computers, statistics, and mathematics

ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT

Halsey, G. D., Jr., J . Phys. Chem., 57,87-9 (1953). Hermans, J. J.. Plastica, 5,352-6, 380-3 (1952).

Imrti, N., and Osawa, F., Busseiron Kenkyu, No. 46, 14-32;

Inoue, T., and Iida, Y., J . Chem. SOC. Japan, Pure Chem Sect.,

Kagawa, I., and Katsuura, K., J . Polymer Sci., 9, 405-15

Katohalsky, A., Endeavour, 12, 90-4 (1953). Katchalsky, A., and Sternberg, M., J. Polymer Sci., 10, 253-5

Klevens, H. B., J . A m Chem. Soc., 74,4624-6 (1952). Klevens, H. B., J . Am. Oil Chemists’ Soc., 30,74-80 (1953). Klevens, H. B., Iiolloid-Z., 128, 61-7 (1952). Kushner, L. M., Duncan, B. C., and Hoffman, J. I., J . Re-

Layton, L. H., Jackson, E. G., and Strauss, U. P., J . Polymer

McBain, J. W., and Das Gupta, S. G., J . Colloid Sei., 8,

Mathews, R l . B., Arch Biochem. Biophys., 43, 181-93 (1953). Mathews, 31, B., and Hirschhorn, E., J . Colloid Sci., 8, 86-96

Hill, T. L., J . Chhsm. Phys., 20,1259-73 (1952k.

NO. 47,49-64 (1953).

74, 145-8 (1953).

(1952).

(1953).

search Natl. BUT. Standards, 49,85-90 (1952).

Sci., 9, 295 (1953).

474-84 (1953).

(1953). \ - - --, -

Matthews, J. B., J . Znst. Petroleum, 39, 265-75 (1953). llehrotra, R. C., Nature, 172,74 (1953). Noguchi, H., Bull. Aichi Gakugei Univ.. KO, 2, 27-30 (1953). Noguchi, H., Busseiron Kenkyc, No. 62, 30-8 (1953).

(36H) Paquot, C., M6m. services chim. &at (Paris), 37, No. 1, 91-102 (1952).

(37H) Parker, k, and Tritsch, L., Modern Plastics, 30, S o . 6, 129-34,

(38H) Price, D., “Detergents,” New York, Chemical Publishing 218 (1953).

Co., 1952.

30, 14-17 (1953). (39H) Rigg, RI. W., and Liu, F. W. J., J . Am. 0 1 2 Chemists’ Soc.,

(40H) Saito, N,, and Sajto. S., Kolloid-Z., 128, 154-8 (1952). (41H) Sat& Keinomke, Yushi Ragaku Kybkaishaz, 1 , 119-24 (1952). (42H) Saxena, A. P., Indian Tertile J., 63,234-7 (1953). (43H) Sisley, J. P., and Wood, P. J., “Encydopaedia of Surface-

Active Agents,” New Tork, Chemical Publishing Co., 1952.

(44H) Stainsby, G., and Alexander, A. E., Australian J . Chem., 6,

(45H) Stevenson. D. G., J. TeztileInst., 44, T12-35 (1953). (46H) Strauss, U. P., Assony, S. J., Jackson, E. G., and Layton,

L. H., J . Polymer Sei., 9, 509-18 (1952). (47H) Strauss, U. P., and Layton, L. H., J . Phys. Chem., 57, 3 5 2 4

(1953). (48H) Tamamushi, B., and Nakadate, S., Rept . Inst. Sci. Technol.

(49H) Trapeenikov, A. A,, and Belugina, G. W., Dokladv Akad.

123-34 (1953).

U ~ Z V . T o k y o , 7, 7-10 (1953).

Xauk S.S.S.R., 87, 635-7 (1952). (50H) Ibid. , pp. 825-7. (51H) S’old, bI. J., Ins t . Spokesman, 16, KO. 8 , 8-16 (1952) (52H) Vold, RX. J., J . Phys. Che~n., 57,26-9 (1953).

ARTHUR ROSE AND RICHARD L. HEINY The Pennsylvania State University, State College, Pa. a. CURTIS JOHNSON JOAN A. SCWlkK Washington University, St. Louis, Mo. Shell Development Co , Emeryville, Calif.

lthough the three major divisions of this papar-computers, statistics, and mathematics- re quite extensively interrelated, this review continues the presentation of each in a

separate section. The subjects of automatic control and computers are showing signs of merging and developing into automation, the nearly automatic Operation Of a complex PrOC- ess. Brief mention is made of operations research and filing and finding activities, although these areas of thought seem to be drifting rather than developing.

ing to survey the specific problems dealt with, and to try to imagine methods of solution for these problems without the use of computers.

One gets the impression that the trend is toward digital computers,

a8 far as new machines are concerned. Certainly the outstand- ing event here is the greater variety of equipment becoming available. Adequately trained and experienced personnel is a bottleneck, both for actual short-range operation problems, and for development of efficient approaches through knowledge of numerical analysis. At least one conference on the specific subject of training has been organized (64A).

HE volume of publications covered for this review is such that only the items of special interest are mentioned in the text. Others are listed in the tables n-ith brief abstracts to

amplify the title or to evaluate the contribution. References judged to be less significant or routine applications have been omitted altogether.

A reading of the papers, abstracts, or even titles of papers on chemical engineering and related subjects demonstrates the rapid adoption of computers as tools. Only two years ago, a paper involving such an application was emphasized as a novelty. Now there are numerous and varied applications within a single year. I n this year’s collection of references the use of computers is often given only casual and incidental mention. In a number of instances preliminary work was done on a small computer, and extended calculations on one of the larger devices. It is interest-

One paper (612) that is difficult to classify expands on the thought that mathematics is not always the best solution to industrial research problems. and suggests that experimental methods are sometimes less expensive and time consuming, and that mathematical analysis often merely confirms designers’ intuition While these ideas are more likely to be applicable to simple problems, the thoughts are worth mentioning as a re- minder to temper enthusiasm with realism.

