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Mathematics. Cornnuters, Operations Research. and Statistics I
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by R. Curtis Johnson, Thompson-Ramo-WooldridgeProducts Co., Los Angeles, Calif. William E. Ball, Monsanto Chemical Co., St. Louis, Mo. William F. Burggrabe, Compumatix, Inc., St. Louis, Mo. Richard 1. Heiny, Dow Chemical Co., Midland, Mich. Joseph 1, Russell, Scientific Design Co., Inc., New York, N. Y . Robert F. Sweeny, Applied Science Laboratories, Inc., State College, Pa.
Add EVOP, optimality,
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random balance, and universal programming
to your vocabulary
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N EARLIER reviews the trends in applications of mathematical techniques in the chemical field have been pointed out, emphasizing the interactions between the areas covered. I n chemical engineering itself, the major developments are an increased emphasis on the fundamentals, such as mass, momentum, and energy transport, which may eventually convert this field to a science and the rapid developments in process control theory and application. I n the first case, the boundary value problem takes on even more significance, and there is increased use of vector and tensor notation. T h e present mathematics review emphasizes the abundance of studies devoted to the fundamentals. T h e int-rest in control theory implies that the mathematics and terminology of the servomechanism specialist may soon become commonplace in chemical literature. Complex variables and transforms will find more willing acceptance in the future. While the demonstrated existence of a solution or the definition of a problem in abstract terminology will always find a friendly reception in the heart of the academician, the practicing engineer continues to be obsessed with the desire for numbers. T h e bridge between unusable theory and practical need has existed for some time in the form of the computer. Thus the numerical method appears more a n d more frequently in the applied mathematics references. T h e use of statistical techniques is increasing
rapidly, and methods which were completely unfeasible in the past are now readily implemented with the use of computers. T h e scope of the operations research portion of the review, new last year, has increased. Economic pressure on industrial concerns, increased demonstrations of the usefulness of operations research in other areas, the development of the techniques used by operations research, and again the availability of the computer all make the future promising for this area. Though it is too early to predict accurately, the technique of dynamic programming may offer unlimited potential; references in this area are few, and applications to practical problems in chemical engineering have not yet been demonstrated conclusively. This review has continued in the same pattern set u p in the past. References of potential interest to practicing chemists and chemical engineers have been covered. I n doing so we have not mentioned many fine pieces of work in the respective areas of interest; these were either too far afield in application or too restricted in scope.
Mathematics As in the past, the use of advanced mathematical techniques has continued to increase. This is the result of the interest in increasingly complex mathematical descriptions of physical systems and the attempt to solve the resulting
equations. These complexities generally arise from the consideration of either additional variables or interaction effects. T h e abstracting of mathematical derivations is a trend that may have caused a few articles to be left out of this review. When this trend is carried to the point where only an original equation plus the author’s solution are presented, it is difficult to evaluate the paper for its interesting or novel applications of mathematics. Of the references listed in this section of the review, about half are concerned directly with the solution of boundary value problems. Thirty three of this difficult class of problems were solved in the more or less normal manner and are grouped together. Tables I, 11, and I11 summarize the important articles dealing principally with mass, momentum, and energy transport phenomena. An article by Sage (SE)discusses the equations and various interactions of these phenomena in the field of fluid mechanics. T h e proceedings of the sixth Midwestern Conference on Fluid Mechanics ( 8 E ) contains several articles of mathematical interest. Although kinetics is generally not considered by itself as far as its mathematical content, an article by Katz ( 3 F ) has some interesting mathematical equations pertaining to reactions catalyzed a t the tube wall. Several reviews in the general fields of molecular properties or kinetic reactions also contained mathematical VOL. 52,
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Mathematical Applications in Energy Transport Application Dynamic
Dynamic response of heat exchangers having internal heat sources Unsteady laminar boundary layers over arbitrary cylinder with heat transfer in incompressible flow Calculation of transients in cross flow heat exchanqer Transient heat conduction in elliptical plates and cylinders Numerical and machine solutions of transient heat conduction problems involving melting or freezing Approximate treatment of unsteady heat conduction in semiinfinite solids with variable thermal properties Dynamics of heat removal from continuous agitated tank reactor Transient heat transfer for laminar forced convection in thermal entrance region of flat ducts Transient response of heated air in enclosure with heat losses Momentum Effects Vortex-flowtemperature separation by superposition of spiral and axial flow Laminar heat transfer in rectangular channels Theoretical explanation of heat transfer in laminar region of Bingham fluid Effect of volume heat source on free-convection heat transfer Turbulent flow in circular tube with arbitrary internal heat sources and wall heat transfer Heat transfer from rotating disk to fluids of any Prandtl nurnber Effect of nonisothermal free stream on boundary layer heat transfer Compressible turbulent boundary layers with heat transfer with pressure gradient in flow direction Heat transfer in laminar flow through tube Variational method for fully developed laminar heat transfer in ducts Miscellaneous Packed thermal diffusion column Heat and mass transfer from wall to fluid in packed beds Wave theory of heat transfer in film boiling Theory of rotating condensation Laminar condensation heat transfer on horizontal cylinder Boundary-layer treatment of laminar film condensation Analysis of aerodynamic heating for reentrant space vehicle Radiator design criteria for space vehicles Heat transfer from annular fin of constant thickness Optimization of conventional-fuel fired thermoelectric generator
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descriptions that were of interest. O n e (6E)discusses some methods used to obtain and relate the Arrhenius rate equation parameters to molecular properties. Others reviewed the molecular structure the foundations of radiaof liquids (ZE), tion chemistry (7E),and, in a somewhat related article (70E), reactor physics in the chemical industry. An interesting analysis of vapor-liquid equilibrium was presented ( 6 A ) . i’iumerical approaches to problems are becoming more prevalent as large-scale digital computers become more available and their use is better understood. A rather new philosophy of expressing continuous problems in discrete form was outlined by Greenstadt ( 4 4 ) . An interesting explanation of the mathematical technique of relaxation, applied on digital computers, was presented by Radd and Tek (7C), while the physical interpretation of the relaxation method was described by Thomsen (IOC). Sugai (9C) has discussed the numerical solution of the Laplace equation when given Cauchy conditions. Use of operational techniques has also been on the increase in applied mathematics, principally because of the impact of automatic control theories on the chemical industry. A revision of Churchill’s ‘(Operational Mathematics” ( I E ) has recently been published; it seems to be considerably improved, with a more expanded treatment of some sections and, in general, better examples throughout the text. Several articles discussing the application of the root locus method \vere found (3A, 4E, 5E), and the interrelationships between frequency and transient responses were discussed (BC, 5 F ) . An interesting listing of some 100 books on instrumentation and automatic control published since 1940 ( 3 E ) emphasizes the rapid growth of this field. Several articles of a probabilistic nature attacked problems vital to the chemical industry. T h e most interesting of these, mathematically, explored the areas of statistical decision theory ( 4 0 ) , the control of subjective evaluations ( 2 0 ) , and the inventory of unknown demand (30). An excellent report (ID) describes in detail the application of the Monte Carlo technique in solving partial differential equations. Several contributions used statistical functions to describe occurrences basically not of a statistical nature (4F, 6F, 7F, 7ZF). As a matter of fact, there seems to be a growing utilization of probabilistic-type distribution functions. Although linear programming problems are generally discussed in the operations research section of this review, a study of the sensitivity analysis of linear programming solutions ( 7 A ) contains a good mathematical treatment of its sub-
