I Computers, Mathematics, I Statistics, and Automation

The use of the. mathernatical tools is so ividrspread that the task 01 .... White, Byron, Instruments B Automation 29,2212-16 (1956). C. Control Syste...
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CHEMICAL ENGINEERING REVIEWS

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FUNDAMENTALS REVIEW

I Computers, Mathematics, I Statistics, and Automation I

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main impression this !.car i s that of integration and irirergro\vth of the several techniques covered fly this review. This has taken placr among the techniques themselves, and bctiveen them and the various phases of applird chemistrl- and chemical engineering. The use of the. mathernatical tools is s o ividrspread that the task 01preparing this rcvien. is approachinq that 01 follo\ving the devt'lopmcnt o f t h e ivhoie of che rn is t r y and chemical engi nt'r ri ng . T h e area of g i ~ a t e s tgrowth has b c r n that 0 1 data processinq b y automatic devices. There has also been considerable application of operations I and linear progi.amming to t h e planninq of production opcra[ioris. .ipparclitI! these applications are more tLasily lated to dollar savinTs o r incoint~ increase than the use of automatic a n d ~natht'matical devices fur solviny tlic problems or physical science. S v v c r theless, the full potentialities oi' aiitoniation as developed froin i ~ i s t r ~ i ~ r i e n t a t i o i i and process control art' probably bciriq delayed ti! lack of adequate u1itic.rstanding of the kinetics inw1vc.d. a n d this in turn by the :reat complcsii!- ( i f these phenomena. tht, inadec~iiiic~~. of' the inanpo\\ei a\ailablt.? and thr fact that long-term eff'ort a n d larqc invcsrmcnt are required for solutions. I n addition to the use of aucc~inaric devices. iisuall~ rlectronic c i i n i ~ ~ ~ i t i ~ r s . the scope o f this rcvitsiv also inclitd(,s activitics that rcquirc onl! i x paper and an acrivr niind, o r h e lattci.. I n t h i s area thc,rt, is increasing cvidrncts of a n ttpsurqc i r i rhc. use of stochastic a n d statistical nit3tliods in place of inotx~convrntional iiia!hrindtical techiiiqurs [or rxpi.t,ssinq piwiscl! t h r iriterrclarions of [tic ~ ' ~ C L C Ja~nSd variab1t.s of a process or prot~lrm. 1 x 2 -

Automation

The major de\,ciopiiieiit in the itcld of process control during the past year has been a n aivakeriiiig i n the chcriiical industries to the possible value of servornec hanisni the or!- a rid coin1ju t c I' s irn u Iation in solving chemical iiidustr). control problems. 1lan)- cherniral and petroleum coinpaiiies havt- o q a n i z e d ne\v groups to undertake \ v o k in this field: or the). have expanded their regular instrument drpai'rmcnts t o ac complish the same purpose. 'I his dr-

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ARTHUR ROSE i s a professor of chemical engineering at Penn State and president of Applied Science Laboratories, Inc., State College, Pa. His work in distillation calculations led him to an early interest in the use of computers as a w a y of speeding up calculations.

R. CURTIS JOHNSON is actively utilizing applied mathematics and automatic digital computers for solving technical problems for the chemical industry. Johnson received his doctorate a t Penn State and i s associate professor of chemical engineering, Washington University, St. Lcuis, Mo.

RICHARD L. HEINY, a graduate of Penn State, works in design and development for Dow Chemical Co., Midland, Mich. He i s now devising practical methods for wider use of statistical techniques in planning and interpreting process development experiments. THEODORE J. WILLIAMS (Ph.D. in Ch.E. from Penn State and M.S. in E.E. from Ohio State) i s now with Monsanto in a newly organized section concerned with special research in automatic control of chemical processes. He is the author of technical papers on process control and automatic computers. J O A N A. SCHILK has a masters degree in chemical engineering from Penn State, where she acquired basic training and experience in the use of automatic digital computers. She is now working a t Shell Development Co., Emeryville, Calif., on engineering assignments.