Possibly the most important contribution of all is the Moniac ($JA), a liquid analog computer that traces the flow of dollars through an economy, and gives hope that more objectivity and

Vol. 46, No. 5 916 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y

FUNDAMENTALS REVIEW

less subjective opinion will eventually determine practical economic measures.

A variety of recent articles review the computer field in a general manner (5A, I IA , %6A, 66.4). A paper by Brooker (8A) is particularly comprehensive, while that of Mac- Williams (48A) reviews historical development. Crumb, Harp, and Keller (R6A, SQA, 44A) deal with the adoption of computers for engineering calculations in a general way. A review has also appeared in German ($SA).

Other general articles deal with specific aspects of digital computers, such as arithmetic processes (SOA), problem preparation (dIA), and components and their functions ( I A , @A, 59A). Three of these papers ( IA , CIA, 69A) were presented at the Computer Symposium program of the American Institute of Chemical Engineers a t St. Louis. A continuation of this symposium a t the Washington, D. C., meeting in March 1954 lists papers on an industrial computation laboratory (46A) and applications in kinetics, thermodynamics of chemical equilibrium, and process control (Table I). The symposium at the Midwest Research Institute in January 1953 (60A) listed papers on a broad range of subjects of interest in engineering and science, and included one review paper (62A) on some chemical engineering applications. The applications of punched cards to spectroscopy were discussed at the 1953 Pittsburgh Conference on Analytical Chemistry and Applied Spectroscopy (S8.4).

It is convenient to have available the published lists of automatic computers in existence to date (23.4, 5dA) and organiza- tions in the field of computers and automation (@A). A “Com- puting Manual” (SYA) is now generally available. Fransson (S2A) has made an interesting direct comparison of analog and digital computers. The results showed short set-up and calculation time for the analog and better accuracy for the digital, as expected.

Various large scale digital machines have been discussed: AVIDAC (14A), ORACLE (IQA), OARAC (d7A) , EDV.4C (SBA), UTU’IVAC (54A). These utilize mercury- acoustic or magnetic drum high speed memory systems and are best suited to problems of considerable complexity, requiring very large memory capacity. In addition, a number of moderate size digital computers, also using magnetic drum memory, are now available: CADAC (21A), the IBM 701 (28A, 89A), the IBM Type 650 Magnetic Drum Calculator (@A) , ALWAC (4YA), Monrobot (5SA) and are well suited to the solution of chemical and chemical engineering problems.

The most significant development is the increasing number of moderate size digital computers being planned, built or used a t universities and research institutes.

I n the small computer field, the most interesting reference (IZA) is to the combination of a desk calculator and an electric typewriter. If such a device could be provided with even limited storage, i t would be invaluable for small nonroutine calculation jobs.

Recent applications of digital computers to chemical and chemical engineering problems are summarized in Table I. The majority of the articles deal with cases in which computers are used as tools to facilitate the solution of the problem within a reasonable period of time, though some unique applications of digital machinery are included.

Only two general references to new analog computers were noted. One (49.4) is an electrical device for the solution of linear simultaneous equations, and the other (I7A) is designed to handle linear and arbitrary nonlinear functions. There are undoubtedly other similar references in specialized publications.

Applications of analog computers are summarized in Table 11. Special Accessories. There are reports on some additional

devices that convert analog data to digital form (4A, 6A, 68A). One paper deals with the reverse operation (6 fA) . A survey of

Reviews.

Digital Computers.

Applications of Digital Computers.

Analog Computers and Applications.

Table 1. Specific Applications of Digital Computers Reference

Bazche1deE;H. R Busch R M and Armstroni, W. P., ‘IN;: ENG. CHEIIZ., 45, 1856 (1953)

Beutler, J. A presented a t the National geeting, Am. Inst. Chem. Engrs., Washington, D. C., March 1954

Black, R. H., Anal. Chem., 25, 74.1 f l R 5 R )

Bucha&n--.k. C J r and Skin- ner, H..’Oil GdA J.;’51, No. 39, 72 (1953)

Chem. Eng. N e w , 31, 4880 (1953)

Cochran, W., and Douglas, A. S., Nature, 171, 1112 (1953)

deWitte L Frankel S P and Porte; i’ D oil ‘G& L, 51, NO. 39’ 7’4 ( i i 5 3 )

Gee, R. ’E et al presented a t the Nagonal Meeting, Am. Inst. Chem. Engrs., Washing- ton, D. C., March 1954

Gumpreoht, R. O., and Sliepce- vlch C. M J . Phys. Chem., 57, $0 (1952)

Guthrie, R. K., et al., Oil Gas J., 51 No. 39 62 (1953)

Gut lhe R K et al Oil Gas J., 51 Nb i 9 ‘$5 (1ih3)

Hickb B ’ L . ’Turner T. E and Wid& W. W., J. bhem. ?hys., 21, 564 (1953)

Hudgens C R andRoss A M Anal. ’ci& 25, 734’ ( i 9 5 i j

Metropolis N et a1 J . Chem.

Oil Gas J., 51, No. 39, 57 (1953)

Phys., Zi, 1587 (19’b3)

Opler, Ascher, IND. ENG. CHEY., 45, 2621 (1953)

Perry, R. H., and Pigford, R. L., IND. ENQ. CHEIM., 45, 1247 (1953)

Rose, Arthur, and Johnson, R. C., Cham. Eng. Progr., 49, 15 (1953)

Sherman, Jack, presented a t the National Meetin m. Inst. Chem. Engrs., %&hington, D. C March 1954

Stevens’,’ R. F., and Brady, J. F., presented a t the National Meeting, Am. Inst. Chem. Engrs,, Washington, D. C., March 1954

prebentid a t the 124ti Mee6: ing, AM. CHEW. Soo., Chicago, I11 Sept. 1953. IND. ENG. C&., 45, 20A (December 1953)

Verzuh F. M presented a t the Fall ’ Gene& Meeting, Am. Inst. Elec. Engrs., Kansas City Mo November 1953. Am.’Inst.”Elec. Engrs. Tech: Paper 53-395.