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ject. Several reports of mathematical interest which d o not seem to fit very well with the others are included in Table IV. Computers I t is difficult to choose just one notable development in computers. Ilowever, certainly a n important development during the past year has been the general advent of “solid-state” digital computers. I t now seems safe to predict that every major computer manufacturer will have a solid state computer for sale
soon. T h e most exciting possibility in applications is not really a development yet, but even the prospect of a universal programming system seems significant. I t is interesting that so many articles dealing with computers have been submitted to one editor that he has found it necessary to call a hiatus for some time. Collections. Among the collections of computer methods and programs are the Proceedings of the Joint NYU-IBM Symposium on Digital Computing in the Chemical and Petrochemical Industries, Proceedings of the Fifth Annual Computer Applications Symposium (14 studies), and a Chemical Engineering Progress Symposium Series ( 3 H ) containing 12 articles. A most ambitious program exchange project has been started during the past year by the American Institute of Chemical Engineers, and a review of progress in this area was presented recently by Carl-
Table II. Mathematical Applications in Momentum Transpor’t“ Application Turbulent flow of non-K‘ewtonian systems Non-NeLvtonian fluid in laminar flow Entrance effects and prediction of turbulence in non-Newtonian flow Viscous flow relative to arrays of cylinders Fluid dynamics and diffusion calculations for laminar liquid jets Longitudinal laminar flow between cylinders arranged in regular array Liquid mixing on bubble-cap and sieve plates Surface dynamics of fluidized beds and quality of fluidization Axial mixing and cxtraction efficiency Residence times and distribution of residence times in dispersed flow systems Rate of mass transfer from gas to moving liquid film Fluid flow between rotating members The study of flow and reaction rates in turbulent flames a See also Table I, 5B, 24B-26B, 33B, 8F.
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Mathematical Applications in Mass Transport” Application
Diffusion-controlled chemical reactions Research on ion exchangers by percolation technique Ion exchange kinetics for systems of nonlinear equilibrium relationships Mass transfer at low flow rates in packed column Steady state absorption of sparingly soluble gas in agitated tank with simultaneous irreversible firstorder reaction Rate of absorption of NO2 in w’ater Evaporation of two-component liquid mixtures Multicomponent distillation on large digital computer a See also Table I, 7B, 75B, 24B-Z6B, 32B; Table 11, IB, 9B, ZOB, 22B.
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son (ZH).Abstracts of program manuals which can be made available first appeared in the May issue of Chemical Engineering Progress and have continued a t the rate of about four each month since. Computing Machinery. A good review of the fundamentals of operation for general purpose digital and analog computers was presented by Freilich (6157). Information on digital differential analyzers a n d a specific classification of various machinery with some specifications were also given. An interesting general article appeared in Fortune ( 7 H ) . Although it is directed toward the business uses of computers, it gives good background a n d general information, lists various available computers with descriptions, and gives some fundamentals of computing in “lay” terms. T h e author’s opinion that most manufacturing companies are making investments in new machinery while running a t a deficit seems quite significant. T h e r e is evidently a good deal of money gambling on a continued accelerated growth in computer use. Washington University workers have developed a special purpose analog computer for matrix calculations ( 3 H ) ,quite a novel gadget. Examples of how a digital differential analyzer can be put to use by the chemical engineer were given by Rutishauser (7H). Books. An up-to-date elementary book on programming a digital computer was published (OH). I t might even be worthwhile reading for the expert programmer. Two Department of Commerce publications appeared. T h e first (5H) is a catalog of available O T S reports concerning computers. T h e second report ( 8 H )is a technical presentation having to
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do with the use of analog computers for chemical engineering problems. Specific Applications, Table V lists articles of interest concerning specific applications to chemical and chemical engineering problems.
Operations Research
A review of recent publications in the field of Operations Research suggests that authors are too often presenting their own theoretical approaches in association with a hypothetical problem on such things as inventory control. T h e techniques and methods of operations research are fundamentally sound enough that the literature needs to begin to employ tests of relevance. There are so many specialties a n d intricacies in this field that the practicing chemist, chemical engineer, or manager must consider that his starting point is a specific problem in place of a specific technique. Consequently, this review concentrates on literature applications a n d newly published basic textbooks in the field. As in the past, practically all applications are of general utility to industry as they encompass a broad business philosophy. Perhaps, the most striking development last year was the abundance of new basic texts in the field. Hopefully, these will serve as forerunners of more successful utilization of potential techniques.
Applications and Techniques. A method of evaluating research and development by Malcolm (73K) is of unusual interest. A technique has been developed for measuring and controlling research and development progress for the Polaris fleet ballistic missile program. Basic data for the computer consists of elapsed time estimates for scheduled activities in a research network. I t has aided technical management in isolating critical and slack time areas, resulted in clarifying the effect of policy changes, and provided probabilities for meeting goals. Life tests a n d performance are pertinent to many areas in the chemical and petroleum fields. Allen ( 7 K ) covers several methods of employing accelerated general life tests a n d their ccrresponding mathema tical foundations. A capacitor problem is treated in detail. A new approach to the determination of replacement costs (75K) utilizes the logarithmic decrement of salvage value in place of the economic life or the obsolescence gradient for determining minimum average costs. Pounds (76K) has developed a statistical scheduling of a highly mechanized production facility to account for equipment maintenance a n d resulting variations in output. A solution to the problem of determin-
ing timing a n d magnitude of purchases of raw materials varying in price from period to period was presented by Fabian ( 8 K ) . A dynamic programming model is employed in the solution. An inquiry into the optimum structure of sales information for a decision by a multiplant, multiproduct organization was made by Chacko ( 3 K ) . I t results in a working hypothesis about the process by which data of a sales organization can be used in decisions. I n terms of people, Craft ( 6 K ) discussed experience of a simulation a n d gaming program given by the American Management Association to develop managers without actual experience. Watson (79K, 20K) has quantitatively related the contribution of the technologist to the financial performance of 40 major chemical a n d petroleum companies. T h e return on invested capital is corrzlated with an independent function which is obtained from a measure of the management participation by technical men and the level of research activities. Table VI lists significant literature articles ,on basic techniques. Also tabulated are additional applications of a more general business character. New Books. As mentioned previously, the appearance of an abundance of basic texts on operations research is perhaps the major development of the year. Bellman (3L) has written what may be the most important book on mathematical methods in operations research to appear in several years. H e presents mathematical bases for answering the question: W h a t do I do now? This is done a t each stage of complex strategy programs in place of employing a preset program throughout. Case
Table IV.
