November 1951 INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY 247 1

In general, punched-card methods will probably not reduce greatly the time spent by the chemical engineer on distillation problems. He will continue to do all the thinking. However, he can get from machine methoda much additional information which will be useful in improving both operation and design but which does not justify the time now required by manual methods.

ACKNOWLEDGMENT The writera wish to thank the chemical engineering staff of

The Daw Chemical Co. for advice ahd d t a n c e , R. N. Shiraa of the Shell Development Co. for helpful suggestiona, and Helen Hecox, Oakland, Calif., and Paul W, Fullerton, Jr., N8w York, of the International Business Machinee Corp., for technical aasistance.

NOMENCLATURE D rate of removal of distillate, moles per hour F H AH, - latent heat of vaponzatjon, calories per mole

3 feed rate, moles per hour = heat content of vapor, calorie8 per mole ( h - 0 at 25’ C.

H = AHVat25’C.)

K - equilibrium conitant L L’ Q S 3 constant in E uation 6, mole er cent T = temperature le-centi axe u = constant in liquation 5, rn% per cent V = vapor rate, rectifying section, moles per hour V‘ = vapor rate, stri pmg section, moles per hour W = rate of removafof bottoms product, moles per hour

= heat loss er theoretical plate, calorie8 per hour

{ = total numLr of components in mixture = liquid heat content, calories per mole

r = ratio of li uid composition to that in product v = +e, z,, liquid composition. mole per cent, leaving nth plate y, - vapor composition, mole per cent, leaving nth plate I = total composition, liquid plus vapor, mole per cent

Subscripts x d = compoaition of distillate liquid xn

= liquid rate, rectifying section, moles per hour = liquid rate, stripping section moles per hour

total heat content of liquid-vapor mixture, calories per mole

compoaition of liouor leaving nth plate from bottom

= compoeition of bottoms liquid

= n for feed plate = number of tray above which feed ie - total heat content of feed

7 = component number

Qt introduced

LITERATURE CITED (1) Bubb, F. W., Nisle, R. Q., and Carpenter, P. G., Petroleum

Tram. Am. Imt . Mining Met. Engra., 189, 143 (1950). (2) Donnell, J. W., and Turbin, R., Petroleum Refiner, 29, No. 10,

108 (1950). (3) Eckert, W. J., J. Chem. Edumtion, 24,54 (1947). (4) Eckert, W. J., “Punched Card Methods in Scientific Computa-

tion,” New York. Columbia University Preas, 1946. (5) Go&, 0. W., and Calvert, J. F., Am. Inst. Elec. Engrs., Tech.

(6) Grosch. H. J., Proaeedinga of Scientific Computation Forum, 1848, International Business Machinee Corp., Endicott, N. Y., 1950.

(7) King, G. W., J. Chern. Education, 24,61 (1947). (8) Krawitz, E., Proaeedinga of Seminar on Industrial Cornputa-

tion, September 1950, International Bdneas Machines Corp., Endicott, N. Y., 1951.

(9) Lewis, W. L., and Matheson, G. L., IND. ENP. CFI~M., 24, 494 (1932).

(IO) Opler, A., and Heitz, R. G., Proceedings of Seminar on Industrial Computation, September 1950, International Businesr Machines Corp., Endicott. N. Y., 1951.

(11) Row, A., and Williams, T. J., IND. ENQ. CHEIM., 42,2494 (1950). (12) Rose, A., Williams, T. J., and Dye, W. S., Proceedings of

Seminar on Industrial Computation, September 1950, International Busineas Maphinee Corp.. Endicott, N. Y.. 1951.

(13) Scarborough, J. B., “Numerical Mathematical Analyah,” Baltimore, Johns Hopkins Presa, 1930.

(14) Sherman, J., and Ewell, R. B., J . C h m . Phys., 46,641 (1942). (16) Stull, D. R., The Dow Chemical Co., Midland, Mich., personal

communication. (16) Thiele, E. W., and Geddee, R. L., IND. ENP. CHEM., 25, a89

(1933). (17) Thomson, G. W., Chem. Rem., 48, 1 (1946). (18) Uitti, K. D., Pelvokmm ReJlnw, 29, 130 (1950). (19) Von Neumann, J., and Goldstine, H. H., Bull. Am. Mafh.

(20) Whittaker, E. T., and Robinson, G., “ C a l ~ u l ~ s of Observations,”

(21) Wilaon, E. B., Jr., Chsm. Rms., 27.17 (1940).