INDUSTRIAL AND ENGINEERING CHEMISTRY

Table I. Theoretical Articles of Interest for Process Control Applications A. Chemical Process Dynamics 1. Distillation columns Aikman, A. R., Instrumenf Soc. Amer. J . 3, 412-6 (1956). Berger, D. E., Short, C. R . , I N D .ENG.CHEM.48,1027-30 (1956). Hengst, K., Meier, A,, Regelungstech. 3,219-25, 243-8 (1955). Kundt, W., Ibid., pp. 194-7. Williams, T. J., Harnett, R. T., Rose, Arthur, IND.ENC.CHEM. 48,1008-19 (1956). Fluid flow Bigliano, R . P., Instrument Soc. Amer. J . 2, 536-9 (1955). Buescher, A. E., Fiedler, G. J., Ibid., pp. 352-60. Catheron, A. R., Hainsworth, B. D., IND. ENC. CHEM.48, 1042-6 (1956). Dungs, H . H., Regelungstech. 3, 254-6 (1955). McGrath, T. F., Fiedler, G. J., Instrument Soc. A m r . J . 2, 234-40 (1955). 3. General Hengst, K., Chem.-Ing.-Tech. 28, 317-8 (1956). Holzmann, E. G., Trans. Am. Soc. Mech. Engrs. 78,251-8 (1956). Kwauk, Mooson, A . I . Ch. E . Journal 2,240-8 (1956). Mandron, John, others, Control Eng. 3,105-21 (September 1956). Zoss, L. M., Ibid., pp. 3,50-5 (January 1956). 4. Heat exchangers Cohen, W. C., Johnson, E. F., I N D .ENC. CHEM.48, 1031-4 (1956). Moz,ley, J. M., Ibid., pp. 1035-41. Paynter, H . M., Takahashi, Yasundo, Trans. Am. Soc. Mech. Engrs. 78, 749-58 (1956). 5. Measuring systems Linahan, T. C., Trans. Am. Soc. Mech. Engrs. 78, 759-63 (1956). Muller-Girard, Otto, Ibid., 77, 591-5 (1955). Williams, T. J., Ibid., pp. 1461-9. 6. Reactors Rilous, Olegh, Amundson, N. R., A . I. Ch. E. Journal 1. 513-21 (1955). Ibid., 2, 117-26 (1956). Sartorius, H., Matuschka, H., Regelungstech. 4, 165-71 (1956). B. Control System Adjustment a n d Optimization 1. Controller setting Coon, G. A., Control Eng. 3,66-76 (May 1956). Ibid., pp. 71-6 (June 1956). 2. optimization Gibson, J. E., Control En,c. 3, 122-9 (September 1956). Herschel, R., Regelungstesh. 4,190-5 (1456). Ibid., pp. 229-33. Lees, Sidney, Trans. Am. SOL.Mech. Engrs. 78, 1339-66 (1956). Reeber, R., Regelungstech. 4, 13-18 (1956). White, Byron, Instruments B Automation 29,2212-16 (1956). C. Control System Dynamics (Nonprocess) 1. Cascade control Franks, R . G., and Worley, C. W., IND.END.CHEM.48, 1074-9 (1956). Gollin, N. W., Control Eng. 3 , 9 4 4 (July 1956). 2. General Theory Davidson, G. M., Nashman, Luther, Conttol Eng. 3, 78-84 (January 1956). Fiedler, G. A,, Automatic Control 4,13-9 (January 1956). Griinwald, E., Regelungstech. 3,142-7, 172-80 (1955). Holzbock, W. G., Control Eng. 3,83-4 (August 1956). Janssen, J. M. L., Aikman, A. R., Ibid., 2, 58-65 (November 1955); Janssen, J. M. L., Zbid., 2,54-9 (December 1955). Oldenburger, R., Rcgelungstech. 4, 213-23 (1956). Paul, S., Ibid., 3,296-302 (1956). Platt, George, Conlrol Eng. 3,63-5 (May 1956). Reswick, J. B., Trans. Am. Soc. Mech. Engrs. 78, 153-62 (1956). Sanders, C. W., Instruments d Automation 28, 1918-19 (1955). Schwent, G. V., McGregor, W. K., Russell, D. W., Instrument Soc. Am. J . 3,218-23; 274-80; 323-9; 426-9 (1956). 3. Hydraulic systems Geyer, L. H., Instruments d Automation 29, 1540-2 (1956). Magnus, K., Rcgciungstech. 3, 276-81, 292-6 (1955). 4. pH sensing systems Dungs, H. H., Chem.-Ing.-Tech. 28, 656-60 (1956). 5. Pneumatic systems Buckley, P. S., Instruments B Automation 28, 1920-2 (1955). Krug, K., Rcgelungstech. 4, 58-61 (1956). 6. TemDerature controllers Ldrenz, I., Regclungstech. 4,66-8 (1956). Ibtd., pp. 79-83. Zoss, L. M., Gollin, N. W., Edelman, R. I., I N D .ENCI. CHEM.48, 1069-73 (1956). D. Data Analysis Methods 1. Analog to digital conversion Flitcher, T. C., Walker, N. C., Instrument Soc. Amer. J . 2, 341-5 (1955). 2.

2. 3. 4.

5.

Deviation ratios Janssen, J. M . L., Regeiungstech. 3, 303-9 (1956). Error coefficients Smith, P. E., Jr., Control Eng. 2,69-74 (November 1935). Frequency response from transient response Goodman, T. P., Reswick, J. B., Trans. A m . Soc. Mech. Engrs. 78, 259-71 (1956). Reynolds, J. B., Jr., Control Eng. 2, 60-3 (October 1955). Teasdale, A. K., Jr., Ibid., 2, 56-9 (October 1955). Use of transient response Laspe, C. G., Instrument Soc. Amer. J . 3, 134-8 (1956).

E. Miscellaneous Theory Articles 1. Amplitude-phase from transfer function Tagajewskaja, A. A., Regelungstech. 4, 68-72 (1 956). 2. Damping Bohm, H., Regelungstech. 4, 83-4 (1956). Schunck, T. E., Ibid., pp. 198-9. 3. Linear programming Beer, Stafford, Process Control t Y Automation 3, 270-4, (1956). 4. Nyquist criteria Lehnigk, S., Regelungstech. 4, 195-7 (1956). 5. Parallel networks Stijssinger, H., Regelungstech. 4,113-7, 139-43 (1956). 6. Products of control values Sauter, E., Regelungstech. 4, 171-2 (1956). 7. Roots of basic equations Goller, E., Regelungstech. 3, 268-72 (1955). 8. Stability, dynamic Engel, F. V. A,, Regelungstech. 4,184-90 (1956).