Todd L. J and Andrews D. H

Applications of Computers Allocation of battery oil and gas

production to individual wells Automatic infrared punched

card identification of mix- tures

Kinetics of coal gasification

Programming of kinetic calculations

X-ray spectrometer analysis of bauxite exploration

Computation of gas volumes from orifice meter charts data

Location of correct x-ray pat- tern by punched card sorting operation (Vand)

Crystal structure analysis with EDSAC.

Computation of resistivity-de- parture curves

Gas phase tubular reactor kinetics with differential fouling of heat transfer surface

Methods for measuring drop size and distribution in aerosols

Petroleum engineering data file on punched oards

Computation of sand volumes from isopach maps

Calculations of microwave spectroscopy

Use of IBM 402.4 Tabulator to simplify use of Fourier synthesis in x-ray atudies

Equation of state oaloulations. modified Monte-Carlo in: tegration

Unsteady state gas flow through porous media, VLE caloula- tions in dultistage field separations; distillation calculations

Ion exchange column calculations

Complex kinetic problems

Theory of unsteady state distillation

Thermodynamic computations: isomerization between n- and and isopentanes

Process control

Calorimetric computations; po- tentiometer with digital re- cording and computing

Solutions of boundary value problems

these devices has been made (dOA), and criteria established for the selection of the proper type of data converter for a given application.

Also of interest are devices that follow curves and convert data therefrom into electrical impulses @A, 16A, 8OA, 66A). Re- lated devices involve a sensing unit (358, 67A) which uses a magnetic pick-off to convert temperature, pressure, and velocity into digital values for use by a computer or to record net weight readings (18A) from dial scales into digital devices. Advances in digital recorders continue (16A). An automatic device measures material properties, computes, and records the viscosity in kettles or pipelines (I3A).

Automation, Operations Research, and Literature Control. References in the area of automation (IOA, IdA, 60A)

May 1954 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y 917


Table II. Applications of Analog Computers Reference Applications of Computers

Bianchi, G., iifetallurgia ital., 45, 123 (1953)

Chance, Britton, et al., Arch. Biochem. Biopkys. , 37, 322 (1952)

Dewey, R. D., Appl . Spectros- copy, 7 , No. 2 , 87 (1953)

Oppenheim A. K J . A p p l .

Paschkis, V.,’and Heisler. M. P., Mechm& 20 112 (1953)

Clcem. Eng. Progr. Symposium Series. No. 5. 65 (1953)

DiEusion phenomena in cor- rosion by differential aeration

Mechanism of catalase action

Plotting of film transmission u s . composition directly from photographs in spectrochemi- cal analysis

Secular equations

Heat flow in piles

Prediction of heat exchanger performance

Geophysical problems

X-ray calculations; Fourier

Conversion of mass spectrom-

High velocity combustion

Design of a heat exchanger involving three fluids

Phase problem in x-ray crystal analysis

synthesis

eter data into digital form

sity, 1952 Pepinsky Ray and Jarnatz, Paul Analog recorder-playback com-

J . A p i l . Phbs., 24, 663 (1963j Waugh D. F and Yphantis, Solute distributions in a centri-

D. A:, J . Ph&. Chem., 57, 312

puter for crystal analysis

fuga (1953)

137,238 (Nov. lib3) analysis problems Wickersham, P. D World O i l , Compressible flow and network

do not yet deal with major specific practical applications. Public utility distribution systems seem to have the nearest approach to automatic control of a complex operation (34A).

Watson (63A) has given an excellent exposition of the purposes and methods of operations research. This states that this technique applies the methods of scientific research t o operating problems outside the conventional fields of science. It sub- stitutes mathematical probability relationships between cause and effect throughout an operation. Operations research gives a working model, on paper, built of complex mathematical formulas. Brown (QA) also presents a paper in this field, in- cluding elementary discussion and a specific example.

The application of machines to literature control is apparently stilI in the approach and basic planning stage (7Aj 45A, bbd). A symposium (IA) on this general subject gave most emphasis to convenient and quick reproduction methods. Machine translation has been in the news, but no technical papers have been published, and routine translation is probably still in the future.

Statistical Methods Judging by the number of recent publications of interest to the

chemical industry, a marked expansion in the use of statistical methods is under way. More impressive than the increase, however, has been the shift in emphasis from using statistical methods for quality control and the analysis of data to using these methods in the design of experiments and even of equipment. Evidence of the increased interest in statistics may be seen in the number of symposia on the subject held during the year (54B-58B), as well as in the number of books and papers. The Gordon Research Conferences now include a week on Statistics in Chemistry. Evidence of the shift in emphasis can be eeen in the number of recent papers in which the design of experiments is the principal topic.

The applications of statistics in statistical mechanics, thermodynamics, structure and behavior of polymer molecules, and in

nuclear phenomena have been considered outside the scope of this review.

Books. A number of books specifically directed toward chemists and chemical engineers were published (17B, 18B, dlB, 96B). The appearance of a11 these in EO short a time suggested a review of older books. Several of these, not mentioned in last year’s review, may be of help to the chemical and chemical engineering experimenter. Four of these have good discussions of experimental design (19B, 2dB, 2SB, 26B). Freeman (19B) and Peach (ZSB) are strictly nontheoretical treatments, while Tippett (26B) and Mood (ZZB), although fairly simple, still require some mathematical background. The other texts all cover approximately the same material-basic statistical theory with applications to analysis and correlation of data, (20B, IgB, I5B).

The papers published during the past year or so seem to fall into four major classes, as indicated in the appropriate part of the bibliography:

Categories of Papers.

1. Very general papers that seem primarily concerned with what can be done with statistics but not in telling how to do it (SQB-46B). These papers are not intended to be of practical value and merely serve as a general introduction to the subject.

2. Articles dealing with quality control. Here the emphasis seems to be strongly on explaining in general terms how statistics are used with very few good examples of actual use (47B-6SB).

3. Papers that present various methods for analysis of data

4. Articles in which the de3ign of experiments is the primary (1B-16B).

interest (67B-38B ).

One thesis includes a section on equipment design (14B), and a monograph on statistics in biological experimentation (+$OB) includes some applications to chemistry. Youden (46B) has begun a bimonthly column in INDU~TRIAL AND ENGINEERING CHEMISTRY on the use of statistics in the chemical industries.