Racman Maroh 21, 1961.

Pope* 50-15 (1950).

Soo., 53, 1021 (1947).

London, Blackie and Son, 1924.

Evaluation of Performance Factors of Fuel-Oxidant Mixtures

STUART R. BRINKLEY, JR. U. S. Bmeoa of Mines, Pittsbargh, fa .

The computation of flame temperature, the analysis of power plant cycles, and related scientific and technical problems in the field of flame and combuetion require a knowledge of the thermodynamic properties of the prod- ucts of combustion reactions. In order to perform the necessary calculations efficiently on automatic equip- ment, systematic, easily programed, computational routines are required.

Generally applicable methods appropriate for applica- tion to automatic equipment are described for the com- putation of the thermodynamic properties of combustion gases. The results of such computation are the equiva- lent, in numerical form, of a Mollier chart for each fuel- oxidant mixture.

The application of these results, employing automatic computational equipment, to the calculationof flame tem- peratures and fuel performance parameters is described.

OR the theoretical description of power plants deriving their energy from the cornbustion of a fuel, it is necessary to solve to an appropriate degree of precision a hydrodynamic problem

requiring for ite solution a knowledge of the thermodynamic prop- erties of the working fluid composed of the products of the com- bustion reaction. It is uaually a good approximation to assume that the properties of the combustion gas are determined by the conditions of thermal equilibrium, and thus it ia possible to em- ploy the methods of classical thermodynamics for their computa- tion. The thermodynamic properties of fuel gases are also of con- siderable importance, since they form the basis for the design of appropriate means for their effective utiliation. The specilica- tion of the operating conditions to produce a gas for use as an in- termediate in a chemical process, such as the synthesis of liquid fuels, may be based upon a study of the variation of the com- position of the synthesis gas with changes in the various proceea variablee.

F

2472 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. 43, No. 11

Although there exist a large number of important scientific and technical applications of the data that can be obtained from the systematic determination of the thermodynamic properties of combustion gases, such applications have been handicapped by the extremely tedious and time-consuming computational meth- ods required. The development of large-scale automatic compu- tational equipment makes feasible the initiation of a systematic program for determining the thermodynamic properties of com- bustion gases and for applying such data to specific problems of scientific and technical importance.

For a long time there has been a need for systematic and economical methods for the calculation of the thermodynamic properties of systems of many constituents, and this need is em- phasized by the application of automatic computational equip- ment. In this paper methods are briefly described that have been routinely employed in this laboratory for the calculation of the equilibrium composition of multicomponent gas mixtures and the evaluation of the thermodynamic properties of the equilibrium mixture. The application of automatic computational equipment to these calculations is discussed. It is possible to utilize punched card equipment in a variety of applications of the results of these calculations to determine performance characteristics of par- ticular fuel-oxidant systems. These methods are illustrated by a discussion of the evaluation of the specific impulse of a rocket propellent.

CARD PROGRAMED ELECTRONIC CALCULATOR

This laboratory is equipped with punched card equipment des- ignated a card programed electronic calculator, supplied by the International Business Machines Corp. This equipment provides automatic computational facilities of moderate speed and very considerable flexibility. It consists of an electronic calculator, controlled by a punched card reader, and supplemented by a moderate amount of internal storage and by printed page and punched card output. The manner in which the equipment is used is determined by the manner in which a set of control panels is wired. By an appropriate choice of control panel circuits, it is possible to operate the equipment as a general purpose digital computer. Alternatively, it is possible to design special control panel circuits making possible the efficient utilization of the equipment as a special purpose computer for the rapid solution of a particular problem.

Specific methods will be given for the application of the IBM card programed electronic calculator to the calculation of the thermodynamic properties of combustion gases and the evalua- tion of the performance characteristics of fuel-oxidant systems.

(3)

resulting in the formation of the dependent constituents from components only. The mass action laws for chemical equilibrium can be put into the form

X I K ~ ( T ) P ~ - ~ ~ ~ x , ~ ~ I (4) where K , is the thermodynamic equilibrium constant, a function of temperature only (it is assumed here that the gas mixture obeys the ideal gas law), for the chemical reaction 3 leading to the formation from components only of the ith dependent con- stituent.