9.

316-20‘

Tolerances Bock, H., Regelungstech. 3,272-5 (1955).

F. Non-linear Systems 1. Backlash Marzendorfer, H., Regehngstech. 4, 10-2 (1956). 2. General theory Goodman, T. P., Paper 56-1RD-17, Am. SOC.Mech. Engrs., Princeton, N. J., March 26-27, 1956; Instruments 3 Automation 29, 1543 (1956). Kalman, R. E., Paper 56-1RD-16, Am. SOC.Mech. Engrs., Princeton, N. J., March 26-27, 1956; Instruments d Automation 29, 1543 (1956). Oldenburger, R., Papor 56-IRD-13, Am. SOC.Mech. Engrs., Princeton, N. J., March 26-27, 1956; Instruments d Automation 29, 1543 (1956). Stout, T. M., Kochenburger, R . J., Control Eng. 3, 55-65 (February); 82-91 (March); 77-85 (April 1956). 3. Optimization Silva, L. M., Trans. Am. SOC. Mech. Engrs. 77, 1317-23 (1955). 4. Sampled data systems Anke, K., Regelungstech. 4, 143-50 (1956). Bertram, J. E., Franklin, G., Control Eng. 2, 107-11 (September 1955).. 5. Use of statistics Mayhood, R. F., Instrument Soc. Amer. J . 3, 76-83 (1956). G. Response Testing Methods 1. Frequency response Hougen, J. O., Lees, Sidnev, I N D . ENO. CHEM.48, 1053-63 (1956). ’ Schafer, O., Fussel, W., Regelungstech. 3, 225-9 (1955). Striile, D., Ibid., pp. 197-200. 2. General methods Bigliano, R. P., Control Eng. 3,72-77 (August 1956). Borck, A., Automatic Control 5, 22-5 (October 1956). Fuchs, A. M., Control. Eng. 3, 174-82 (September 1956). James, E. W., Ibid., pp. 169-73. Kaufman, A. B., Instruments d Automation 28, 2104-8 (1955). 3. Pulse testing Lees, Sidney, Hougen, J. O., IND. ENG. CIIEM. 48, 1064-8 (1956).

H. Transients 1.

2.

3.

Distillation Jackson, R. F., Pigford, R . L., I N D . ENC. CHEM.48, 1020-6 (1956). Rose, Arthur, Johnson, C. L., William?, T. J., Ibtd., pp. 1173-9. Fluid flow Alstad, C. D. and others, A . I . Ch. E. Journal 1 , 417-25 (1955). Alves, G . E., Ibtd., 2,143-7 (1956). Davis, W. R., Automalrc Control 4, 24-6 (January 1956). Wissler, E. H., Isbin, H. S., Arnundson, N. R., A . I. Ch. E. Journal 2, 157-62 (1956). Propellant ignition Cook, M. A., Olson, F. A., A . I. Ch. E . Journal 1,391-400 (1955)

VOL. 49, NO, 3, PART II

MARCH 1957

555

Information for optimizing process operations b y computations which previously took engineers a d a y or more to complete, will be provided in minutes b y a built-in general purpose analog computer and log sheet printer being built into the new model IV 24,000 bpd fluid catalytic cracker at Esso Standard Oil Co's Beloit Refinery near Havana.

Table II. A.

Use of Analogs to Simulate Process Dynamics and Control

'

Distillation Columns Jackson, R . F., Pigford, R. L., I U D .E s c . CHEM.48, 1020-6 (lL156). Rose, Arthur, Johnson, C. L.., \Villiams, T. J., ILzd., pp. I 1 73-9. LVilIiams, T. J., Harnett, R . T.. Rose, Arthur, IBD. EYC..( : r i k \ i , 48, 1008-19 (1056).

B . Computers i n Control Loops Hussey, J . L., Instruments B Automation 29, 2000-4 (1956). James, E. \V, and others, Conlroi Eng, 3, 83-6 (November l f l i h )

D.

j

Fluid Flow or Pneumatic System7 Kayan, C. F., Trans. Am. Soc. .Ilrch. Bigrs. i 7 , 543-8 (lOi5). Shearer, J. L., Paper 56-1RD-1, Am. Soc. Mech. Engrs., I'rinreron, N. J., March 16-37, 1 9 5 0 ; Instrrrmcnls B .Ititomnti'on 29, 1545 (1956). \Vissler, E. H., Isbin, H. S.,.\niiitidx)n, S . I[;. C H E \ i . 48, 1177 (l(l.56). Rose, Arthur, iVilliam~,r , J.. Ibid., 4 i , 2284 (195.5). Voqel, I>. C., Krueqer, R. F.,

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National Laboratory, ORNI.-l572. Dame\vood, G., Oil Gas J . 54, Electrical No. 65, 98 (1956). network

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\\'iasler, E. H., Isbin, H. S., KEAC: Amunclcon, N. R . , .i. I . Ch. E . J o u r n d 2 , 157 (1356).