Quality Control. The articles concerned with quality control seem to fall into two classes: those which present the methods of applying statistics with limited examples (47B, 50B, 63B) and those which present the applications with few details as to the methods (48B, 51B, 52B) Dorenfeld et al. (49B) give a fairly detailed discussion of the ufie of statistics in improving the performance of a commercial fractionator.

The analysis of data can take several forms: analysis of variance, determination of confidence limits, significance of trenda, and significance of differences. While this aspect of the use of statistics is extremely important, and probably more widely used than any of the others, only a few papers were published that had the primary purpose of presenting methods for the statistical analysis of data (2Bj BB, 11B). Of these, the one by Liebhafsky et al. (11B) was the most interesting as it presented an operating rule for analytical chemists with a good discussion of the development and limitations of the rule. This should have application outside analytical chemistry. A number of other papers appeared in which the statistical inter- pretation of data was secondary t o reporting the data (IB-YB, QB, IbB, lSB, 15B). Dorenfeld (7B) , Frazier (OB), and Pohley (13B) give detailed discussion of the statistical analyses and correlations performed.

I n several other publications ( IB , 3B, lOB, I@, 16B) analysis of data is discussed in conjunction with the main topic of the paper.

Design of Experiments. Factorial design of experiments can be used to greatly reduce the number of experiments that must be performed to complete a particular investigation. Harrington (36B) presents an excellent nonmathematical discussion of the methods and advantages of factorial design and fractional replication. He presents no applications. Box (UB, 98B) discusses factorial design with the object of deter- mining an optimum set of operating conditions. Grohskopf (34B) gives a more technical discussion using as an example a

Data Analysis.

918 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y Vol. 46, No. 5

FUNDAMENTALS REVIEW

chemical reaction in which four variables were studied. Several investigators have reported experiments that were designed statistically (31B, SRB, S6B). Brownlee ($OB), Youden (S7B, %B), Connor (3OB), and Gore (SSB) have also presented methods for the design of experiments.

Probably the most important contributions are those that are specific and factual, and likely to be instructive and encouraging to those who are new to the application of statistical methods, after the first flush of enthusiasm is dampened by practical difficulties. Many of the contributions of the last year have this characteristic, but certain papers ( f B , 9B, IOB, 55B) and books (18B, 19B, dSB, 26B) seem worthy of special mention as educational tools.

Educational Tools.

Mathematical Techniques In this section, papers dealing with material more elementary

than differential equations or with the use of ordinary differential equations in kinetics, or partial differential equations in thermodynamics and the mathematics of statistical mechanics are not reported. The papers that are included concern the analysis of physical situations by partial differential equations and the solutions of partial differential equations by various methods. The most widely used and important of these is the Fourier analysis. The generalized series solution is of great importance in ordinary differential equations. Special cases of this lead to Bessel and Legendre functions.

Another topic of particular interest to chemical engineers is the calculus of finite differences, since difference equations arise naturally in the mathematical analysis of multistage equipment. In addition, i t is sometimes necessary to use finite difference methods in the solution of differential equations unsolvable by analytical means. This approach is used in the setting-up of problems for digital computers.

The chemical engineer has in general sadly neglected the study of dimensional similitude and dimensiona1 analysis. Yet practically every plant has grown by scale-up from test tube to pilot plant to full scale. Papers dealing with original thinking in these topics have been reported.

Numerous other topics are of more specialized utility. The main advantage of vector calculus, for example, is in allowing the representation of complex expressions by simpIer notation. The solutions of certain types of equations are also greatly simplified by the use of vector relationships. Other topics falling in the catagory of extremely convenient tools are the various transforms, particularly Laplace and Fourier, tensors, and matrices.

Boundary Value Problems. Practically all applied problems in chemical engineering and chemistry can be analyzed by partial differential equations. Most of the papers reviewed devoted the theory section to an explanation of the solution of the partial differential equations.

The equation arising most frequently in the analysis of operations involving heat transfer or diffusion is

v2T = q5

The Laplacian operator, VI may refer t o one, two, or three dimensions, and several coordinate systems may be employed. The function, +, is dependent on the nature of the problem. When 4 = 0, the equation usually refers to steady state heat transfer or diffusion. When $ = (Y (aT/&9), this term is an accumulation term in the equation of continuity. When $ = a(dZT/ae2), the wave equation results.

This equation presents a typical boundary value problem. The most common solution, when possible, is by separation of variables to obtain several possible solutions. Boundary conditions are imposed to eliminate several of these solutions, and the

remaining terms are usually expanded in a series. The nature of the solution is entirely dependent on the number of dimensions and the nature of @. Numerical constants are determined by substitution of boundary values. The possible variations are infinite.

Most of the papers involving use of this equation do not present new forms of the equation or new methods of solution; the solutions differ widely, however, because of the different boundary conditions. Thus, standard Fourier sin, cos, or sin- cos series are sometimes obtained with and without exponential decay terms. At times only exponential series result; these can be expressed in hyperbolic form in some cases. Some authors prefer the shorter complex variable terminology. Systems in cylindrical coordinates will yield Fourier-Bessel expansion series. With systems such as the semi-infinite solid, where a transient change is restricted to a relatively small portion of the whole, the Fourier integral approach is preferable. In such cases the solution is often an error function. All of these problems fall into the eigenfunction classification-Le., every series has its particular eigenfunctions-and the eigenvalues are the numerical solutions. Occasionally a paper will report a new equation for which eigenvalues are not available in tables. In such cases they must be calculated, usually with the aid of an automatic computer.

In all cases, of course, it is first necessary to derive the equations. Perry and Pigford (21 C) present partial differential equations for the combined accumulation, diffusion, and reaction in a study of kinetics of gas-liquid reactions. A good illustrative example of use of equations of state, Continuity, diffusion and energy balance, and of application of boundary conditions is given by Hirscbfelder, Curtiss, and Campbell (IOC) in a study of the theory of flame propagation. Danckwerts (6C) and Kramers and Alberda (15C) show a use of the equation of continuity and a one-dimensional boundary value problem in studying continuous flow systems. Frequency response analysis was used by Kramers and Alberda. The frequency response method is being used now, even in the determination of such variables as thermal con- ductivity, which are customarily studied by steady state means. The one-dimensional diffusion equation was used by Barrer (BC) in his study of gas flow in capillary systems. He claims better correlations than by steady state means.