Equation 1 comprises a set of linear equations equal in number to the number of components of the system (usually, but not necessarily, equal to the number of different elements of the system). Equation 4 comprisee a set of nonlinear equations equal in number to the number of dependent constituents of the system. Together, the two sets are sufficient to determine com- pletely the composition at equilibrium of a multicomponent gas mixture. For fuel-rich carbon, hydrogen, oxygen, nitrogen sys- tems, an appropriate choice of components is

j = CO, Hz, H20, N2

with the dependent constituents,

i = CO,, 0 2 , 0, OH, H, NO, N, NHI, CH,

For fuel-lean carbon, hydrogen, oxygen, nitrogen systems, an a p propriate choice of components is

j = COz, H20, Op, N2

with the dependent constituents,

i = CO, Hz,O, OH, H, NO, N

The simultaneous solution of Equations 1 and 4 is most con- veniently carried out by an iterative method. If an approximate set of values of the mole fractions of the components, zJ , is selected, Equation 4 permits the calculation of the mole fractions of the dependent constituents, x 2 . The resulting set satisfies Equation 1 if the correct values of X, were selected. If the func- tion F, does not vanish, an improved set of values of xi is obtained by application of the Newton-Raphson method (19). If the func- tion F , is expanded in a Taylor series about the approximate set of values of rJ , with neglect of derivatives of second and higher orders, a set of linear equations results

CALCULATION OF EQUILIBRIUM COMPOSITION OF COMBUSTION GASES

A systematic procedure for determining the equilibrium com- position of multicomponent gas mixtures has been described in detail elsewhere (9, 3, 6, 7, 11). The conditions for thermody- namic equilibrium can be stated in a form that is particularly ap- propriate for routine calculation. The conservation of each ele- ment by the system requires that

Fj = 0 (1)

where i

and where xi and xi are the mole fractions in the equilibrium mix- ture of the i th dependent constituent and j t h component, respec- tively; q j is the mole fraction of the j th component in the hypo- thetical stoichiometrically equivalent mixture consisting of com- ponents only; uij are the coefficients of the chemical reactions

where the rth and ( r + 1)th approximations to the composition are related by

( 6 )

which can easily be solved for the fractional corrections, hi('), to the rth approximation to the mole fractions, xi, of the compo- nents. The superscript r denotes that the quantity is to be evalu- ated with the rth approximation to the composition of the system. The coefficients of Equation 5 are given explicitly by

z 9 ( r + 1 ) = zJ(r) [I + h J ( r ) ]

= p i Vj' - Ujjl

Vi) = U<j' ( U i - 1) xi i

where 6 j j f is the Kronecker delta.

November 1951 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

The relations for the determination of the equilibrium com- position, expressed by Equations 1 through 7, are given in a very general form and are widely applicable to a variety of specific systems. The same general computational methods are em- ployed to determine the equilibrium composition of different speciIic'systems. The relations to be employed for several par- ticular systems have been written down explicitly elsewhere (6).

The work required for the solution of Equation 6 oan sometimes be reduced by employing in each iteration fixed values of the par- tial derivatives of the Taylor expansion, calculated with the ini- tial approximation to the cornposition of the system (see last para- graph, 9). This approximation may increase the number of iter- ations required for conversion, but may markedly decrease the amount of computatiod required during each iteration. How- ever, the storage facilities of the computational equipment to be described below are usually insufficient to permit retention, through a series of iterations, of the necessary quantities.

The Newton-Raphson method for determining the equilibrium wnposition is baaed upcn a first approximation obtained from a Taylor series expansion of the equilibrium conditions. A zero order approximation, converging more slowly but requiring a con- eiderably simpler computational routine, iB obtained by retain- ing only the leading term of the Taylor series expansion, Equa- tion 5. This simple iteration procedure is equivalent to employ- ing Equation 2 to determine the mole fractions of the components and Equation 4 to determine the mole fractiona of the dependent constituents. However, the range of utility of this simple itera- tion prooedure is usually too limited to justify the formulation of the necessary computational program.

PUNCHED CARD METHODS FOR CALCULATION OF EQUILIBRIUM COMPOSITION

The calculation of the equilibrium composition by means of Equations 1 to 7 consists of a series of algebraio steps that crtn be reduced to a sequence of elementary arithmetic operations. Although it is possible to carry out the calculations with the aid of parallel-type punched card equipment fa), it is far more con- venient to employ a sequence calculator. The card programed calculator ib employed in this laboratory for this purpose in the manner indieated by the block diagram of Figure 1.