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Digital Computers: ACE: Automatic Computinq Engine, pilot model, National Pli?sics Laborator), Teddington. Enqlanrl C:KC: I ill:\. -sis (Siedcr-Tare equation). The procedure is sct u p for easy computation Ivith a desk calculator. A fairly large number of articles have appeared, in lvhich some statistical analysis of data \vas presented? but in which the analysis w a s stricti>- srcondary

MATHEMATICS AND STATISTICS to the subject of the paper. Some of these articles are listed in Table VIII. Other Applications a n d Developments. Lieblein and Zelen (35C) have done an extensive investigation of the fatigue life of deep-groove ball bearings. They describe the fatigue life with Weibull’s distribution and treat the data with order statistics to take account of the fact that the failures are caused by the smallest extreme-values in the fatigue life distribution. Their discussion is very complete, but technical. One investigator (37C) uses a statistically defined mixing coefficient to describe the excellence of mixing in mixing chemicals into soils. This should be useful in other applications, too. Sprowls (38C) built a n automatic coin flipper for sampling demonstrations which could also be used to determine order of experiments, time of sampling, etc. A discussion of cost estimation (34C) includes good qualitative relationship between standard deviation of estimate and the cost of the estimate. The data necessary to obtain a quantitative estimate are not normally available, but if the estimater realizes that the relationship is there, he can make qualitative estimate of the relative reliabilities of his own estimates. Another author (362) describes the use of ratio delay in random sampling to determine how effective a work force is. Books a n d General References. Probably the best new book on statistics for the industrial experimenter, by Bowker and Lieberman (40C), uses mathematical notation extensively and may well frighten the reader at first. However, the concepts are simple and clearly put, once the reader has accustomed himself to the symbols. I t has the usual sections on basic concepts, quality control, and analysis of data. Unusual emphasis is placed on sampling inspection, and great pains are taken to distinguish between the use of u’ (the

population standard deviation) and of u (estimate of the population standard deviation). Tests and procedures are very clearly laid out. Snedecor (49C) hag published a new edition of “Statistical Methods” with more emphasis on sampling and on the treatment of small samples. Kendall’s book (45C) on “Rank Correlation Methods” has a very good introduction to the subject, and generally presents the information clearly with detailed mathematical proofs included separately from the text. . A new basic text has appeared (39C) that is well suited as a general reference on statistical methods. A humorous expos6 of fallacies which may be, and often are, perpetrated in the name of statistics has been published (43C). Two books of tables of random numbers and correlated random deviates (42C, 48C) may be useful in design. Two journals of statistical abstracts (&C, 47C) include references to the use of statistical methods in all types of industry, but are of particular interest to the chemical industry. One of these (44C) includes many abstracts of foreign articles (but no Russian). Both place considerable emphasis on statistical techniques and operations research for use by management. Two articles attempt to summarize statistical methods for application to chemical engineering. Daniel’s approach (47C)is more qualitative, but is good background information, particularly on design. Volk (50C) treats the subject much more quantitatively, including many brief examples, tables, and good directions for use. Besides the usual basics, he includes chi-square test, rank correlation, and sign correlation, and gives proper emphasis to the analysis of variance. Mandel and Linnig (467)have published a review entitled “Statistical Methods in Chemistry,” covering the years 1952 through 1955. Symposia a n d Meetings. A number

Table IX. Institution University of California (Los Angeles) Cornell University Georgia Institute of Technology Harvard University University of Illinois State University of Iowa Marquette University University of Michigan University of Michigan New York University North Carolina State College University of Oklahoma Purdue Univenity Rutgers University Queen’s University

of schools have offered intensive short courses in statistical methods particularly designed for the industrial experimenter; some of these are listed in Table IX. Many of these are becoming annual affairs. Information concerning future courses can be obtained from the sources described in the references. Several symposia on statistical methods have also been announced (57C, 52C). The Gordon Research Conferences again included one week on “Statistics in Chemistry and Chemical Engineering.” Related Fields a n d Nonexperimental Applications of Statistics. The usual application of statistical methods in chemical engineering has been in the design of experiments and analysis of data. However, there seems to be an increase in the use of statistical methods to analyze and describe physical or chemical systems, and several interesting examples have appeared. Klinkenberg (62C)describes statistically five types of flow systems, ranging from flow through intermittent stages to uniform flow with a finite rate of mass transfer, and compares the holding-time distributions. This paper should be valuable to those designing reactors for reactions in which mixing and holding time are important. Noyes (67C)combines probability functions describing molecular reactivity with a ntatistical description of diffusion in liquids to derive expressions for reaction rate constants. The results are interesting, but much more must be known about liquid behavior before they become generally useful. So0 (69C)examines the properties of particles suspended in a gas. H e uses Lagrangian distribution functions to describe the variations in velocity of both the gas and the particles. A statistical model of dry blending has been set up by Adams (543 and expected variances for such a model have been calculated. He assumes that blending is described by Poisson’s distribution.

Special Short Courses on Statistical Methods

Principal Topic Statistical methods in industry Applied industrial statistics Quality control by statistical methods Box method of experimental design Quality control by statistical methods Quality control by statistical methods Quality control by statistical methods Quality control by statistical methods Box method of experimental design Sixth statistical quality control seminar Statistical methods for research workers General statistics Advanced quality control Advanced quality control by statistical methods Intensive training in statistical methods

Length, Days 10 4

9 3 10 10 10 10

3

5 7 10 10

7 10

Reference Ind. Quality Control 12, No. 11, 92 (1956). Ibid., No. 10, 18 (1956). Ibid.,No. 7, 30 (1956). Chem. Eng. News 34, 1570 (1956). Ind. Quality Control 12, N o . 7 , 30 (1956). Ibid., 13, No. 2, 24 (1956).