Series Solutions. A good illustration of the two most common types of series, the Fourier and the Fourier-Bessel, is to be found in the analysis of water vapor saturation in gas-measuring burets by Stein and Reid (S6C). I t is practically impossible to saturate a buret in a reasonable length of time when water stands on the confining liquid, but saturation is rapidly attained when the wall is wet. Solutions to equations of solid diffusion lead to exponential series and error functions (SSC). Fluid dynamics equations and the Laplacian arise in the study of hydroextraction by Haruni and Storrow (K). Studies in liquid-liquid extraction (lac) lead to exponential and sing series. In extrusion studies, equations similar to two-dimensional heat flow with accumulation are solved in terms of hyperbolic series (4C).

Fourier-Bessel series are obtained in mixing and distribution of liquids in high velocity air streams (1 YC), in material transfer in turbulent gas streams (.2SC), and in the separation of gases by free double diffusion (SOC). Another study of diffusion in a fluid in a tube ( l 4 C ) leads to both Fourier-Bessel series and integral expressions. A new eigenfunction problem was obtained by Waugh and Yphantis (3SC) in a study of transient solute distributions in an ultracentrifuge. In preliminary work for finding methods of measuring drop size and distribution in aerosols, Gumprecht and Sliepcevich (7C) studied the scattering of light by large spherical particles. The resulting series require the use of complex variables, Bessel functions, and Legendre polynomials. An illustration of one form of the wave equation is given by Sofer, Dietz, and Hauser (S6C) in a study of the curing of resins.


ENGINEERING, DESIGN, AN PROCESS ~ ~ ~ ~ ~ ~ ~ M ~ N ~

I n some cases a series is used, not as a solution to an equation as much as a method of “curve fitting.” A typical example is in x-ray crystal structure analysis by Fourier synthesis. The equations are shown by Hudgens and Ross (11C) . At times a problem which would appear to lead to a series solution is solved without the use of series. In the study of steady-state burning of a semi-infinite solid or liquid, the one-dimensional heat flow equation, with accumulation and heat generation factors, yields a simple, nonseries solution when moving coordinates are used

As mentioned in the introduction, vectors are use- ful in abbreviated terminology and in simplified handling of certain mathematical operations. While workers in other fields, such as electrical and mechanical engineering, mechanics, and aerodynamics, make extensive use of vectors, the chemists and chemical engineers do not seem to have progressed to this point. Vector terminology was found in papers on mixing (SJC, %C), double diffusion (%E), and in diffusion in a spherical system

While several references in last year’s review made use of Laplace transforms, very little new material has been found. The use of these transforms simplifies the solution of certain ordinary and partial differential equations. Examples were found in frequency response analysis in automatic control (5C), diffusion (14C) , and coalescence (18C).

Finite Differences. Finite difference equations are the natural mathematical result of analysis of steady state stagewise equipment. In such cases the formal calculus of finite differences may sometimes be used to solve the equations. Almost always a graphical solution is possible-for example, in the McCabe-Thiele and Ponchon-Savarit methods in distillation. Such typical finite difference equations of continuity (with graphical solutions in some caees) are presented for ion exchange (QC, 28C, 40C), for leaching (27C), and for distillation

In nonstagewise equipment the mathematical analysis leads to ordinary or partial differential equations, and the transient operation of stagewise equipment leads to difference-differential equations or simultaneous differential equations. Such cases are often not solvable by standard means, and finite difference equations are often substituted for the purpose of numerical analysis. This method is used in the batch distillation work of Rose and Johnson (Z&C). Papers by Opler (SOC) and by Acrivos and Amundson ( I C ) illustrate the forms for equations to be suitable for computers.

Piumerical methods illustrated include the method of relaxation (Common to heat transfer) in hydroextraction (SC) and the Runge-Kutta method in the study of coal gasification (SC).

Many applications of dimensional analysis are t o be found in the literature; practically all are standard approaches, usually to fluid flow or heat transfer, and do not involve new mathematical thinking. The fev papers mentioned below fall into two groups: one includes papers written about dimensional analysis and similitude; the other includes papers which seem to go beyond the usual cookbook approach and to include original thought.

Discussions of dimensional similitude are to be found in papers on continuous flow system (GC), mixing (25C, RGC), gas-liquid reactors (S?C) , and furnaces (gBC).

Silberberg and Mclletta (SlC-34C) published a four-article survey of basic fundamentals and methods of dimensional analysis, presented for the understanding of the average engineer. Kayser ( 1 S C ) presents a recent approach using “directional” dimensions. In this system such groupings as the Reynolds number are not truly dimensionless. The result is a much clearer understanding of the significance of such terms as Reynolds number.

(23C). Vectors.

(12C). Laplace Transforms.

(19C) .

Dimensional Analysis.

Automatic Computers

(1.4) Aiken, H. H., presented at the 46th Annual Meeting, Am. Inst. Chem. Engrs., St. Louis, hIo., December 1953.

(2A) AM. CHEW Soc., Symposium on Statistics in the Design of Experiments before the Div. of Ind. Eng. Chem., 124th Meeting, Chicago, Ill., September 1953.

(311) Anal. Chem., 25, 36 A (April 1963). (4A) Aviation Week, 58, No. 20, 88 (1953). (5-4) Bauer, W. F., Ind. Math., 3, 13 (1952). (6A) Bennett, R. R., and Low, H., Electronics, 26, 164 (November

1953). (711) Berry, bI. M., Perry, J. W., and Kent, Allen, Battelle Me-

morial Institute, Columbus, Ohio, “Machine Literature Searching IV.”

(8A) Brooker, R. A., Brit. J. Appl. Phys., 4,321 (1953). I Brown, A. A., Ind. Math., 3, G (1952).

