DEVELOP. 7, 410-415 (1968)
DYNAMICS AND CONTROL OF A BATCH REACTOR
Dynamics of a particular batch reactor are analyzed, using a pseudo-steady-state approach. The exothermic reaction involves separation of zirconium cladding from a highly enriched uranium fuel element by hydrochlorination in a gas-fluidized bed of inert solids. A practical stability criterion is derived and verified with experimental data. Control system simulation suggests that temperature control by manipulation of reactant feed composition is preferable to that obtained by manipulation of wall cooling.
Lowell B. Kopjel, Argonne National Laboratory, Argonne, Ill. IND.ENG. CHEM.PROCESS DESIGNDEVELOP. 7, 416-421 (1968)
PERIODIC OPERATION OF POLYMERIZATION REACTORS
The periodic operation of both condensation and free-radical polymerization reactors under sinusoidal perturbations in monomer feed concentration is considered with the aim of giving increased flexibility in the polymer molecular weight distribution (MWD). For the examples considered such operation yields both MWD widening (-16%) and narrowing (-67,) in the condensation case, but only widening in the free-radical cases ( - 5 0 ~ 0 ) when compared with steady-state operation.
W. H. Ray, University of Waterloo, Waterloo, Ont., Canada IND.ENG. CHEM.PROCESS DESIGNDEVELOP.7, 422-426 (1968) 92
INDUSTRIAL AND ENGINEERING CHEMISTRY
R. L . Laurence and G. Vasudevan, The Johns Hopkins University, Baltimore, M d .
IND.ENG.CHEM.PROCESS DESIGNDEVELOP.7, 427-433 (1968)
CHEMICAL REACTORS. REACTOR VOLUME
INFLUENCE OF PACKING O N EFFECTIVE
When an empty vessel with a low ratio of length to diameter is used as a continuous flow chemical reactor, the flow pattern involves a considerable amount of backmixing. The empty vessel may behave as the perfect mixer or the mixer with dead space. Putting a tower packing in such a vessel will have a twofold effect: The effective volume will be lessened and the amount of backmixing will be reduced. The net effect depends upon packing characteristics, packing size, and void volume. The flow pattern will depend upon the packing characteristics and can be evaluated i n terms of the axial dispersion model or the series of tanks model. This paper evaluates the two effects of adding packing for various types of packings. A. R. Shuki, T . E. Corrigan, and M . J . Dean, Ohio State University, Columbus, Ohio IND.ENG.CHEM.PROCESS DESIGN DEVELOP. 7, 433-434 (1968)
THERMAL CRACKING OF PROPANE. PRODUCT DISTRIBUTION
KINETICS AND
The thermal cracking of propane was studied in a flow apparatus between 625’ and 85OOC. and at atmospheric pressure. Fairly complete product distributions, including those for Cq, Cj, and CS hydrocarbons, were established and a reaction scheme was deduced. The order of the propane decomposition was determined by comparing experiments with different degrees of feed dilution and was found to vary with conversion and temperature. When the rate was fitted by means of a first-order kinetic expression, the rate coefficient decreased with increasing conversion. This socalled “inhibition” was expressed mathematically by considering the rate coefficient to be a hyperbolic function of conversion. The activation energy of the first-order rate coefficient increases with conversion. Finally, rate equations based on the radical nature of the process are discussed.
Alfons G . Buekens and Gilbert F. Froment, Rijksuniversiteit Gent, Gent, Belgium IND.ENG.CHEU.PROCESS DESIGN DEVELOP. 7, 435-447 (1968)
REMOVAL OF NITROGEN FROM ARGON WITH TITANIUM-METAL SPONGE
The rate of reaction of titanium-metal sponge with nitrogen in argon-nitrogen gas mixtures was studied at 900’C. The reaction rate is dependent on the partial pressure of nitrogen in the gas phase. At least three titanium-nitrogen solid phases are formed as the reaction proceeds, and the rate-controlling mechanism is believed to be the diffusion rate of atomic nitrogen through the TiN, (6) phase. A single relation has been developed which describes the titanium-nitrogen reaction kinetics of argon-nitrogen mixtures. Additional mathematical relationships were developed to permit estimation of the required size and useful lifetimes of titanium sponge gettering beds designed to remove a nitrogen impurity from otherwise pure argon.