Ibid., 12, No. 10, 18 (1956). Ibid., No. 11, 92 (1956). Ibid., 13, No. 3, 27 (1956). Ibid., 12, No. 10, 18 (1956). Cham. Eng. News 34, 5042 (1955). Ind. Quality Control 12, No. 8,48 (1956). Ibid., No. 9, 28 (1956). Ibid., No. 11, 92 (1956).

Tappi 39, No. 9, 138A (1956).

VOL. 49, NO. 3, PART II

MARCH 1957

561

FUNDAMENTALS REVIEW Some rechniques not particularly related to statistical methods, bur \v hich have a semimathematical basis and can often be used hand in hand ivith sratistical methods, are becoming popular i n the chemical industr!.. No extensive reiview of these techniques is intended here. but a fr\v refrrences on rach t r c h nique are included ivhere the emphasis is on applications of interest to chemical engineers. Operations Research. Operations research is defined i n \,arioiis rvays but is usually the methodical rnuineration of alternative choices folloived by some scheme for recognizing and selecting the best (or probably best) one. T h e choices can be solutions to design problems. different orqanizarional decisions. combinations of inventory quantities. and times. etc. The .American Institute of Chemical Engineers included an Operations Research S ~ - ~ n p o s i u(77c') m rvith ten papers describing operations research and examples of applications [see especially 70B-12B,J5B. J7B] in its September meeting a t Pittsburgh. .\ number of papers hac.? describrd the value of operations research and ho\v t o institute operations research programs ( X C , 58C- 67C.oK'. 70C'). T\vo of these (GIG', 70C) include some examples of application. 1lcC:loslie~~ a n d 'Trefethein ( t i iC) have published a n excellent book on "Operations Research for AZana yeinen t .' ' Linear Programming. Linear programming is one tool of operations research. and i n \ - o l v r s the problem of opriinizing a linear iinction of a sei of' variables. 'Three articles ( i.iC. 5!/C'. &%') describe linear programming. mention possible applications. and present simple examples of use: a ib~trrli(71C.') presents a more coiiiplicatcd and vei'!. instructiw exaniplc in the s o l u t i o n of gasoline refinin? and blcndinq problenii. Game Theory. Kropa (Ii3C) dcscribes tlie application of game theor!. rules to some of he pr~~blcnis associatrd \virh process dc\-clopnicnt. Ijarticularl! those reqiiiriiiq tlir rise of sulijectivc~ j Lid p e n I . Decision Making. llood ( h i ) ( .) prysents a n o\.ersiniplif?cd picturc of the application of mathernatical tools t o the, problems 0 1 decision iliaking. Ho\vc i t is a good introduction io the existence of the field.

Mathematics There is no doubt that tlie chemical engineering field, as far as its reseai.ch is concerned! is liecominq more fundamental. The ph!.sical chemists and ph!-sicists have ahandoned their tradit i o n a i fields of tIi~~rriiod~-na~riics~ kinetics, mechanics. and fluid dynamics 10 pursue

562

more exoric intellecrual concepts such as photoluminescence and ultimate par-

ticles. The chemical erigineer has no choice but to determine his ou'n thermodynamic properties and rates of' reaction. and to rtxfine the basic relationships \vliich. reduced to empirical "hand bookrry." form the backbone of u ni t operations. I n studying the fundamentals thr engineer has found many ne\v and stirnulating pursuits, not the least of ivhich is advanced mathematics. I n contrast to the papers of just a feiv years ago: o u r journals no\v are liberally clutterrd \\.ith Bessel functions. Inatricrs. transforms. vectors. and the like. References to the uses of analog and digital coniputers. and statistical techniques are commonplace. T h e rraiisition to the ne\v mathematical domain is rapid. and rdther painful to some older engincws and educators. I n previous rc-vie\vs it [vas notcd that the greatest nurnbcr of rnatliematicall~~ interesting papers \\ere dcvoted t o boundary value problrins. 'The trend continues. Of' the .34 papers selectcd for rel'erence here, all but about six are in this class. 'This is not surprising. Directly or indirectly no\\! otic of the most important practical probie~iis i n chemical rnqineeriny is that of reactor design. Designers \ \ - o d d like to tx, able to abandon the '.tubrs-in-paralIel" method of reactor scale-up. Continuously regenerated liiiid c a t a l ~s t beds are becoming niorr popular. Scaleu p from laborator!, to pilot plant ( a step \ve hope e\-entuall~-t o a\-oid Ii! iise of eoinl)utcrs) to priidiiction ~ ) l a n t in cht~nical rcactors is no siniplr dimensional analysis p ~ ~ j l ~ l c i;I n . coinplrtr understandine o l [lie kinetics. thet~niod~-na~nics. Huid dynainics. ditfltsion . and 11 ea I tra nslc i. [ ' ' mcjmrn i u ~ r i . c'nrrq!-. and Inass trans]""~t"i cou]'leci Lvitli system geonicrr). nnd ci)nrrril loops is tlie ultimatc nrcd. '1.0 this end vatious rrsearc1ii.r~ a r c diliqrntl!- carryinq oil t \\.ark in all these firlds. i v i t h considcrable emphasis o n intc~.rrlated inathematical relaiicinsliips. Each additional coinplication ma!. recliiire rnailicriia~ic~i1 techniclucs n c i v t c this field. 'I'lic standaid boundary value prol)lrii1 is a sinqlc diiFcrc~nrial eqiiation. or 21 set ol equations. \vhicli ma! tic ordinar). or partial\ subject to initial conditions or boundary values. \ V h r n ordinary equations are involved: a solution ma!. occasional1~- be found i n closrd form. IVith linear ordinar!- equations tht, solution morc frequentlj. cannot he solved in closed forin! but is solvable i n infinite series. I n t h i s \\.a!. such functions as Bessel a n d Legendre arise. l ' h r largest number of houndar!, value priiblt~rnsare dcscritxd I,! partial diflercntial equations stcad\ o i ' (in-