Chem. Eng., 60, 226 (July 1953). Chem. Eng. News, 30, 4563 (1952). Ibid., 31, 76 (1953). Ibid., p. 95. Ibid., p. 1008. Ibid., p 2330. Ibid., p. 2718. Ibid., p. 2910. Ibid., p. 2914. Ibid., p. 3246. Chem. Processing, 16, 64 (Sovember 1953). Clippinger, It. F., Dimsdale, B., and Levin, J. H., J. SOC.

Ind. AppZ. Math., %,I, 91 (1963). Computer Research Corp , Hawthorne, Calif., “CADAC

102-A Specifications.” Computers and Automation, 2, No. 2, 5; No. 4, 24 (1953). Ibid., 2, KO. 8 , 4 (1953). Crumb, C. B., Jr., Mech Eng., 74, 635 (1952). Digital Computer Sewstetter, Office of Taval Research, Wash-

Elec. Eng., 72, 273 (1953). Ibid., p. 484. Electronics, 26, 200 (July 1953). Felker, J. H., EZPctronics, 26, 150 (March 1953). Fortune, 45,100 (March 1952). Fransson, K. F., S.A.E. Journal, 60,28 (December 1952). Fuohs, O., Chem.-Ing.-Tech., 25, 377 (1953). Gas Age, 108, 17 (Kov. 22, 1951). Gen. Eke. Rev., 56, KO. 1, 11 (1953). Gluck, S. E., Elec. Eng., 72, 159 (1953). Gruenberger, Fred, “Computing Manual,” 3rd ed., Madison,

Hallett, L. T., Anal. Chem., 25, 15-4 (April 1953) Harp, ITr. hI., PetroZecmt Refiner, 32, KO. 4, 159 (1953). Hollander, G. L., presented at the Ninth Annual Flectronics

Conference, Chicago, Ill., September 1953. Householder, A. S., presented at the 46th Annual Meeting,

Am. Inst. Chem. Engrs., St. Louis, &Io., December 1953. International Business Xlachines Corp., New York, I B X Form

22-6060-0, 1953. Johnson, E. C., Ind. Math., 3, 92 (1952). Keller, A., Mech. Bng., 75, 891 (1953). Kent, Allen, Berry, hI. M., and Perry, J. VI., Battelle Memo-

rial Institute, Columbus, Ohio, “Machine Literature Search- ing 111.”

Landee, F. A,, presented at the National Meeting, Am. Inst. Chem. Engrs., Washington, D. C , Xarch 1954.

Logistics Research, Ino., Redondo Beach, Calif., “The ALWAC .”

lIacWilliams, IT, H , Jr., Elec. Eng., 72, 116 (1953). Many, A., Oppenheirn, V., and Amitsur, S., Reo. Sci. Instr.,

24, 112 (1953). Midwest Reseaich Institute, Kansas City 2, No . , “Sym-

posium on Industrial Application of Automatic Computing Equipment,” 1953.

hldler ”I, Waddell, B. L., and Patmore, J., Electronics, 25, 127 (October 1952).

k f o d . Ind., 24, 42 (December 1952).

ington 25, D. C.

‘Tis., University of Wisconsin Press, 1952.

920 I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY Vol. 46, No. 5

FUNDAMENTALS REVIEW

(53A) Oil Gas J., 51, No, 39, 59 (1953). (54A) Ibid., p. 61. (55A) Perry, J. W., Advances in Chem. Ser. No. 8 , 106 (1953). (56A) Proc. Inst. Radio Engrs., 41,1219 (1953) (300 pp.). (57A) Product Eng., 23, 191 (April 1952). (58A) Ibid., 24, 234 (May 1953). (59A) Rajchman, J. A., presented at the 46th Annual Meeting,

(BOA) Ridenour, L. N., Sci. American, 187, No. 3, 116 (1952). (61A) Roach, A. E., Chem. Eng., 60,324 (February 1953). (62A) Rose, Arthur, and Schilk, Joan, “Symposium on Industrial

Applications of Automatic Computing Equipment,” Mid- west Research Institute, p. 75 (1953).

Am. Inst. Chem. Engrs., St. Louis, Mo., December 1953.

(63A) Watson, A. N., Chem. Eng., 60,324 (February 1953). (64A) Wayne University, Conference on Training Personnel for the

(65A) Weissler, G. L., Einarsson, A. W., and McClelland, J, D., Computing Machine Field, June 22-23, 1954.

Rev. Sci. Inatr., 23, 209 (1952).

Statistical Methods-Analysis of Data

(1B) Box, G. E. P., AnaZyst, 77, 879 (1952). (2B) Brownlee, K. A,, presented as part of the Symposium on

Statistics in the Design of Experiments before the Division of Ind. Eng. Chem. at 124th Meeting, AM. CHEM. SOC., Chicago, Ill., September 1953.

(3B) Danckwerts, P. V., Chem. Eng. Sci., 2, 1 (1953). (4B) Daniel, C., and Riblett, E. W., presented as part of the Sym-

posium on Statistics in the Design of Experiments before the Division of Ind. Eng. Chem. at 124th Meeting, AM. CHEM. SOC., Chicago, Ill., September 1953.

(5B) David, D. J., and Oertel, A. C., Australian J. Appl. Sei., 4, 235 (1953).

(GB) Day, R. V., Horchler, D. H., and Marks, H. C., IND. ENG. CHEM., 45, 1001 (1963).

(7B) Dorenfeld, A. C., McGovern, L. J., and Wharton, G. W., Petroleum Refiner, 31, No. 12, 147 (1952).

(8B) Duncan, D. B., presented as part of the Symposium on Statistics in the Design of Experiments before the Division of Iud. Eng. Chem. at 124th Meeting, AM. CHEM. SOC., Chicago, Ill., September 1953.

(9B) Frasier, D., Klingel, A. R., and Tuba, R. C., IND. ENG. CHEY., 45, 2336 (1953).

(1OB) Grohskopf, H., Ibid., 45, 1260 (1953). (11B) Liebhafsky, H. A., Pfeiffer, H. G., and Balis, E. W., Anal.