Miscellaneous Mathematics Publications Application Upper and lower bounds for special eigenvalues Structure analysis of torispherical shells Gas film lubrication study-theoretical analysis of slider bearings Formation of gas bubbles at submerged orifices Problems in industrial reactions Structure analysis of high pressure apparatus Gas lubrication study-numerical solution of Reynolds equation for finite slider bearings Analysis of complex kinematic chains with influence coefficients Cross-flow air-water contactors ; numerical methods Stability of steady exothermic chemical reactions in simple nonadiabatic systerx Graphical and analytic criteria developed to select best continuous stirred tank reactor system
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examples a r e intriguing and demonstrate that his principles of optimality may be generally useful to those in chemical a n d allied fields. A great many problems fall within the scope of this concept and it is highly recommended as a stimulus to problem solving. A major work that presents a broad view of the mathematical techniques used in operations research has been written by Saaty (74L). Required reader background fluctuates highly, but its concentration on linear, nonlinear, and dynamic programming should be of great utility to workers for both the theory and applications presented in a bibliography (13L). Chorofas (5L) has written a text for transmission of simulation, gaming, linear programming, a n d matrix methods to mid-management. I t appears to lack unity and integration. Wiener (77L) has written an excellent text on nonlinear problems in random theory for the analyst inclined toward sophisticated mathematics. Texts covering production scheduling a n d inventory control have been written by Magee (70L) a n d Vazsonyi (76L). Vazsonyi presents a more mathematical description of programming methods than Magee, who confines his text more to a discussion of general problems a n d practical methods of solution. Mathematical theories of inventory a n d production problems were covered by h - r o w (ZL). An outstanding asset is the wide variety of models it brings to bear on inventory problems. I t represents a n excellent source on the application and use of the theory of stochastic processes. Table V I 1 lists other text material of consequence together with brief comments on their content a n d estimated utility. Of particular import is the comprehensive listing of available unclassified literature i n the field brought together by the Case Operations Research group.
S fa t i s tic s T h e most important developments in the application of statistical methods to chemical engineering problems in the past year have been in three areas: T h e application of statistical design to production problems in the form of evolutionary operation. T h e beginning of use of a system of experimental design called random balance. Several reports describing in language intelligible to the ordinary engineer the application of more sophisticated mathematical techniques to the analysiq of data. These points are discussed a t greater length below. Again last year the use of
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statistical methods in the chemical industry appeared to be becoming more extensive, as well as more sophisticated. As in the past, the use of statistical methods in statistical mechanics, thermodynamics, structure and behavior of polymer molecules, and nuclear phenomena has been considered outside the scope of this review. Design of Experiments. T h e most important development in the past year has been the extensive use of evolutionary operation, a n adaptation of statistical design that permits study of a n operation on a plant scale without varying conditions enough to upset the operation or the product. Koehler (77M, 78M) gave a good, simple description of Evop a n d of the philosophy behind its use. Box a n d H u n t e r (3M) have set u p the calculations involved in two- and threevariable programs and carried these calculations through several cycles. Their description should be easy for the practicing engineer to use. A good, simplified discussion of Evop has been published by Whitwell ( 2 1 M ) . This article
Table V.
is suitable as a n introduction to the subject for interested persons not directly concerned with the calculations involved. SVhile it appears that Evop is in fairly widespread use in the chemical industry, as yet there is very little information published concerning the actual financial benefits obtained from its use. However, it is the most promising technique yet proposed for the improvement of an operating plant process without undue effort or expense, and as such it should be carefully studied by chemical engineers concerned with production operations. T h e articles mentioned above are adequate guides to the application of Evop for engineers with some statistical background. A second important development in the design of experiments has been the rise of interest in random balance design. 'This technique, originated by Satterthwaite, has the distinguishing characteristic that any systematic choice of experimental conditions is deliberately avoided in favor of a strictly random set of conditions. By so doing, Satterthwaite claims
Applications of Automatic Computers Simulation and Dynamic Distillation Ref. Behavior Rcf Dynamic behavior of ( I I J ) Multicomponent calcula- ( 7 F , 21, 175, tions" 3OJ-32J) fixed bed reactorsa Nonideal equilibria calcu- ( 3 8 J ) Stirred tank reactora (784 lationsa Unsteady state behavior (245) of chemical processesb Flash vaporization calcu- ( 3 9 J ) lationsa Simulation of distillation ( 2 5 J ) columns* Heat and Mass Transfer Frequency response= (8C) Heat exchanger design" ( 7 J , 45J) Optimization Natural convectiona (34J) Heat and mass transfer in ( 3 7 4 Optimizationa (20J) dryinga Economic designsn (27.~1 Cooling towersa (9G) Back diffusion in extrac- (22B,4 8 J ) Thermodynamics and Equilibrium tiona Antoine equation pro- ( 7 J ) grama Reactor Design Chemical equilibrium ( 76J) Pyrolysis design" (3J) calculationsa Activation energy calcu- ( 7 0 J ) Theory of corresponding ( 2 7 J ) lationsa states for multicomReactor design" (5G, 57J) ponent systemsa Kinetics of ethane py- ( 4 3 J ) rolysisa General and Miscellaneous Unsteady state flow ( 9 J ) throuph Dorous mediaa General appiications" Pipelincdesign' (75J, 79J, 22J, Generala 3.3J Generalb Generala$* Hydraulic circuits and ( 2 3 J ) Maintenance records and componentsa calculationsa Translation and Information Survey of use in oil industr ya Retrieval Growth and interaction Linquistic computera (6J) of droplets in a vapor Machine translationa (73J, 29J) s tr e ama Language coding for (26.7) Use of flowsheets in calmachine searching" culationsa Translation encoding of ( 4 7 J ) Applications of relaxation abstractsa proceduresa Information retrievala (46J, 49J) Teaching tools in engiGAT procedure in trans- ( 5 2 J ) neei-inga lationa Teaching aids in engi5 Digital application. Analog application. neeringb
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The magnetic decision element (right), developed by the Applied Mathematics Division, U. S. Naval Ordnance Laboratory, contains two units in one package which can replace two vacuum tubes shown on the left
Table VI. Techniques and Applications of Operations Research Subject Techniques Comparison of maxima seeking methods New method to determine stochastic decision rules for sequential planning Status of multistage linear programming Optimal detection of failure in complex system Method of formulating linear models for subsequent computer analysis
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Applications Specification control, simulation obsolescence-inventory economics Inventory location and control Order size,. "iob lot manufacturing Discrete time-inventory. prob_ lems Seasonal demand-inventory economics Data processing; decision theory
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Table VII.