The e uipment is oontrolled through control anel circuits to perform L e elementary operations of arithmetic &cluciiig square root) on lOai 't numbers under the control of orders punched into order car$ and read in succession by the pro am control unit. Each order consists of a 10-digit number (war$ that s ci- fies the storage locations (addresses) of the factors upon wgch the operation is to be performed, the addreae of the storage looa- tion to which the result is to be transferred, the nature of the o er- ation to be performed, and in addition special instructions, i r r e quired, for shiftin the factors. The program for any algebraic problem consists ofthe set of order words which, carried out in se- quence, will lead to the solution of the robiem a t hand. The preparation o€ the pro ram for a particugr problem does not in- volve any control panefwiring.

A number of program controls are also provided. The load op- erations permit conditional or unconditional readin into storage of numencal information required for the solution ofthe problem. In the present example, these orders are employed to read into storage the values of the stoichiometric and equilibrium constants to be employed for a articular composition a t specified tempera- ture and raesure. $he unch and print orders permit printing and punc&ng into cards t i e final results of the calculation.

In problems of an iterative nature, it is neoessary to provide alternative computational routines and a t some point in the cal- culation to select the appropriate routine on the basis of criteria developed by the calculation itself. Thus, in the present example, if the function, Fi, does not vanish, it is desired to improve the estimate of zj and to oarry out an additional iteration. If the function Fi does vanish, it is desired to arrange the resulta in proper location for punching and to set up the maohine for oon- sideration of the next problem. In the general purpose

2473

computer, two different portions of the order card may contain instruction words. The machine normally obtains its instruc- tions from the normal insfuction field. However, the conditional transfer order makes possible a discriminatory transfer to an al- ternative transfer instruction field if, when conditional transfer ia ordered, the number in a specified storage location is negative, and the computer may be required to continue to obtain its in- struotions from the transfer field for any predetermined length of time. Transfer does not occur if the discrimination number is positive. By means of this order, it is possible to provide com- pletely automatic control over the course of the calculation with- out the neceaeity for intervention by the machine operator a t any time. The control panel circuits for the general purpose com- puter have been described elsewhere in detail (8).

ARITHMETIC ORDERS 1 C-A+8 2 C-A-8 3 C - A X 8 4 C=A+S 5 c-91

CONTROL ORDERS Conditional transfer Lord Olccliw b d Print Punch

Figure 1. Block Diagram of 10-Digit General Purpoae Calculator

The basic sequence for determining the equilibrium composi- tion of a system containing components of carbon, hydrogen, oxygen, and nitrogen requires about 125 order cards. Each or- der requires about one third second. An average of three itera- tions is required in a systematic program of calculations for each determination of the composition. Each determination thus requires a little less than 3 minutes, a t a unit cost of approximately $1.50.

EVALUATION OF THERMODYNAMIC PROPERTIES

The thermodynamic properties of the gas mixture are easily evaluated, using well-known thermodynamic formulas (1 ), after the composition has been determined. The mean molecular weight is given by

M = CMo (8 )

where

where Mj is the molecular weight of thejth component. The spe- cific enthalpy of the gas mixture, referred to the stoichiometri- cally equivalent mixture of the elements in their standard states a t 0 ' K., is given by

where i t is sesumed that the gas mixture obeys the ideal gas law, and where the summation is over all of the constituents of the mix- ture. In t h i R expression, ( p - %)x is the change in enthalpy when 1 mole of the kth constituent is taken from its standard state to the temperature of the mixture, and ( A,%)* is the energy increment for the formatiou of 1 mole of the kth cunstituent at the standard state from the elements in their standard states. The specific enthalpy is a function of pressure because the com-

2474 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. 43, No. 11

position is an implicit function of pressure. capacity at constant prassure is given by

The specific heat

(11)

tion with general purpose arithmetic orders to carry out, at high speed, subroutines that are comparatively simple in character.

The determination of the thermodynamic properties of a typi- cal mixture of known composition requires about 100 order cards. The calculation takes somewhat less than 1 minute to perform at a unit cost of about 60 cents.

where Cpk is the molar heat capacity a t constant pressure of the kth constituent at the temperature of the mixture and the sum is over all constituents. The adiabatic exponent can be calculated from the relation

where R is the gas constant. The specific entropy s is given by

where i$ is the molar entropy of the kth constituent a t unit pres- sure and at the temperature of the mixture. The density of the misture is easily evaluated from t8he equation of state.