Ch~m., 23, 1531 (1951). (12B) Mueller, M. F., and Laeder, A. D., Chem. Eng., 60,193 (October

1953). (13B) Pohley, F. M., presented as part of the Symposium on Statis-

tics in the Design of Experiments before the Division of Ind. Eng. Chem. at the 124th Meeting, AM. CHEM. Soc., Chicago, Ill., September 1953.

(14B) Ryan, J. F., Ph.D. thesis, The Pennsylvania State University, 1953.

(15B) Schwarz, C. E., and Smith, J. M., IND. ENG. CHEM., 45, 1209 (1953).

(16B) Youden, W. J., Analyst, 77, 874 (1952).

Statistical Methods-Books

(17B) Bowker, A. H., and Goode, H. P., “Sampling Inspection by

(18B) Clark, C. E., “An Introduction to Statistics,” New York,

(19B) Freeman, H. A,, “Industrial Statistics,” New York, John

(20B) Grant, E. L., “Statistical Quality Control,” New kork,

(21B) Hald, A., “Statistical Theory with Engineering Applications,”

(22B) Mood, A,, “Introduction to Theory of Statistics,” New York,

(23B) Peach, P., “Industrial Statistics and Quality Control,” 2nd

(24B) Simon, L. E., “An Engineer’s Manual of Statistical Methods,”

Variables,” New York, McGraw-Hill Book Co., 1952.

John Wiley & Sons, 1953.

Wiley & Sons, 1942.

McGraw-Hill Book Co., 1946.

New York, John Wiley & Sons, Inc., 1952.

McGraw-Hill Book Co., 1950.

ed., Raleigh, N. C., Edwards & Broughton Co., 1947.

New York, John Wiley & Sons, Inc., 1941.

(25B) Snedecor, G. W., “Statistical Methods,” 4th ed., Ames, Iowa,

(26B) Tippett, L. H. C., “The Methods of Statistics,” 4th ed., New Iowa State College Press, 1946.

York, John Wiley & Sons, Inc., 1952.

Statistical Methods-Design of Experiments

Box, G. E. P., Analyst, 77, 879 (1952). Box, G. E. P., presented as part of the Symposium on Statis-

tics in the Design of Experiments before the Division of Ind. Eng. Chem. at the 124th Meeting, AM. CHEM. SOC., Chicago, Ill., September 1953.

Brownlee, K. A., Chem. Eng. Progr., 49, 617 (1953). Connor, W. S., and Youden, W. J., presented as part of the

Symposium on Statistics in the Design of Experiments before the Division of Ind. Eng. Chem. at 124th Meeting, AM, CHEM. SOC., Chicago, Ill., September 1953.

Daniel, C., and Riblett, E. W., Ibid., 124th Meeting, Chicago, Ill., September 1953.

Frasier, D., Klingel, A. R., and Tuba, R. C., IXD. ENG. CHEM., 45, 2336 (1953).

Gore, W. L., presented as part of the Symposium on Statistics in the Design of Experiments before the Division of Ind. Eng. Chem. at 124th Meeting, AM. CHEM. Soo., Chicago, Ill., September 1953.

Grohskopf, H., IND. ENQ. CHEM., 45, 1260 (1953). Harrington, E. C., Jr., Tappi, 36, l37A (May 1953). Schwars, C. E., and Smith, J, iM,, IND. ENG. CHEM., 45, 1209

Youden, W. J., Anal& 77, 874 (1952). Youden, W. J., and Connor, W. S., ChPm. Eng. Progr., 49, 549

(1953).

(1953).

Statistical Methods-General

(39B) Acton, F. S., J. Chem. Ed., 30, 128 (1953). (40B) Ann. N . Y. Acad. Sci., 52,789 (1950). (4tB) Chem. Eng., 60, No. 12, 118 (1953). (42B) Ferencz, P., and Lloyd, B. H., Ibid., 60,219 (April 1953). (43B) Ferencs, P., and Lloyd, B. H., Ibid., 60, 207 (May 1953). (44B) IND. ENG. CHEM., 45, 65 A (November 1953). (45B) Wagner, H. B., presented as part of the Symposium on Sta-

tistics in the Design of Experiments before the Div. of Ind. Eng. Chem. at the 124th Meeting, AM. CHEM. SOC., Chi- cago, I&, September 1953.

(46B) Youden, W. J., IND. ENG. CHEM., 46,107 A (1954).

Statistical Methods-Quality Control

(47B) Blaedel, W. J., and Meloche, V. W., presented as part of the Symposium on Recent Advances in Analytical Chemistry before the Div. of Chem. Ed. a t the 124th Meeting, AM. CHEM. Soc., Chicago, Ill., September 1953.

(48B) Day, R. V., Horchler, D. H., and Marks, H. C., IXD. ENG. CHEM., 45, 1001 (1953).

(49B) Dorenfeld, A. C., McGovern, L. J., and Wharton, G. W., PetroZeumRefiner, 31, No. 12, 147 (1952).

(50B) Johnson, R. A,, presented as part of the Symposium on Recent Advances in Analytical Chemistry before the Div. of Chem. Ed. a t the 124th Meeting, AM. CHEM. SOC., Chicago, Ill., September 1953.

(51B) Mayer, M. J., Morton, J. W., and Laufer, S., J. Agr. Food Chem., 1,404-8 (1953).

(52B) Mueller, M. J. and Laehder, A. D., Chem. Eng., 60, 193 (October 1953).

(53B) North, C. H., Olmstead, L. E., and Toomey, R. D., Anal. Chem.. 25, 527 (1953).

Statistical Methods-Symposia

(54B) AM. CHEM. SOC., 123rd Meeting, Los Angeles, Calif., March 1953.

(55B) AM. CHEM. SOC., Symposium on Statistics in the Design of ExperimentR before the Div. of Ind. Eng.*Chem., 124th Meeting, Chicago, Ill., September 1953.



(56B) Am. Inst. Chem. Engrs., presented at the 45th Annual Meet-

(57B) Am. Soc. Testing Materials, Philadelphia, Pa., Symposium on

(5SB) Chem. Eng. Neus, 31, 288 (1953).