Texts in Operations Research Ref. Comment Integrated collection of research (7L) papers Theoretical technique source Language of economist Introductory for early math simplex method Maze of computational operations Bridge between specialists and management; weak Comprehensive listing of available unclassified literature in field Published notes General text
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This process analog computer, developed by Phillips Petroleum Co.'s instrumentation laboratories, continuously monitors reaction conditions at the Marlex polyethylene plant
to screen a large number of variables with a small number of experiments. Many professional statisticians ( 2 3 M ) have been quick to criticize random balance. Admittedly random balance is much less efficient than more systematic designs, especially for cases involving a small number of variables a n d a relatively small range on these variables. However, it is a badly needed step in the direction of reducing the number of experiments needed to screen variables and to draw some conclusions. Random balance, its advantages, disadvantages, a n d the conditions for its application have been described by Satterthwaite and others (79M, ZOM) and Budne (4M-6M). Available techniques for the analysis of data from random balance experiments were covered by Anscombe (ZM). At present the analyses are largely graphical ; this leaves much to be desired. Two applications were described by Budne ( 6 M , 7 M ) and Satterthwaite ( Z O M ) , but these are not adequate to draw any conclusions concerning the value of random balance. Simple discussions of design have been published by Hunter (75M) a n d by Johnson (76M),who is particularly concerned with cellulose experimentation. A good qualitative discussion of the value of statistics in experimental strategy has been published by Anderson ( 7 M ) . H e makes a point of the proper selection of variable ranges a n d makes good suggestions for the transformation of variables to enable a regular statistical design to cover only the operable ranges of variables. Youden's series of articles
(22)M has included a simple but good explanation of Evop (No. 6) a n d a brief discussion of random balance (No. 8). Several articles with very good descriptions of the application of statistical designs are listed in Table V I I I . Those by Hatfield (74M) a n d Dotson (77M) are probably most worthy of study, as they each handle a complicated situation with a large number of variables. Analysis and Correlation of Data. Several articles in which the analysis and/or correlation of experimental data were done statistically are listed in Table IX. These may be of value to experimenters with similar problems. A good description of fitting straight lines or parabolas to data was presented by Greyson and Cheasley ( 6 N ) . They say the method can be extended to polynomials. T h e treatment is quite complete and includes numerical examples. Two articles are concerned with the solution of equations such as may be developed in correlating data. One of these ( 7 N ) presents a method for finding maximum or minimum solutions of equations in many variables. T h e locus of the maxima or minima is expressed as a funcR is then tion of R, where R2 = &,2. plotted against y. T h e function y = f ( x , ) must be known. T h e second article by Marquardt ( 9 N ) presents methods of solving nonlinear models, such as arise from theoretical rather than empirical approach to correlating data. T h e methods are illustrated with the V a n Laar and Antoine equations. T h e above three reports should be of considerable use in correlating experimental data, a n d each is a clear a n d comVOL. 52, NO. 4
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Table VIII. Applications of Experimental Design Ref. Purpose of Investigation Use of Experimental Design Investigate bacterial deteriora- Used half-replicate of 24 as each run took 28 ( 8 M ) tion of woolen felts days. Very good discussion; a lot of time and money were saved Predicting properties of pre- Used Box design on four variables (30 points in (QW cipitated CaHPO, powder three blocks). Fitted equations of type y = Z b,xi. Complete discussion Study of perforated plate dis- Used Box design for four variables (28 points). ( 7 O M ) tillation columns Good description of design and comparison of results with previous classical study Density transients in a fluidized Factorial design far six variables at up to five ( 7 I M ) bed levels. Found two- and three-factor interactions. Complete statistical analysis Pulping of wheat straw 4 X 5 factorial in randomized complete blocks, ( 121M) two replications. Used orthogonal polynominals in linear and curvilinear regression analysis; also used three dimensional models Factors in separating finely 44 factorial split into pseudofactors to make a 28 ( 73B) Ran in four blocks. Found two- and threedivided solid on a vibrating screen factor interactions. Good description of design and analysis Phosphate fertilizer production Box design far five main variables (32 points) ( 7 4 1 M ) augmented by partial factorials to check nine minor variables. Developed a 15-term quadratic to fit data. Good handling of a complicated situation
plete guide for its use by the practicing engineer. Three others (7117, 7711: 72,V) present simple rules (with some short cuts) for the calculation of standard deviation and confidence limits. These are appropriate for the experimenter with little math training who wants to apply some statistical analysis to his data. Quality Control. T h e setting up of a control program to minimize costs associated with laboratory analyses, re-
jection of good material, and acceptance of poor material has been described by Davies ( I P ) . The discussion treats only batch situations, but it should be readily extended to continuous operations. Three articles (2P,6P, 7 P ) present qualitative discussions of the advantages of statistical techniques in quality control. Of these, Shewell (7P) also pays attention to the problem of minimizing control costs. A control chart using geomctric
moving averages was described by Roberts (5P). I n this each point is weighted, thc most recent receiving the most weight. These charts are considerably more complex than ordinary control charts, but their use may be justified if it is important to detect small changes quickly. Sampling for specifications and quality control in situations where it is difficult to get a representative sample was discussed by Maltenfort (3P). H e points out many common faults in the setting of specifications. Mandel (4P) presented a complete guide for setting u p a system for interlaboratory evaluation of testing methods. T h e system is illustrated by a study involving 14 laboratories and 14 materials. Special Applications. Several authors have used statistical methods in efforts to reduce or control costs. Milsum (7Q) described the distributions of errors and loads in minimizing costs associated with a regulator subject to both a steady reference input and a load disturbance. T h e use of elementary statistics to reduce costs in materials handling and inventory problems was reported by Robbins (SQ). H e warned that many withdrawal-from-stock distributions are not normal, and that inventory costs may be inflated unless this is taken into account. McElrath (5Q) pointed out that so-called “inefficient” statistics may sometimes be more economic to use than the more sophisticated “efficient” statistics. H e gives several simple examples. Plans for testing equipment life were set up by Liebernian (44). His study is not easy to read, but it is complete and can be used by engineers Lvith no statis-
Table IX.