INDUSTRIAL AND ENGINEERING CHEMISTRY

MATHEMATICS AND STATISTICS

of Bessel transforms has not been previously noted. Other papers dealing with heat transfer which involve Bessel functions and Laplace transforms are by Ting (290) and by Sellars, Tribus, and Klein (230). An interesting paper on water hammer by Skalak (250) makes use of double Laplace transforms. Fourier transforms were employed by So0 (270) in a n evaluation of statistical properties of momentum transfer in two-phase flow. An interesting boundary value problem in radiant heating of dispersed particles is presented by Sleicher and Churchill (260). The Laplace transforms obtained are complicated, and use is made of approximate transforms to simplify the expression so that inversion is possible. A good discussion of simplification of the transforms is included. The use of Laplace transforms on ordinary differential equations converts them to algebraic equations, whereas the transformation of a partial differential equation usually leads to a n ordinary differential equation. Keller and Ballard (770) worked with graphical solutions of the unsteadystate heat transfer equation with a generation term in predicting temperature changes in frozen liquids. T h e equation derivation and the graphical methods are carefully shown. There is also a discussion of the calculation of thermal conductivity, specific heat, and density from transient measurements, which has not been used to advantage in chemical engineering work. Another mathematical treatise from Minnesota appeared in the work of Bilous and Amundson ( 4 0 - 6 0 ) as a result of their studies in chemical reactor stability and sensitivity, and optimum temperature gradients in tubular reactors. Transforms, matrices, and transforms of matrices are used. Most of the equations are nonlinear, and a n analog computer was used in solution. I n one paper ( 6 0 ) considerable use is made of the calculus of variations, which has not been noted in previous reviews. As another approach to the heat and mass transfer problems, Acrivos ( 7 0 ) described the method of characteristics. This appears to be essentially a systematic method of treating partial differential boundary value problems to take greatest advantage of properties of partial derivatives along contours, and then devising a numerical procedure on this. Not as much appeared in vectors and tensors as has been reported in the past. Only two papers are mentioned here. Pigford, Troose, and others (270) discuss neutron lifetimes and void coefficients for research reactors. An extensive appendix shows the use of

vector terminology in this field. Lyche and Bird (200) work from the usual vector and tensor equations in a study of the Graetz-Nusselt problem for a power law non-Newtonian fluid. T h e solution yields a typical series, and new eigenvalues had to be determined. The remaining papers fit i n t o , the “miscellaneous” group. The paper on activity coefficients a t infinite dilution by Gautreaux and Coates (730) appears to be of mathematical as well as of thermodynamic interest. The handling of partial derivatives is interesting and out of the ordinary. A discussion of graphical solutions on dimensionlcssnumber plots by Fitch (700) is informative. T h e paper gives some insight into the nature of graphs, and the use of vector addition and of cqmpensating change vectors is interesting. Two papers in “mathematical economics” were considered of interest. Cichelli and Brinn ( 8 0 ) make use of Lagrangian multipliers in optimum heat exchanger design. Typical finite difference equations of economics are well illustrated in the cost analysis paper by Jelen (760). Another paper in finite differences, by Knudsen (780),deals with batch extraction. This is of interest because some finite difference equations are actually solved, and the results are reported graphically.

Acknowledgment This year for the first time a systematic effort was made to obtain suggestions and material for this review from experts in the fields involved. In this connection, grateful acknowledgment is made to R. B. Bird, John R. Bowman, N. H. Ceaglske, H. E. Hoelscher, Leon Lapidus, W. R. Marshall, Jr., J. M. Mozley, and Ascher Opler.

