[Luyben] Process Mod.. - Student subdomain for University of Bath

More documents

Recommendations

Info

NUMERICAL METHODS 97 T Newton-Raphson convergence. Generalizing Eq. (4.9), we get the recursive iteration algorithm : where Tn + 1 = new guess of temperature Tn = old guess of temperature f, = value off,,, at T = T. fi = value of the derivative off, df/dT, at T = T. (4.10) The technique requires the evaluation off ‘, the derivative of the function f(r) with respect to temperature. In our bubblepoint example this can be obtained analytically. AT, = PC;: - p = x &WT+BI) + (1 _ $ &‘z/T+Bz) -xA,P, -(l -x)&P; = T2 (4.11) If the function .were so complex that an analytical derivative could not be obtained explicitly, an approximate derivative would have to be calculated numerically: make a small change in temperature AT, evaluate fat T + AT and use the approximation (4.12) A digital computer program using Eqs. (4.10) and (4.11) is given in Table 4.2. The problem is the same bubblepoint calculation for benzene-toluene performed by interval-halving in Table 4.1, and the same initial guesses are made of temperature. The results given in Table 4.2 show that the Newton-Raphson algorithm is much more effective in this problem than interval halving: only 4 to 5
98 COMPUTER SIMULATION TABLE 4.2 Example of iterative bubblepoint calculation using “ Newton-Rapbson” algorithm C C MAIN PROGRAM SETS DIFFERENT VALUES FOR INITIAL C GUESS OF TEMPERATURE C SPECIFIC CHEMICAL SYSTEM IS BENZENE/TOLUENE c AT 760 MM HG PRESSURE C PROGRAM MAIN COMMON Al,Bl,A2,B2 C CALCULATION OF VAPOR PRESSURE CONSTANTS C FORBENZENEANDTOLUENE C DATA GIVEN AT 100 AND 125 DEGREES CELCIUS A1=L0G(2600./1360.)/((1./(125.+273.)) - (L/(100.+273.))) BkLOG(2600.) - A1/(125.+273.) A2=L0G(1140./550.)/((1./(125.+273.)) - (1./(100.+273.))) B2=LOG(1140.) - A2/(125.+273.) C SET LIQUID COMPOSITION AND PRESSURE x=0.5 P=760. WRITE(6,3)X,P 3 FORMAT(’ X = ‘,F8.5,’ P = ‘,F8.2,/) TO=80. DO 100 NT=1,3 WRITE(6,l) TO 1 FORMAT(’ INITIAL TEMP GUESS = ‘,F7.2) T=TO CALL BUBPT(X,T,DT,P,Y,LOOP) WRITE(6,2) T,Y,LOOP 2 FORMAT(’ T = ‘,F7.2,’ Y = ‘,F7.5,’ LOOP = ‘,13,/) 100 TO=T0+20. STOP END iterations are required with Newton-Raphson, compared to 10 to 20 with interval-halving. If the function is not as smooth and/or if the initial guess is not as close to the solution, the Newton-Raphson method can diverge instead of converge. Functions that are not monotonic are particularly troublesome, since the derivative approaching zero makes Eq. (4.10) blow up. Thus Newton-Raphson is a very efficient algorithm but one that can give convergence problems. Sometimes these difficulties can be overcome by constraining the size of the change permitted to be made in the new guess. Newton-Raphson can be fairly easily extended to iteration problems involving more than one variable. For example, suppose we have two functionsf,(,,, X2j and f,(,,, X2) that depend on two variables x1 and x2. We want to find the values of x1 and x2 that satisfy the two equations f1(x1* x2) = 0 and hfx,, xz) = 0