Applications in the Analysis of Data Use of Statistical Mcthods Ref. Compared two paper testing ma- (2N) chines; complete analysis of variance; results were very comparable Correlated results from t\vo types of (3.V) instruments ; calculated numbtlrs of tests on each required to get desired confidence limits Correlated results from two types of (4,V) instruments ; developed correlation of type logy = Q (log x) 5 Set up analysis calculations includ- (5119 ing complete statistical analysis for a Burroughs E 101-1 digital computer; analysis was six-point assay with four standards and 1 6 unknowns Used elliptical confidence limits to (SA’) compare slopes and intercepts Correlated two types of vapor pres- (lo.+’) sure data from six laboratories with equation Y = QX 6; assigned confidence limits to slopes and intercepts
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analog computer can be used to solve complex
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m dChemical Engineering Fundamentals Review (3A) Ellingsen, W. R., Ceaglske, N. H., Zbid., 5, 30-6 (1959). (4A) Greenstadt, J., ZBM J . Research Deoeloj. 3, No. 4, 355-63 (1959). (5A) Lay, J. E., Trans. Am. Soc. Mech. Engrs. 81C, 213-22 (1959). (6A) Lu, B. C. Y., Li, J. C. M., Ting, T. W., IND. ENG. CHEM. 51, 219-22 (1959). --, \ - -
(7A) Shelty, C. M., J . Znd. Eng. 10, No. 5, 379-86 (1959). (8A) Toor, H. L., Chiang, S. H., A.I.Ch.E. Journal 5, 339-44 (1959). BOUNDARY VALUEONLY (1B) Acrivos, A., Chem. Eng. S G ~9, . 242-9 (1 950) \ - ~ - ' I -
(2B) Bird, R. B., Chem. Zngr. Tech. 31, 569-72 (1959). (3B) Bogue, D. C., IND. ENC. CHEM. ' 51, 872-8(1959). f4B'i Boon. E. F.. Tal, S. E.. Chem. Iner. ' Tech. 31; 202-12 (1959). ' (5B) Brunner, M. J., Trans. Am. SOC. Mech. Engrs. 81C, 223-9, (1959). (6B) Callinan, J. P., Berggren, W. P., Ibid., 81C, 237-44 (1959). (7B) Chang, Y . P., Zbid., 81C, 1-12 (1959). (8B) Dorweiler, V. P., Fahien, R. W., A.Z.Ch.E. Journal 5, 139-44 (1959). (9B) Gilbert, T. J., Chern. Eng. Sct. 10, 243-53 (1959). (10B) Han, L. S., Trans. Am. SOC.Mech. Engrs. 81C, 121-28 (1959). (11B) HaDDel. J.. A.1.Ch.E. Journal 5 , ' 17h-7 (ij59). ' (12B) Hirai, E., Zbid., 5, 130-3 (1959). (13B) Keller, H. H., Somers, E. V., Trans. Am. SOC. Mech. Engrs. 81C, 151-6 (1959). (14B) Lightfoot, E. N., A.Z.Ch.E. Journal 4, 499-500 (1958). (15B) Lorenz, M., Emery, A . H., Jr., Chem. Eng. SL. 11, 16-23 (1959). Il6B) Pearson, J. R. A.,Zbid., 10, 281-4 (1959). (17B) Randall, I. E., Sesonske, A., A.Z.Ch.E. Journal 5, 150-4 (1959). (18B) Rice, W. J., Wilhelm, R. H., Zbid., 4, 423-9 (1958). (19B) Richardson, J. F., Clzem. Eng. Sci. 10, 234-42 (1959). (20B) Scriven, L. E., Pigford, R. L., A.Z.Ch.E. Journal 5, 397-402 (1959). (21B) Siegel, R., Sparrow, E. M., Trans. Am. SOC.Mech. Engrs. 81C, 280-90 (1959). (22B) Sleicher, C. A , , Jr., A.Z.Ch.E. Journal 5 , 145-9 (1959). (23B) Sparrow, E. M., Gregg, J. L., Trans. Am. Sot. Mech. Engrs. 81C, 249-51 (1959). (24B) Zbid., pp. 113-20. (25B) Zbid., pp. 291-6. (26B) Zbid., pp. 13-18. (27B) Zbid., 26-E,161-5 (1959). (28B) Sparrow, E. M., Loeffler, A. L., Jr., A.Z.CI2.E. Journal 5, 325-30 (1959). (29B) Van de Vusse, J. G., Chem. Eng. Sci. 10 229-33 (1959). (30B) Walz, A., J . Research Natl. Bur. Standards 63B, 53-60 (1959). (31B) Wevers. C. J. H., Chem. En,g. Sci. ' 10; 171-89 (1959). (32B) Yagi, S., Wakao, N., A.I.CI1.E. Journal 5 , 79-85 (1959). (33B) Yang, K . T., Trans. Am. SOC.Mech. Engrs. 26E, 171-8 (1959). '
i
One of the advanced features of the RW-300 digital control computer is its modular construction which permits rapid, convenient maintenance
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tical background. T h e theory of extreme values has been reviewed briefly by Mandel (SQ); a very good bibliography is included. Fluctuations in a chemical reactor were treated statistically by Aris ( 7 Q ) . T h e treatment is illustrated with a second order reaction; the description is complicated but should be of value in reactor design. T h e random downstream migration of molecules in chromatography was described (34) with the use of rand o m walk. Probability theory was used (9Q) to establish theoretical limits of precision in x-ray emission spectrography. T h e precision of a method for determining gamma-ray density was evaluated by Cove11 (ZQ) using a Poisson distribution function. Books. Three books concerned with design of experiments have appeared. A short, readable book (5R)explains the philosophy of design and describes the choice on designs to fit particular situations. A selection of reports presented at a symposium o n design in 1957 was edited by Chew (4R). T h e main value here lies in five articles describing the industrial applications of design. Kitagawa and Mitome (8R) have given a complete listing of factorial designs. A volume intended as the text for a n introductory course in statistical methods was published by Lindgren (9R). Statistical quality control was described (ZR) in strictly nonmathematical terms for management people. T h e analysis of straight-line data is the
subject of a book by Acton (7R). H e states that his objective is to enable the experimentalist, rather than a professional mathematician, to apply these techniques; i n this he succeeds. He does treat briefly the analysis of nonstraight-line data. This book is wellwritten and should be of great value. A text on the handling of extreme values, by Gumbel ( 7 R ) , is a book for specialists, but it covers the subject well. I n addition to the books above, four have appeared about which the reviewer has no information except that they would seem to warrant attention (3R, SR, 70R, 77R). I n addition to published articles, many symposia are held each year. T h e journal Industrial Quality Control publishes information on almost all such symposia a n d short courses about three months in advance. T h e Gordon Research Conferences include a conference on statistical methods in chemistry and chemical engineering. The 1959 Gordon Conference was unusually good in that the discussions were generally concerned with very recent advances a n d were kept at a fairly high technical level.
Bibliography Mathematics
ANALYSIS (1A) Arpaci, V. S., Clark, J. A., Trans. Am. Soc. Mech. En rs. 81C, 253-66 (1959). (2A) Dodge, D. fV., Metzner, A. B., A.Z.Ch.E. Journal 5, 189-204 (1959).