M . L. Kyle, R. D . Pierce, L. F. Coleman, and J. D. Amtzen, Argonne National Laboratoy, Argonne, Ill. 60439
IND. ENG.CHEM.PROCESS DESIGN DEVELOP. 7,447-453 (1968)
EFFECT OF SLIP AND PHASE CHANGE ON SOUND VELOCITY IN STEAM-WATER MIXTURES AND RELATION TO CRITICAL FLOW
Experimental data and a n approximate analysis indicate that both slip and phase change occur within a wave traveling in steam-water mixtures and that the wave velocity is affected by these events. Slip results from the density difference between the two phases and the consequent difference in mechanical response to the wave pressure gradient. Some phase change occurs when rarefaction waves pass through steam-water mixtures. Whether vapor or liquid is the continuous phase and the degree of fineness of suspension of the discontinuous phase both influence the extent of phase change and slip and hence the wave velocity. As a result sound velocity depends upon the degree of mixing in the mixture and is not a thermodynamic property. The potential range of values of wave velocity, and hence critical flow fluid velocity, is very large. V . J . DeJong and J . C. Firey, University of Washington, Seattle, Wash. IND.END.CHEM.PROCESS DESIGN DEVELOP. 7,454-463 (1968) MIXING OF FLUIDS IN TANKS OF LARGE LENGTH-TO-DIAMETER RATIO BY RECIRCULATION
Measurements were made to determine the time required for mixing of miscible fluids, by recirculation, in tanks 4 and 6 inches in diameter and 15 to 40 feet long. The recirculation rate was varied from 2 to 100 gallons per minute. With a pulse input concentration, mixing times from 0.5 to 6.7 hours were required for the ratio of the completely mixed concentration to the maximum detectable concentration to reach 0.998. A theoretical method, using the axial-dispersed “plug-flow” model, was developed for predicting mixing times. Based on dispersion intensity data collected on onepass runs through the tanks, mixing times could be predicted fairly well ( ~ k 2 0 7 ? ~a)t circulation rates high enough to give Reynolds numbers greater than 10,000 in the tanks. At lower circulation rates, times predicted by using this dispersion intensity data were high. Predicted mixing times based on dispersion intensity data for straight pipes were normally too low. James E. Harrell, Jr., and Josefh J . Perona, Oak Rzdge National Laboratory, Oak Ridge, Tenn. 37830 IND.ENG.CHEM.PROCESS DESIGNDEVELOP 7,464-468 (1968)
Ionization 0
Radical formation
0
Crosslinking Chain scission
0
LIQUID-PHASE OXIDATION OF CYCLOHEXANOL
Homopolymerization
The liquid-phase oxidation of cyclohexanol, in multistage agitated flow reactors under atmospheric pressure and within the temperature range of 100’ to 13OoC., was studied. Cyclohexanol was oxidized to hydrogen peroxide and cyclohexanone, accompanied by decomposition of hydrogen peroxide and oxidative degradation of cyclohexanone. The rates of these reactions correlated as follows, for cyclohexanol conversion of less than 15%. (Rate of k z ( H z 0 2 ) 2 . (Rate oxidation of cyclohexanol) = k l ( H 2 0 2 ) 1 / 2 of decomposition of H s O Z ) = ka(HLO2) k 4 ( H 2 0 2 ) 5 . (Rate
+
+
of degradation of cyclohexanone) = kb ((>=o)’.
Graft copolymerization
-Whatever the effect of high energy radiation on polymers, it is covered in this book. Eighteen papers examine all aspects of polymer irradiationby x-rays, beta rays, electron beams, or ultraviolet l i g h t - m d report new research.
Operat-
ing variables of reactors were optimized by using dynamic programming to minimize the loss of hydrogen peroxide. Sachio Ishimoto, Tadahisa Sasano, and Kenji Kawamura, Tetjin Products Development Institute, Iwakuni City, Japan IND.ENG.CHEM.PROCESS DESIGN DEVELOP 7,469-475 (1968) ASPECTS OF THE FORWARD DYNAMIC PROGRAMMING ALGORITHM
Computational features of the forward dynamic programming algorithm as applied to a nonautonomous optimal control problem are discussed and a detailed numerical example is presented. J.H. Seinfeld and Leon Lapidus, Princeton University, Princeton, N . J. IND.ENG.CHEM.PROCESS DESIGN DEVELOP. 7,475-478 (1968) COMMUNICATION. DISCRETE DYNAMIC PROGRAMMING ALGORITHMS
Systems engineering problems which can be cast in the shortestroute format and a class of discrete optimal control problems are considered to provide better understanding of the elements of the optimal control problem by comparison to the shortest-route problem from operations research and to redefine the optimal control problem for further application of available computational algorithms in systems engineering. J . H. Seinfeld and Leon Lapidus, Princeton University, Princeton, N . J. IND.ENG.CHEM.PROCESS DESIGNDEVELOP.7,479-480 (1968)
The first two papers survey radiation mechanics i n polymers and the chemical nature of reactive species produced. The remainder report newwork in all of theareas listed above. 275 pages cloth bound (1967) $8.00 postpaid in U.S. and Canada; plus 20 cents foreign and PUAS. Set of L.C. cards free with library orders.
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