BIBLIOGRAPHY Automation

Carr, J. W., Control Eng. 3, No. 1, 63 (1956). Carr, J. W., Perlis, A. J . , Ibid., 3, No. 2,84 (1956). Ibid.,No. 3, 99 11956). Carroll, J. M . , ‘Electronics 29, No. 6, 122 (1956). Chm. Eng. News 34, 3542 (1956). De Carlo, C. R., Chem. Eng. Progr. 51.487 (1955). Donovan, R . G., Schrage, R. W., Oil G Q ~ J54,131 . (May 28, 1956). Fa man, D. L., Ibid., 54, No. 56, 140 r1956). Freilich, A. H., Regional Meeting, Western Petroleum Refiners’ Assoc., Wichita, Kan.. June 1955 (Tech. 55-18). Hashbarger, H. A., Sample, N. L., Preprint 1, Symposium on Scientific Aids to Management, Pittsburgh Meeting, Am. Inst. Chem. Engrs., September 1956. Klingel, A. R., Jr., Corvett, C. L., Ibzd.,Preprint 4. Marshall, B. O., Jr., Ibtd., Preprint 3. Marshall, W. R., Jr., private communication; Adler, C. R., others, Chem. Eng. Progr. 50, 14 (1956). ~, Organick, E. L., Oil GQS J . 54, No. 50,162 (1956). Rio, Joseph, Sharkey, .4. F., Preprint 2, Symposium on Scientific Aids to Management, Pittsburgh Meeting, Am. Inst. Chem. Engrs., September 1956. Rubinoff, M., Beter, R. H., Control En,g. 3, No. 11, 115 (1956). Schrage, R. W., Preprint 5, Symposium on Scientific Aids to Management. Pittsburgh Meeting, Am. Inst. Chem. Engrs., September 19 56. Symonds, G. H., “Linear Programming, the Solution of Refinery ProSlems,” Esso Standard Oil, New York, 1955. Wass, C. A. A , , “Introduction to Electronic Analogue Computers,” McGraw-Hill, New York, 1955. Wilkes, M. V., “Automatic Digital Computers,” Wiley, New York, 1956. I~

Statistical Methods-Design

(1A) Brown, R. B., Beter, R. H., Control Eng. 3,70-6 (Dec. 1956). (2A) Ceaglske, N. H., “Automatic Process Control for Chemical Engineers,” Wiley, New York, 1956. (3A) Cohen, W. C., Johnson, E. F., IND. ENG. CHEM. 48, 1031-4 11956). \ - - - - I .

Automatic Computers

(4A) Duncan, R. A., Automation 3, 67-71 (February 1956). (5A) Lowenstein, Milton, Control Eng. 3, 71-7 (October 1956). (6A) Moore, B. C., Ibid., pp. 85-90 (August 1956). (7A) Mozley, J. M., IND. END. CHEM. 48, 1035-41 (1956). (8A) Peterson, K. W., Automation 3, 52-7(June1956). (9A) Reed, W. E., Control Eng. 3, 92-9 (October 1956). (10A) Tivy, V. V., Instrument SOG.Amer. J . 3, 40-5 (1956). (11A) Wing, Paul, Jr., Zbid., 2, 332-7 (1955).

(1C) Barnett, M. K., Hamilton, P. M., Mead, F. C., Jr., A. I. Ch. E . Journal 2,235-40 (1956). (2C) Daniel, Cuthbert, “Proceedings of Third Berkeley Symposium on Mathematical Statistics and Probability,” vol. V,87-98, University of Calif. Press, Berkeley, 1956. (3C) Hembree, E. E., Tappi 39, 91-3 (1 956). (4C) Lashof, T. W., Mandel, J., Worthington, V., Ibid., pp. 532-43. (5C) Matson, V., Ibid.,pp. 77-83. (6C) Robinson, J. B. D., Analyst 81, 316-19 (1956). (7C) Simpson, J. S., Lees, K. A., Ibid., DD. 562-5. (8C) Ydiden, W. J., IND. ENG. CHEM. 47, NO. 12, 89A-90A (1956); 48, NO. 2, 98A-100A; NO. 4, 107A-9A; NO. 6, 104A-5A; NO. 8, 59A-60A; NO. 10, 6lA2A (1956).

VOL. 49, NO. 3, PART II

MARCH 1957

563

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FUNDAMENTALS REVIEW Statistical Methods-Quality Control and Sampling Techniques .4m. SOC. for Quality Control, “National Convention Transactions-1 956,” Baltimore, Md., 1956. Beall, G.: Tabpi 39, 26-9 (1956). Beaudry, J. P., Can. Chem. Processing 40, No. 7, 80 (1956). Bowker, .4. H.: “Proceedings of Third Berkeley Symposium on Mathematical Statistics and Probability,” vol. V, 75-85, Lniversity of Calif. Press, Berkeley, 1956. Chem. Eng. 63, No. 7, 264-8 (1956). Debing, L. M., Harrington, E. C.. Jr., IND.ENG.CHEM.48, No. 5. 36.4-9A (1956). Hartwell, F. C., Purcell, \V. R.; Tappi 39,238-42 (1956). Jackson, J. E., Ind. Quality Control 12, NO. 7,4-8 (1956). Leone, F. C., Tappi 39, hTo. 1. 1414-3.4 I1 956 i. (18C) Newchurch, ‘E. J.,‘.4nderson, J. S., Spencer, E. H., Anal. Chem. 28, 154-7 (1956). (19C) Roberts, F. J., ISD.EXG.CHLM.48, NO. 4, 27A-30A (1956). (20‘2) Walter, J. T.. Industrial Labs. 7. 6-8 (June 19j6). (21C) Walter, J. T., Petroleum Rrjiner 35, No. 2,106-8 (1956). (22C) Ward, R. V., Can. Chem. Processing 40, NO.2,29-32 (1956).

Statistical MethodsAnalysis of Data (23C) Davis, D. S., Chem. Eng. 63, No. 4, 206 (1956). Deitz, V. R.’, J . Research Natl. BUY. Standards 57, 159-70 (1956). Ergun, S., IND. ENG. CHEM.48, 2063-8 (1956). Hicks, C. R., Ind. Quality Control 13, No. 2, 17-20 (1956). Zhid., No. 3, 5-8 (1956). I h i d . , N o . 4 , 13-16 (1956). Levenspiel, O., \Veinstein, N. J., Li, J. C. R., IND. ESG. CHEU. 48, 324-6 (1956). Nelson, L. S., J . Chem. Educ. 33, 126-31 (1956). Vance, F. P., Tingey, F. H., Chem. Eng. Progr. 52, 375-80 (1956). Wooding, M’. M., Tappz 39, 417-25 (1956 ). (33C) Yang, d.,Ibid., 39, No. 8 , 146.4-8.i (1956).