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NUMERICAL METHODS (1C) Abramowitz, M . , Cahill, W. F., Wade, C., J . Research Natl. Bur. Standards 62, 101-5 (1959). (2C) Barbor, R. P., Larkin, J. D., others, A.Z.Ch.E. Journal 5, 37-46 (1959). VOL. 52, NO. 4
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) m d Chemical Engineering Fundamentals Review
(3C) Dekker, W. A., Snoeck, E., Kramers, H., Chem. Eng. Sei.11, 61-71 (1959). (4C) Dusinberre, G. M., Trans. Am. SOC. Mech. Engrs. SIC, 61-7 (1959). (5C) Kirkpatrick, E. T., Ibid.,81C, 54-60 i 1959). (6C) Murray, W. D., Landis, F., Ibid., 81C, 106-12 (1959). (7C) Radd, M . E., Tek, M. R., A.1.Ch.E. Journal 5 , 111-15 (1959) (8C) Schechter, R. S.,Wissler, E. H., IND.ENG. CHEM.51, 945-8 (1959). (9C) Sugai, I., I B M J . Research Develop. 3, 187-8 (1959). (1OC) Thomsen, J. S.,A.1.Ch.E. Journal 5. 268-9 11959). (11C) Tien, ‘ Cl, ’Thodos, G., Zbid., 5 , 373-8 (1959). (12C) Yang, K. T.,, Szewcgyk, A., Trans. Am. SOC.Mech. Engrs. 81C, 251-2 (1959). PROBABILITY (1D) Ehrlich, L. W., J . Assoc. Computing Machinery 6, 204-18 (1959). (2D) Howe. W. G.. Peacor,, W.,. J . Ind. Eng. 10, 138-43 (1959). (3D) Morris, W. T., Ibid., 10, 299-302 (1959). (4D) Wagner, H. M., Ibid., 10, 25-32 ( 1959). REVIEWSA N D BOOKS (1E) Churchill, R. V.: “Operational Mathematics,” 2nd ed., McGraw-Hill, New York, 1958. (2E) Dahler, J. S.,A.1.Ch.E. Journal 5, 212-22 (1959). (3E) ZSA Journal 6, No. 3, 60-1 (1959). (4E) Jawor, T., Control Eng. 6 , No. 10, 96-102 (1959). (5E) Ibid.,NO. 11, pp. 119-22. (6E) Johnston, H. S., Pitger, K . S., A.1.Ch.E. Journal 5 . 277-84 (1959). (7E) Kuppermann, k, J . Chem. ‘Educ. 36, 279-85 (1959). (8E) “Proc. 6th Midwestern Conf. Fluid Mechanics,” Univ. of Texas, Austin, Tex., 1959. (9E) Sage, B. H., A.1.Ch.E. Journal 5, 331-8 (1959). (1OE) Tomlinson, W. R., Jr., IND.ENC. CHEM.51, 1339-44 (1959). SPECIAL TECHNIQUES (1F) Amundson, N. R., Pontinen, A. J., Tierney, J. W., A.I.Ch.E. Journal 5 , 295-300 11959). (2F) Fanning, R. J., Sliepcevich, C. M., Ibid., 5 , 240-4 (1959). (3F) Katz, S., Chem. Eng. Sei. I O , 202-11
(4G) Hayes, W. B., 111, Hardy, B. W., Holland, C. D., A.1.Ch.E. Journal 5 , 319-24 (1959). (5G) Horn, F., Kuchler, L., Chem. Eng. Tech. 31, 1-11 (1959). (6G) Jung, H., Ibid.,31, 195-201 (1959). (7G) Michael, W. A., Z B M J . Research Decelop. 3, 256-9 (1959). (8G) Modrey, J., Trans. Am. Soc. Mech. Engrs. 2GE, 184-8 (1959). (9G) Schechter, R . S., Kang, T. L., IND. ENG.CHEM.51, 1373-6 (1959). (10G) Spalding, D. B., Chem. Eng. Sci. 11, 53-60 (1959). (11G) Trambouze, P. J., Piret, E. L., A.1.Ch.E. Journal 5 , 384-90 (1959). Computers GENERAL (1H) Bello, Francis, Fortune 60, 128 (October 1959). (2H) Carlson, W. M., 52nd Ann. Meeting, Am. Inst. Chem. Engrs., San Francisco, Calif., December 1959. (3H) Chem. Eng. News 37, 22-23 (Aug. 10, 1~. 9 -5 _ 9,. ).
(4H) Chem. Eng. Progr. Symjosium Ser. 55, No. 21 (1959). (5H) “Computers, 1937-58” (Catalog of Tech. Repts. CTR-371), Office of Tech. Services. u. s. DeDt. of Commerce. Washington 25, D. C: (6H) Freilich, Arthur, ISA Journal 6, 54-65 (July 1959). (7H) Rutishauser, R . W., IND. ENG. CHEM.51, 52A-54A (July 1959). (8H) Williams, T. J., “Application of Analog Computers to Various Combustion, Flame, and Fluid Dynamics Problems,” PB 151775, Office of Tech. Services, U. S. Dept. of Commerce, Washington 25, D. C. (9H) Wrubel, M. H., “A Primer of Programming for Digital Computers,” McGraw-Hill, New York, 1959. APPLICATIONS
. Petrol. Rejner 37, NO. 12, 165-72 (1958): (6J) Brown, A. F. R., 135th Meeting, ACS, Boston. Mass.. Ami1 1959.
(1959). \ - - - - / .
(4F) Overcashier, R . H., Todd, D. B., Olney, R. B., A.I.Ch.E. Journal 5, 54-60 (1959). (5F) Resenbrock, H. H., Control Eng. 6 , NO. 7, 116-20 (1959). (6F) Said, A. S.,A.1.Ch.E. Journal 5 , 223-4 (1959). (7F) Said, A. S., Zbid.,5 , 69-72 (1959). (8F) Siegel, R., Sparrow, E. M., Trans. Am. SOC.Mech. Engrs. 81C, 29-36 (1959). (9F) Sparrow, E. M., Siegel, R., Ibid., 81C, 157-67 (1959). (10F) Swanson, B. W., Somers, E. V., Ibid.,81C, 245-8 (1959). (11F) Wolfe, W. A., Ibid., 81C, 19-23 (1959). (12F) Zwietering, T. N., Chem. Eng. Sei. 11, 1-15 (1959). MISCELLANEOUS (1G) Appl. F. C., Zorowski, C. F., Trans. Am. Soc. iMech. Eners. 26E. 246-50 (1959). (2G) Galletly, G. D., Ibid., 81B, 51-66 (1959). (3G) Gross. W. A,. IBM J. Research D’eaelop. 3, 237-55 (i959). Y
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366
(1OJ) Computer Applied, Brit. Chem. Eng. 4, No. 1, 2 (1959). (1 1J) Deans, H. A., Lapidus, L., 52nd Ann. Meeting, Am. Inst. Chem. Engrs., San Francisco, Calif., December 1959. (12J) DeSilva, 3. R., Pickrell, W. S., Petrol. ReJiner 38, 159-60 (March 1959). (13J) Dostert, L. E., 135th Meeting, ACS, Boston, Mass., April 1959. (145) Farrar, G. L., Oil Gas J . 57, 116-21 (.4pril 13, 1959). (15J) Giles. B. L., 40th Natl. Meeting, Am. Inst. Chem. Engrs., Kansas City, Mo., May 1959. (16J) Goldwasser, S. R., IND.ENG.CHEM. 51, 595-6 (1959). (175) Graven, R . G., Petrol. Rejner 38, 209-16 (May 1959). (18J) Homan, C. J., Tierney, J. W., 52nd Ann. Meeting, Am. Inst. Chem. Engrs., San Francisco, Calif., December 1959.