(17C) Longwell, J. P., and Weiss, M. A., Ib id . , 45,667 (1953). (18C) Melaak, Z. A., Quart. Appl . Math., 11,231 (July 1953). (19C) Rleyer, P., Chem. Eng. Sci., 2, 53 (1953). (2OC) Opler, A., IND. EKG. CHEM., 45,2621 (1953). (21C) Perry, R. H., and Pigford, R. L., Ibid., 45,1247 (1953). (22C) Pfeiffer, P. W., Chem. Eng. Sci., 2,45 (1953). (23C) Reid, W. P., J . Phys. Chem., 57, 242 (1953). (24Ci Rose. A,. and Johnson. R. C.. Chem. Eno. Proor.. 49. 15

ing, Cleveland, Ohio, December 1952.

Statistical Methods for the Detergent Laboratories, 1953.

Mathematical Techniques

Acrivos, -4., and Amundson, S. R., IND. ENG. CHEM., 45,

Barrer, R. M., J . Phys. Chem., 57, 35 (1953). Batchelder, H. R., Busche, R. M., and Armstrong, W. P.,

Carley, J. F., and Strub, R. A, , Ib id . , 45,970 (1953). Ceaglske, N. H., and Eckman, D. P., Ib id . , 45, 1879 (1953). Danckwerts, P. V., Chem. Eng. Sci., 2, 1 (1953). Gumprecht, R. O., and Sliepcevich, C. M., J . Phys. Chem., 57,

Haruni, M. hI., and Storrow, J. A., Chem. Eng. Sci., 2, 164

Hiester, N. H., et al., IND. ENG. CHEM., 45, 2402 (1953). Hirschfelder, J. O., Curtiss, C. F., and Campbell, D. E., J .

Hudgens, C. R., and Ross, A. M., Anal. Chem., 25,734 (1953). Jost, W., Chem. Eng. Sci., 2, 199 (1953). Kayser, R. F., IND. ENG. CHCW., 45, 2634 (1953). Klinkenberg, A., Krajenbrink, J. J., and Lauwerier, H. d.,

Kramers, H., and Alberda, G., Chen. Eng. Sci., 2, 173 (1953). Licht, W., and Pansing, W. F., IND. EKG. CHEX., 45, 1885

467 (1953).

IND. ENG. CHEX., 45, 1856 (1953).

90 (1953).

(1953).

Phys. Cham., 57,403 (1953).

Ib id . , 45, 1202 (1953).

(1953).

- I

(1953). (25C) Rushton, J. H., and Oldshue, J. Y., Ibid. , 49,161 (1953). (26C) Ib id . , p. 267. (27C) Scheibel, E. B., Ibid. , 49, 354 (1953). (28C) Schlinger, W. G., and Sage, B. H., IND. ENG. CHEM., 45, 657

(29C) Schwars-Bergkampf, E., Chem.-Ing.-Tech., 25, 177 (1953). (30C) Schwertz, F. A, IA-D. ENG. C m w , 45, 1592 (1953). (31C) Silberberg, I. H., and McKetta, J. J., Jr., Petroleum Refiner,

(3%) Ib id . , p. 147 (RIay 1953). (33C) Ibid. , p. 101 (June 1953). (34C) Ib id . , p. 129 (July 1953). (35C) Sofer, G. A, Diets, A. G. H., and Hauser, E. A,, IND. E m .

CHEM., 45,2743 (1953). (36C) Stein, T. W., and Reid, R. C., Anal. Chem., 25, 1919 (1953). (37C) Van Krewlen, D. IT., and Hoftyeer, P. J., Chem. Eng. Scz.,

(38C) Vermeulen, T.. IND. E m . CHmi., 45, 1664 (1953). (39C) Waugh, D. F., and Yphantis, D. A,, J . Phys. Chem., 57, 312

(40C) Zeegers, J. A,, Chem. Eng. Sci., 2, 74 (1953).

(1953).

32, 179 (April 1953).

2,145 (1953).

(1953).

F mg A. M. OPPENHEIM

University of California, Berkeley, Calif.

T H O M A S BARON Shell Development Co., Emeryville, Calif.

The most significant feature of last year’s literature is the large number of books, essays, proceedings of symposia, and review papers providing a convenient and up-to-date summary of the present status of fluid mechanics. of classical hydrodynamics was cul-

minated by Lamb’s well-knonrn trea- tise which characterizes the method

of approach of the last century. The significant feature of the progress made since then is the emphasis of the effect of depar- tures from idealizations of classical hydrodynamics.

The advances made in this direction over the first half of this century are summarized most authoritatively by Prandtl, the founder of the mechanics of real fluids. The English-speaking world should welcome the translation of Prandtl’s last book en- titled “Essentials of Fluid Dynamics” (10A). In words of Batchelor (Proc. Phys. Soc., 66, 518, 1953):

The book fills admirably the long-felt need for description of fluid floV it is in nature, It consists of four different and effectively self-containedchapters precededby a short introductory discussion of pressure relations applying to fluids in equilibrium. The first of these chapters is concerned with the dynamics of frictionless fluids. The second describes the motion of viscous fluids and the associated phenomena of boundary layers and turbuIence. The third section deals with the high speed flow of gases not in great detail and, as elsewhere, avoiding mathematics if possible but nevertheless conveying the essential ideas involved in the flonr of a compressible fluid. $ ~ ~ & ~

quently bear little resemblance to real fluids. chapter is on miscellaneous topics. T o my knowledge, no other

922 I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY Vol. 46, No. 5

HE: organization of this review is essentially the same as last year’s. The subject matter is scanned in a sequence corre- sponding to the degree of progress made up to date in the in-

terpretation and analysis of each topic, starting from viscous flow and leading up to flows associated with chemical reactions. HOY- ever,we changed our method of approach; the survey contains this time only a few representative papers of each field selected primarily for their interest to the chemical and process industry. Consequently, the number of reported papers has been consider- ably reduced.

The literature of 1953 is prolific in contributions to fluid mechanics in that an unusual number of books of fundamental importance appeared. Because of their significance, they are discussed in the first section.

T

Books and Proceedings of Symposia Classical hydrodynamics deals with ideal fluids which fre-

The development

computers, statistics, and mathematics

Documents