PUNCHED CARD METHODS FOR EVALUATION OF THERMODYNAMIC PROPERTIES

Except for the term in the expression for the specific entropy arising from the entropy of mixing, the relations for the thermo- dynamic properties involve only a sequence of the elementary op- erations of arithmetic. The quantity, x log x , could be evalu- ated with the general purpose computer by the introduction of a punched card table of logarithms into the program cards with utilization of the selective load order to control entry into the table. Alternatively, the quantity would be evaluated by a card programed application of the series expansion. Each of these alternatives is excessively slow.

or---- I !

control

Order cards

- - ~ -___------- ~

output

Answer cards

ARITHMETIC ORDERS CONTROL ORDERS 1 C = A t B Conditionai transfer 2 C = A - 0 Load 3 C = A X B Selective load 4 C = A t B Prinf 5 C=-AlogA Punch

Figure 2. Block Diagram of &Digit Special Calculator

The card programed calculator can be efficiently employed for the evaluation of the thermodynamic properties of gas mix- tures in the manner illustrated by the block diagram of Figure 2. Control panel circuits are employed that permit the elementary operations of arithmetic (not including square root) on 8-digit numbers under the control of orders punched into order cards and read in succession by the program control unit. Each order is en- tirely analogous to those employed with the circuits for 10-digit general purpose arithmetic, and the same provisions are made for program control. By limiting the arithmetic to 8 digits and by eliminating the square root order, it is possible to reserve sufficient internal control facilities to permit the provision of a special order for the quantity - A log A , internally programed by control panel wiring. In this way, the quantity icr computed at high speed by the electronic calculator unit in a time comparable with that required for the arithmetic orders. This application illus- trates the possibility of employing internal program in combina-

CALCULATION OF EQUILIBRIUM COMPOSITION AND THERMODYNAMIC PROPERTIES OF COMBUSTION

GASES

The methods described here have been employed in an exten- sive series of calculations of the equilibrium composition and thermodynamic properties of combustion gases, carried out on the electronic numerical integrator and calculator (ENIAC) ( 4 ) . This computer is a high-speed sequence cltlculator with limited internal storage. It operates on a single address order system and, consequently, the programs differ in detail from those em- ployed in this laboratory. Because of its high speed, it is pos- sible to compute an equilibrium composition in an average time of about 4 seconds and to determine the thermodynamic proper- ties in an average time of about 3 seconds.

EVALUATION OF PERFORMANCE FACTORS OF F U E L OXIDANT SYSTEMS

The results of the evaluation of the thermodynamic properties at equilibrium of gas mixtures consist of a table in punched card form that is in every way analogous to the familiar Mollier chart. The table can be exploited in the analysis of working cycles in exactly the same way that such charts are employed (9, IO), utiliz- ing the punched card analogs of the usual graphical procedures. Although the cycle is very simple, the procedures are completely illustrated by the calculation of the specific impulse of a rocket fuel from a punched card table of the thermodynamic properties of the combustion gases.

The Mollier chart for the combustion gases of a rocket fuel is illustrated by Figure 3, The conditions of the combustion cham- ber are determined by the ordinate equal to the specific enthalpy, H I , of the intact fuel-oxidant mixture (referred to the same base as that employed for the combustion gases) and the constant pres- sure line for the chamber pressure, pc. The intersection of these two lines determines the abscissa describing the chamber adia- batic. For optimum operating conditions of a rocket, the ex- haust conditions are determined by the abscissa, 8, and the con- stant pressure line for the ambient pressure, PO. The intersection of these two lines determines the corresponding value of the specific enthalpy, H,, of the exhaust gases. The difference in specific enthalpy

(14) ( A H ) a = H. - Hf

is related to the specific impulse, I, by

1 I ENTROPY

Figure 3. Calculation of Specific Impulse from Mollier Chart

November 1951 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 2475

where g is the acceleration due to gravity.

(16) triggered by the finder card. The computer automatically col- lates the finder card with the table deck to determine the point of entry in the table, forms the proper differences, and carries out the required direct or inverse interpolation by means of the fourth

The punched card table of thermodynamic properties may be utilized in the calculation of specific impulse in the following atepa :

difference Stirling formula. The-choice between direct and in- verse interpolation ia effected by means of the position of a tog- gle switch on the control unit.

1. In the table of h(po, T), perform an inverse interpolation to determine the adiabatic flame temperature, To = T(p., HI).