INDUSTRIAL AND ENGINEERING CHEMISTRY
(19J) H u m , R . J., McIntire, R. L., Austin, K . L., 40th Natl. Meeting, Am. Inst. Chem. Engrs., Kansas City, Mo., May 1959. (2OJ) Johnson, R. C., Roberts, S. M., Pfretschner, C. A., 52nd Ann. Meeting, Am. Inst. Chem. Engrs., San Francisco, Calif., December 1959. (21J) Kellett, J. W., Perky, -4.S., 52nd Ann. Meeting, Am. Inst. Chem. Engrs., San Francisco, Calif., December 1959. (225) Kniebes, D. V., Wilson, G. G., 40th Natl. Meeting, Am. Inst. Chem. Engrs., Kansas City, Mo., May 1959. (235) Knutson, L. E., Automatic Control 10, 56-61 (February 1959). (245) Lamb, D. E., 52nd Ann. Meeting, Am. Inst. Chem. Engrs., San Francisco, Calif., December 1959. (25J) Lamb, D. E., Pigford, R. L., Ibid. (265) Leibnowitz, J., Frome, J., others, 135th Meeting, ACS, Boston, Mass., April 1959. (275) Leland, T. W., Mueller, W. H., IND.ENG.CHEM.51, 597-600 (1959). Rosinski, J., Engel(28J) Lieberman, .4., hart, T., 41st Katl. Meeting, Am. Inst. Chem. Engrs., St. Paul, Minn., September 1959. (29J) Lukjanow, Ariadne, 135th Meeting, ACS, Boston, Mass., April 1959. (30J) Lyster, V. N., Sullivan, S. L., Jr., others, Petrol. ReJiner 38, 221-30 (June 1959); 151-6 (July 1959). (31J) McIntire, R . ’L., Automatic Control IO, 56-61 (May 1959). (32J) Maddox, R . N., Erbar, J. H., Oil Gas J . 57, 116-19 (April 27, 1959). (335) Martch, H. B., Jr., 40th Natl. Meeting, Am. Inst. Chem. Engrs., Kansas City, Mo., May 1959. (34J) Martini, W. R., Carr, J. W., 111, Churchill, S. W., 39th Natl. Meeting, Am. Inst. Chem. Engrs., Atlantic City, N. J.. March 1959. (35J) Mauchley, 5. W.,IND.ENG.CHEM. 51, 7-8 (1959). (365) Naphtali, L. M., Chem. Eng. 66, 179-84 (Oct. 19, 1959). (37J) Nissan, A. H., Hansen, D., 52nd Ann. Meeting, Am. Inst. Chem. Engrs., San Francisco, Calif., December 1959. (38J) O’Brien, N. G., Turner, R . L., 40th Natl. Meeting, Am. Inst. Chem. Engrs., Kansas City, Mo., May 1959. (395) Organick, E. I., Hollingsworth, B. J., Petrol. Rejiner 38,172-3 (May 1959). (40J) Parisot, P. E., Chem. Eng. 66, 137-44 (Sept. 7, 1959). (415) Perry, J. W., Melton, J. L., 135th Meeting, ACS, Boston, Mass., April 1959. (42J) Radd, M. E., Tek, M. R., A.1.Ch.E. Journal5, 111-15 (1959). (435) Snow, R., Peck, R. E., Von Fredersdorff, C . G., Ibid.,5 , 305-9 (1959). (445) Stevens, W. F., 52nd .4nn. Meeting, Am. Inst. Chem. Engrs., San Francisco, Calif., December 1959. (455) Taborek, J. J., Chem. Eng. Progr. 5 5 , 45-8 (October 1959). (46J) Taube, Mortimer, Ibid., 55, 60-3 (January 1959). (47J) Tierney, J. W., 52nd Ann. Meeting, Am, Inst. Chem. Engrs., San Francisco, Calif., December 1959. (485) Vermeulen, T., Jacques, G. L., others, 39th Natl. Meeting, Am. Inst. Chem. Engrs., Atlantic City, N. J., March 1959. (49J) Wall, Eugene, Chem. Eng. Progr. 5 5 , 55-9 (January 1959). (50J) Williams, T. J., Lauher, V. A , , ’ 52nd Ann. hfeeting,’ Am. Inst. Chem. Engrs., San Francisco, Calif., December 1959.
an (515) Worley, C., Znstr. and Control Systems 32, 1022-3 (1959). (52J) Zarechnak, Michael, 135th Meeting, ACS, Boston, Mass., April 1959.
Operations Research TECHNIQUES AND APPLICATIONS (1K) Allen, W. R., Operations Research 7, 303-12 (1959). (2K) Brooks, S. H., Zbid., 7,423-36 (1959). (3K) Chacko, G. K., Zbid., 7, 313-22 (1 \ -959’1. .--,(4K) Charnes, A., Cooper, W. W., Management Sci. 6, 53-9 (1959). ( 5 ~ ) o c ~ l l eJ.y ,L., J . Znd. Eng. 10, 296-8 {17J>/.
(6K) Craft, C. J., Zbid., 10, 335-63 (1959). (7K) Dantzig, G. B., Management Sci. 6, 53-72 (1959). (8K) Fabian, T., Fisher, J. L., others, Operations Research 7, 107-22 (1959). (9K) Gluss, B., Zbid., 7, 468-77 (1959). (10K) Grassi, R. C., Gradwohl, A. J., J. Ind. Management 10, 352-3 (1959). (11K) Hansemann, F., Operations Research 7, 483-98 (1959). (12K) Llewellyn, R . N., J. Znd. Eng. 10, 176-80 (1959). (13K) Malcolm, D. G., Roseboom, J. H., Clark, C. E., Operations Research 7, 646-69 11959). (14K) Morse, P. M., Zbid., 7, 67-78 (1959). (15K) Olmer, F. J., Management Sci. 6, 111-22 (1959). (16K) Pounds, W. F., Kraii, E. P., J. 2nd. Eng. 10, 17-20 (1959). (17K) Randolph, P. H., Zbid., 10, 118-21 (1 \ -959) .-_,.
(18K) Schupak, M. B., Operations Research 7,45-57 (1959). (19K) Watson, K. M., Chem. Eng. Progr. 55, 37-44 (February 1959). (20K) Zbid., pp. 38-48 (July 1959).
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BOOKS (1L) Arrow, K. J., Hurwicz, L., Uzawa, H., “Studies in Linear and tNon-Linear Programming,” Stanford Univ., Press, Stanford, Calif., 1958. (2L) Arrow, K. J., Karlin, S., Scarf, H., “Studies in the Mathematical Theory of Inventory and Production,” Stanford Univ. Press, Stanford, Calif., 1958. (3L). Bellman, R., “Dynamic Programming,” Princeton Univ. Press, Princeton, N. J., 1957. (4L) Berge, C., “Theorie des Graphes et ses Applications,” Dunod, Paris, 1958. (5L) Chorofas, D. N., “Operations Research for Industrial Management,” Reinhold, New York, 1958. (6L) Dorfman, R., Samuelson, P. A., Solow, R . M., “Linear Programming and Economic Analysis” McGraw-Hill, New York, 1958. (7L) Gass, S. I., “Linear Programming: Methods and Applications,” McGrawHill, New York, 1958. (8L) Heady, E. O., Candltf, W., “Linear Programming Methods, Iowa State College Press, Ames, Iowa, 1958. (9L) Lindsay, F. A . , “New Techniques for Management Decision Making,” McGraw-Hill, New York, 1958. (1OL) Magee, J. F., “Production Planning and Inventory Control,” McGraw-Hill, New York, 1958. (11L) Operations Research Center, Massachusetts Institute of Technology, “Notes on Operations Research,” Technology Press, Cambridge, Mass., 1959. (12L) Operations Research Group, Case Institute of Technology, “A Comprehensive Bibliography on Operations Research,” Publications in Operations Research, No. 4,Wiley, New York, 1959.
r dChemical Engineering Fundamentals Review
3L) Riley, V., Gass, S. I., “Linear Programming and Related Techniques: A Comprehensive Bibliography on Linear, Non-Linear, and Dynamic Programming,” Johns Hopkins Press, Baltimore, Md., 1958. 4L) Saaty, T. L., “Mathematical Methods of Operations Research,” McGraw-Hill, New York, 1959. 5L) Sasieni, M., Yaspan, A., Friedman, L., “Operations Research: Methods and Problems,” Wiley, New York, 1959. 6L) Vazsonyi, A , “Scientific Programming in Business and Industry,” Wiley, New York, 1958. 7L) Wiener, N., “Nonlinear Problems in Random Theory,” Technology Press, Cambridge, Mass., and Wiley, New York, 1958.