Page 1 and 2:
I WILLIAM WIlllAM 1. LUYBEN PROCESS
Page 3 and 4:
The Series Bailey and OUii: Biochem
Page 5 and 6:
PROCESS MODELING, SIMULATION, AND C
Page 7 and 8:
ABOUT THE AUTHOR William L. Luyben
Page 9 and 10:
CONTENTS Preface Xxi 1 1.1 1.2 1.3
Page 11 and 12:
CONTENTS .. . xlu 5.7 Batch Reactor
Page 13 and 14:
CONTENTS xv 10.3 10.4 Performance S
Page 15 and 16:
comwrs xvii 151.3 Transpose of a Ma
Page 17 and 18:
CONTENTS Xix 20.4 Sampled-Data Cont
Page 19 and 20:
. Xxii PREFACE Another significant
Page 21 and 22:
PROCESS MODELING, SIMULATION, AND C
Page 23 and 24:
2 PROCESS MODELING, SIMULATION. AND
Page 25 and 26:
4 PROCESS MODELING, SIMULATION, AND
Page 27 and 28:
6 PROCESS MODELING. SIMULATION, AND
Page 29 and 30:
8 PROCESS MODELING, SIMULATION, AND
Page 31 and 32:
10 PROCESS MODELING, SIMULATION, AN
Page 33 and 34:
12 PROCESS MODELING, SIMULATION, AN
Page 35 and 36:
CHAPTER 2 FUNDAMENTALS 2.1 INTRODUC
Page 37 and 38:
FUNDAMENTALS 17 big, complex system
Page 39 and 40:
FUNDAMENTALS 19 z=o I z + dz z=L FI
Page 41 and 42:
FUNDAMENTALS 21 The minus sign come
Page 43 and 44:
FUNDAMENTALS 23 Molar flow of A lea
Page 45 and 46:
FUNDAMENTALS 25 For liquids the PP
Page 47 and 48:
FUNDAMENTALS 27 Flow of energy (ent
Page 49 and 50:
FUNDAMENTALS 29 The units of this f
Page 51 and 52:
FUNDAMENTALS 31 FIGURE 2.9 Laminar
Page 53 and 54:
FUNDAMENTALS 33 Then Eq. (2.48) bec
Page 55 and 56:
FUNDAMENTALS 35 Wenzel in Chemical
Page 57 and 58:
FUNDAMENTALS .37 This exponential t
Page 59 and 60:
FUNDAMENTALS 39 a direction 0 = 45
Page 61 and 62:
EXAMPLES OF MATHEMATICAL MODELS OF
Page 63 and 64:
EXAMPLES OF MATHEMATICAL MODELS OF
Page 65 and 66: EXAMPLES OF MATHEMATICAL MODELS OF
Page 75 and 76: EXAMPLES OF MA~EMATICAL MODELS OF C
Page 81 and 82: EXAMPLES OF MATHEMATICAL MODELS OE
Page 107 and 108: 88 COMPUTER SIMULATION Our discussi
Page 109 and 110: 90 COMPUTER SIMULATION lation packa
Page 111 and 112: 92 COMPUTER SIMULATION For this rea
Page 113 and 114: 94 COMPUTER SIMULATION TABLE 4.1 Ex
Page 115: 96 COMPUTER SIMULATION Clearly, the
Page 119: I()() COMPUTER SIMULATION Settingfl
Page 589 and 590:
ANALYSIS OF MULTNARIABLE SYSTEMS 57
Page 591 and 592:
ANALYSIS OF MULTIVARIABLE SYSTEMS 5
Page 593 and 594:
ANALYSIS OF MULTIYARIABLE SYSTEMS 5
Page 595 and 596:
Page 597 and 598:
Page 599 and 600:
FIGURE 16.5 Multivariable INA (- 1,
Page 601 and 602:
Page 603 and 604:
Page 605 and 606:
ANALYSIS OF MULTIYARIABLE SYSTEMS 5
Page 607 and 608:
Page 609 and 610:
Page 611 and 612:
Page 613 and 614:
DESIGN OF CONTROLLERS FOR MULTIVARI
Page 615 and 616:
Page 617 and 618:
Page 619 and 620:
Page 621 and 622:
Page 623 and 624:
Page 625 and 626:
DESIGN Ok CONTROLLERS FOR MULTIVARI
Page 627 and 628:
Page 629 and 630:
show all

[Luyben] Process Mod.. - Student subdomain for University of Bath

Create successful ePaper yourself

Delete template?

Save as template?