Statistical Methods-Other Applications and Developments (34C) Hicks, J. S., Steffans, L. R., Chem. Eng. Progr. 52, 191-4 (1956). (35C) Lieblein, J., Zelen, M., J . Research S a t l . Bur. Standards 57, 273-316 119561 j._,. (36C) Smith, A . N., Petroleum ReJner 35, NO. 2, 135-7 (1956). ( 3 7 C ) Smith, J. C., IND.ENG.CHEM.47, 2240-4 (1955). (38C) Sprowls, R. C., A m . Statistician 10, YO. 2, 12-13 (1956).

Statistical Methods-Books and General References (3%) Adams, J. K., “Basic Statistical Concepts,’’ McGraw-Hill, New York, 1955.

564

(40C) Bowker, A. H., Lieberman, G. J.. “Handbook of Industrial Statistics,’’ Prentice Hall, Englewood Cliffs. N. J.. 1955. (41C) Daniel, ’Cuthbert, IND.ENG.CHEM. 48, 1392-402 (1956). (42C) Fieller, E. C., Lewis, T., Pearson. E. S., “Correlated Random Normal Deviates,” Cambridge University Press, Cambridge, Eng., 1955. (43C) Huff, D., “How to Lie with Statistics,’’ W. \V. Korton Bi Go., New York, 1954. (442) International Journal of .4hstracts on Statistical .$lethods in Industry, International Statistical Institute. (45C) Kendall, 51. G . , “Rank Correlation hlethods,” 2nd ed.; Hafner Publishing Co., S e w York. 1955. (46C) Mandel, J., Linniq, F. J . , ilnal. Chem. 28, 7-0-7 (1956 ). (47C) “Quality Control and Applied Statistics,” Interscience, Ne\v York. (48C) Rand Corp., “A Million Random Digits with 100,000 Sormal Deviates,” Free Press, Glencoe, Ill., 1955. (49Ci Snedecor. G. W.. “Statistical Methods,” 5th ed., Iowa State College Press, Ames, Iowa, 1956. (5OC) Volk. W., Chem. Eng. 63, No. 3, 165-90 (1956).

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Statistical Methods-Symposia and Meetings (51C) Army Chemical Center, Md., 2nd Ann. Statistical Engineering Symposium, April 26-27, 1956. (52C) University of \$’isconsin, “Applied Statistical Methods in Chemical Engineering,” December 1957.

Statistical Methods-Related Fields; General Comments (53C) Acrivos, Andreas, Chem. Eng. 63, No. 8,215-6 (1956). (54C) Adams, J. F. E., Baker, A. G., Trans. Inst. Chem. Engrs.. (London) 34. 91-8 (1956). (55C) Bacharach, ‘A. L:, Analyst 81, 73-4 (1956). (56C) Carlson,’ W. M., Chem. Eng. Progr. 52,241-3 (1956). (57C) Chem. Eng. Progr. 52, No. 10, 80-2 ( l- 9 5 - 6_i (58‘2) Churchman, C. LV,, IND. ENG. CHEM. 48, 393 (1956). (59C) Fenech, E. J., Acrivos, Andreas, Chem. Eng. Sci. 5, 93-8 (1956). (60C) Flood, M. M., Industrial Labs. 7, 50-5 (September 1956). ( 6 l C ) Greaney, T. J., Hamilton, J. P., Chem. Eng. Progr. 51, 504-6 \

~

2’

(\ 1- 9- -5-5/).

(62C) Klinkenberg, .A,, Sjenitzer, F., Chem. Eng. Sci. 5 , 258-70 (1956). (63C) Kropa, E. L., IND.ENG.CHEM.48, 388-92 (1956). (64C) McCloskey, J. F., Trefethern, F. h-. (ed.) “Ouerations Research for Management,” Johns Hopkins Press, Baltimore, Md., 1954. (65C) Mellecker, J. B., Chem. Eng. Progr. 52. No. 3. 85M (1956). hfurphy, W. J., IND.END. CHEM. 48, 69A (March 1956). Noyes, R. M., J . A m . Chem. Soc. 78, 5486-90 (1956). Schwan, H. T., Wilkinson, J. J., Chem. Eng. 63, No. 8 , 211-14 (1956).

INDUSTRIAL AND ENGINEERING CHEMISTRY

(69C) Soo, S. L., Chem. Eng. Sci. 5 , 5’-6(1956 ). ( 7 0 ~ Stikon,’ ) P., IXD. ENG. CIIEM. 48, 402-7 (1956). (71C) Syrnonds, G. H., Ibid., 48, 394--401 (1956).

Mathematics (1D) Acrivos, Andreas, IND.EKG.CHEM. 48, 7’03-10 (1956). ( 2 D ) -4mundson. N. R.. Ibid.. pp. 26--35. (3D) Ibid., pp. 35-43. (411) Bilous, Olegh, Amundson, K. I Xfunro. W. D., Amundson, N. I