Statistics DESIGNO F EXPERIMENTS
--
(1M) Anderson,? r-
(2i 195:
(3M) E 1, 77-95 ( (4M) Budnc NO. 10, 5-1U (5M) Zbid., No. ‘ (6M) Zbid., No. ‘ (7M) Budne, T. (1959). (8M) Carlson, J. A., Conkey, J. H., Tap@ 42, 196-202 (1959). (9M) Comstock, A. J., Jurnack, S. J., Moonev. R . W.. IND. END. CHEM. 51, 325% (1959). ’ (10M) Dechman, D. A., Van Winkle, M., Zbid., 51, 1015-18 (1959). (11M) Dotson, J. M., A.Z.Ch.E. Journal 5,169-74 (1959). (12M) Ernst, A. J., Clark, T. F., others, Tappi 42, 235-43 (1959). (13M) Fowler, R. T., Lim, S. C., Chem. Eng.Sci. 10, 163-70 (1959). (14M) Hatfield, J. D., Rindt, D. W., Slack, A. V., IND. ENG. CHEM. 51, 677-82 (1959). (15M) Hunter, J. S., Control Eng. 6, No. 9, 137-41 (1959). (16M) Johnson, A. F., Tappi 42, No. 7, 176A-8A (1959). (17M) Koehler, T. L., Chem. Eng. Progr. 55, NO. 10, 76-9 (1959). (18M) Koehler, T. L., Tappi 42, 261-4 ( 1959). (19M) Satterthwaite, F. E., Technometrics 1, 111-37 (1959). (20M) Satterthwaite, F. E., Shainin, D., Conference, Northeastern Section ACS, Wellesley, Mass., April 1959. (21M) Whitwell, J. C., Tappt 42, 467-73 (1959). (22M) Youden, W. J., IND.ENG. CHEM. 50, NO. 12, 77A-8A (1958); 51, NO. 2, 81A-2A; NO. 4, 81A-2A; NO. 6, 79A-80A; NO. 8, 65A-6A; NO, 10, 79A-80A (1959). (23M) Youden, W. J., Kempthorne, O., others, Technometrics 1, 156-94 (1959). ANALYSIS AND CORRELATION OF DATA (1N) Brooke, M., Petrol. ReJner 38, No. 7 , 139-42 (1959). (2N) Capott, R. J., Newman, S. B., Mandel, J., Tap$; 42, 480-6 (1959). (3N) Chesley, K . G., Jones, E. D., Traux, R. L., Ibid., 42, 244-5 (1959). (4N) Zbid., pp. 299-300.
(5N) Ferrari, A., Russo-Alesi, F. M., Kelly, J. M., Anal. Chem. 31, 1710-7 (1959). (6N) Greyson, M., Cheasley, J., Petrol. Refiner 38, No. 8 , 135-42 (1959). (7N) Hoerl, A. E., Chem. Eng. Progr. 55, NO. 11, 69-78 (1959). (8N) Loscalzo, A. G., Benedetti-Pichler, A. A., Anal. Chem. 30, 2018-9 (1958). (9N) Marquardt, D. W., Chem. Eng. Progr. 55, NO. 6, 65-70, (1959). (10N) Mottlau, A. Y., Petrol. Refiner 38, NO. 11, 147-8 (1959). (11N) Schechter, R. S., Wissler, E. H., Zbid., 38, No. 3, 189-91 (1959). (12N) Wissler, E. H., Schechter, R. S., Zbid., 38, NO. 4, 135-8 (1959). QUALITY CONTROL (1P) Davies, 0. L., Technometrics 1, 49-61 (1959). (2P) Kressman, J. G., Tappi 42, No. 8, 38A-62A (1959). (3P) Maltenfort, G. G., Zbid., 42, No. 4, 56A-60A (1959). (4P) Mandel, J., Lashof, T. W., A S T M Bull. NO. 239. 53-61 (1959). (5P) Roberts, S: W., Technometrics 1, 23950 (1959). (6P) Sheppard, H. R., Ginsburg, H., A S T M Bull. NO. 236, 31-4 (1959). (7P) Shewell, C. T., Anal. Chem. 31, No. 5, 21A-7A (1959).
SPECIALAPPLICATIONS (1Q) Aris, R., Amundson, N. R., Chem. Eng. Sci. 9, 250-62 (1959). (2Q) Covell, D. F., Anal. Chem. 31, 1785-90 (1959). (3Q) Giddings, J. C., J . Chem. Educ. 35, 588-91 (1959). (4Q) Lieberman, A., Znd. Quality Control 16, NO. 2, 14-18 (1959). (5Q) McElrath, G. W., Bearman, J. E., Zbid., 16, NO. 3, 10-14 (1959). (6Q) Mandel, J., A S T M Bull. No. 236, 29-30 (1959). (7Q) Milsum, J. H., Trans. Am. Soc. Mech. Engrs. 81D, 254-62 (1959). (8Q) Robbins, M. D., Chem. Eng. 66, NO. 13, 128-30 (1959). (9Q) Zemany, P. D., Pfeiffer, H. G., Liebhafsky, H. A., Anal. Chem. 31, 1776-8 (1959). BOOKS (1R) Acton, F. S., “Analysis of StraightLine Data,” Wiley, New York, 1959. (2R) Allen, D. H. W., “Statistical Quality Control,” Reinhold, New York, 1959. (3R) Bowker, A. H., Lieberman, G. J., “Engineering Statistics,” Prentice-Hall, New York, 1959. (4R) Chew, V., ed., “Experimental Designs in Industry,” Wiley, New York, 1958. (5R) Cox, D. R., “Planning of Experiments,” Wiley, New York, 1958. (6R) Duncan, A. J., “Quality Control and Industrial Statistics,” rev. ed., R. D. Irwin, Inc., 1959. (7R) Gumbel, E. J., “Statistics of Extremes,” Columbia Univ. Press, New York, 1958. (8R) Kitagawa, T., Mitome, M., “Tables for the Design of Factorial Experiments,” rev. ed., Dover, New York, 1958. (9R) Lindgren, B. W., McElrath, G . W., “Introduction to Probability and Statistics,” MacMillan, New York, 1959. (10R) Schefft, H., “The Analysis of Variance,” Wiley. New York, 1959. (11R) Williams, E. J., “Regression Analysis,” Wiley, New York, 1959. VOL. 52, NO. 4
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