2. In the table of s(pa, T), perform a direct interpolation to determine the adiabatic, S - s(pe, !Pa).

3. In the table of B(p0, T), perform an inverse interpolation to determine the exhaust temperature, TO = T ( p , S):

4. In the table of h(p0 T), perform a &re& lnte olation to determine the specific enthalpy, H., of the exhaust,% - h(p0, To). At the conclusion of these s tep , the application of Equation 16 to determine the specific impulse is immediate. This method of evaluation is independent of the assumption of the validity of the polytropic expansion law.

Anmar - card u r d 8 ORDERS CONTROLS

Interpolate dim! internal, triggered Interpolate inverse by finder card

Figure 4. Block Diagram of Interpolation Calculator

More complicated cycles involve the same kind of operations in the punched card table of the thermodynamic properties of the combustion gases, supplemented in most cases by similar con- siderations involving a supplemental punched card air table.

The interpolations are efficiently carried out by the card pro- gramed calculator, utilizing special control panel circuits that make the equipment an interpolation calculator. This method of operation is illustrated by the block diagram of Figure 4. In this case, the table deck, preceded by a “finder” card listing the argument for direct interpolation of the function for inverse in- terpolation, is read by the card reader of the program control unit. The table deck contains no control punohes of any kind. All control is internal, accomplished by control panel wiring and

- The time required to carry out the interpolations for flame

temperature and specific impulse is completely determined by the time required for the reader to read the table, and it is negligible in comparison to the time required for the construction of the ta- ble of thermodynamic properties.

ACKNOWLEDGMENT

This research is part of the work being done a t the U. S. Bureau of Mines on Project NA onr 25-47, supported by the Office of Naval Research and the Army Air Forces. The calculations on the electronic numerical integrator and calculator of the equilib- rium composition and thermodynamic properties of combustion gases were carried out under arrangements made with the Office of the Chief of Ordnance, Department of the Army.

LITERATURE CITED (1) Beattie, J. A., C h m . Reus., 44, 141-92 (1949). (2) Brinkley, 9. R., Jr., J . C h m . Phya., 14, 663-4, 686 (1946).

(4) Brinkley, S. R., Jr., and Lewis, B., Chem. Eng. News. 27, 2540-1 (1949).

(5) Brinkley, 8. R., Jr., and Lewis, B., U. S. Bur. Mines, Rep$. Inuest. 4806, in press.

(6) Brinkley, S. R., Jr., and Smith, R. W., Jr., “Proc. Scientific Computation Forum 1948,” pp. 77-82, New York, Interna- tional Business Machines Corp., 1950.

(7) Brinkley, 8. R., Jr., and Smith, R. W., Jr., “Proc. Seminar on Scientific Computation 1949,” pp. 58-63, New York. Inter- national Business Machines Corp., 1950.

(8) Brinkley, 8. R., Jr., Wagner, G. L., and Smith, R. W., Jr., “Proc. Technical Computation Forum 1950,” New York, International Business Machines Corp., 1951.

(9) Hottel, H. C., and Eberhardt, J. E., C h m . Revs., 21, 439-60 (19371,

(10) Hottel, H. C., Williams, G, C., and Satterfield, C. N., “Thermo- dynamic Charta for Combustion Procemes,” New York, John Wiley & Sons, 1949.

(11) Kandiner, H. J., snd Brinkley, S. R., Jr., IND. ENG. CHEM., 42, 850-5 (1950).

(12) Scarborough, J. B., “Numerical Mathematical Analysis,” pp. 178, 187, Baltimore, Johns Kopkins Press, 1930.

(a) IW., 15, 107-10 (1947).

R B O l I V l D Apfl 12, 1961.

Monte-Carlo Method for Solving Diffusion Problems

GILBERT W. KING A r t h r D. Little, he., and Massachnsetts Institate of Technology, Cambridge, Mass.

HE Monte-Carlo method is a new means of solving problems in physics and engineering made available by large scale computing machines. An automatic computer should not be

considered as a glorified slide rule that will do engineering calcula- tione as they have been done in the past, but as an organism that will attack the problems in a new and powerful way. Clae- &a1 mathematics is only a tool for engineera and physicists and is

not inherent in the realities with which they attempt to -,a]. It has been customary to idealize and simplify the mechanisms of the physical world in the form of differential and other types of equations of classical mathematics, because solutions or metho& of attack have been discovered during the last few hundred years with means generally available-namely, pencil, paper, and logarithm tables.

T