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class and home problems
Th e object of this colum n is to enh ance our readers ' collection of interesting and novel problemsin chem ic al engin eering. Problems of th e type that can be used to motivate th e stude n t by presentinga particular principle in class, or in a new light, or that can be assigned a s a novel home problem, arerequest ed as well a s tho se that are more traditional in nature, which elucidate dif f icult concep ts.Please subm it th em toProfessorJam es o.Wilkes and Professor T. c.Pap on as tasiou, ChE Department,Un ivers ity of Michigan, A nn A rbor, M148109.
AMUNDSON'S MATRIX METHODFOR BINARY DISTILLATION REVISITED
J.J.J. CHENUnivers ity ofAucklandAuckland, N ew Zealand Further, 1'1 and 1'2' the roots of Eq . (3), will also
satis fy Eq . (5). Th us
Eqs. (5) , (6), and (7) will be applied to binary distilla-tion. However, fir st we need to formalize some ofAmundson 's treatment.
A mundson'!' expressed the binary distillat ion prob-~ lem as a matrix difference equation . In thispa per, matrix power equations will be used to solveand simplify the same problem, making it suitablefor illustrating the ap plication of matrices in coursesof engineering math em atics or separations processes.
rt = a.rl + PrJ = a.r2 + P
(6)(7)
SOME RELATIONSHIPS IN MATRICESCons ider the matrix A of orde r 2:
A = [ a l a 2] (1)b I b2
whose characteristic equation, IA - rIl = 0, is
la~~ r b: ~ rl= 0 (2)or
r2 _(aI+ b2) r +( a Ib2- a2 bI)= 0 (3)where I is the unit matrix of the order of A. From theCayley-Hamilto n theorem, the matr ix A also satis-fies its own characteristic equatio n. Thus
A2 - (a l + b2) A +(a Ib2 - a2 bl ) I = 0 (4)where 0 is the zero matrix of the order of A.
By using equations like Eq . (4) for higher powersand substituting from the lower power equations, itca n be show n that
(5)where a an d p are numerical constants which de-pend on the matrix A a nd expone nt p.
Copyr ight ChE Division. ASEE 1991
50
BINARY DISTILLATION
Following Amundson , by assu ming constant vola-ti lity, the equilibrium line is
xY= - - (8)
A +Bxand the operating line is
y = mx+ b (9)Taking the top product composition as d, the us e ofa total conde nser gives YI = d. The plate numbers arecounted from top to bottom. The liquid leaving plate1 is obtained from Eq. (8). Thus
AYIXl = (10)
- BYl + l
and Y2' obtained from the substitution of XI from Eq.
Jo hn J. Chen is an Associate Professor in chemicaland materials engineering at the University of Auck -
....... ...,.;:,;;;~........... land. New Zealand.
Chemical Engineering Edu cation
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Simp lifying
I'I (Yn - Yp+n)+[Yp+ n(1- BYn)- (mA- Bb)Yn - b]r2(Yn - Yp +n)+[Yp+ n(1- BYn}- (rnA- Bb)Yn - b]
(21)
and assuming constant vola t ility a nd molal over-flow, the only unknown s are a a nd B. Furthermore, ~may be solved explicit ly in terms of a by using Eq .(18). Thus
[- BYnYp+n +Yp+ n - (rnA- Bb)Yn - b]aP= (19)Yn -Y p +n
It is now possible to eliminate a and ~ in Eq s. (6) and(7), and to evaluate p, the number of pla tes. Thefactors 1'1 and 1'2 a re the roots of th e characteris ticequation of the square mat rix given in Eq . (16 ), andthey may be readily shown to be
1'],1'2 =t[(mA - Bb+ 1 ) ~( mA - Bb+ l)2 - 4mA] (20)Dividin g Eq . (6) and Eq. (7), substitu t ing the valuefor ~ from Eq . (19), a nd eliminating a
(13)
(12a)
(12b)
*YI+n a lYn +a2YI+n=~=YI +n blyn +b 2
wh ere
(10) into the operating line equation [Eq. (9)], isY2 = mXI + b (lla)
(rnA - Bb)YI + b (llb)Y2 = - By] + 1
Now we can defin e y , the composition of thevapour leavin g the ( p+n)t~+plate, as
The plate number from which we begin the ste p-ping-off process is n. The value of yp+n when p = 1(i .e., Yl+) ' is thus given by
(18)
APPLICATION
(22)
Th e roots for the characteristic equat ions for aboveand below the feed are , resp ectively
(23)
(24)
(25)
(26)1'2 = 0.40920
I'f n --!-1'2
x
y = 0.75x+ 0.249
Y= 1.3773x +0.001886
1'] = 0.75130
Thus, the number of pla tes p betwen tray number(p -i n ) and n is given by
I'I (Yn - Yp+n)+ [Yp+n(1- Byn)- (mA- Bb)Yn - b]Cn-;-------'----;----T--'--~--~-----~1'2(Yn - Yp+n)+ [Yp+ n(I - BYn)- (rnA- Bb)Yn - b]p = - ----''--------'---'-----"--'------- - - ------'-
We shall now apply Eq. (22) to th e same problemconsidered by Amundson in solving the distillationof a 0.40 mole fractio n benzen e mixed with toluen eintroduced at its bubble point. Th e equilibr ium curveis give n by
Y=----0.41+0.59x
The top produce is 0.995 benzen e, and the bottom is0.005 benzen e. The operating lines above and belowthe feed are, respectively
and
[ Yp+n*]= [(rnA - Bb) b1]P [YIn] (16)
Yp+n
-
results in 9.54 plates above feed position.
Below the feed position, Eq. (22) may be a ppliedby taking n =9.54, i.e., Y9fi4 =0.549. We obtain y +~I . fi4by subs t itu t ing values for A, B, m , and b from Eqs.(23) and (25), and usin g 1'1 =0.99744 and 1'2 =0.56615.We obtain y +9.54 by substit uting x = 0.005 in to theeq ui lib rium fine to give yp+9fi4= 0.01 21. Thus
Applying Eq . (22) to abo ve the feed posi ti on ,n =1, y\ =0.995, inserting the a ppropriate va lues forA, B, m, a nd b from Eq s. (23) and (24) with referen ceto Eqs. (10 ) and (11), a nd usi ng r l = 0.75 130 and1'2 = 0.40920, we obtain Yn+\ by substitut ing the feedcomposition into the operating line as the feed is in -troduced at it s bubble point. Thus Yn+l = 0.549 (or0.553 using the lower operating lin e). ... '
..7 ,
1'1 = 0.99744
Appl ying the va lues1'1 ~ 0.75130
YI = 0.995
B = 0.59A = 0.4 1
1'1 = 0.99744
Y 9,54 = 0.549
B = 0.59
A = 0.41
1'2 =0.40920
Yp+l = 0.549
ill = 0.75b = 0.249
1'2 = 0.56615
Yp+9 .54 = 0.01 21
ill = 1.3773
b = 0.001886
(27) POWER OF SPREADSHEETSConti nued from page 49.stude nts . In sigh t into the relative importance ofva ria bles and sensitivity of results to changes in theinput can be gathered from such an exercise. Theease of changi ng input data al so allows instructorsto efficient ly check ca lcula tio ns made with differentcombinations of indep enden t va ria bles.
CONCLUSIONSOur experience with sp readsheet computing has
proved to us that it is feasible to provide ins tructionon spreadsheet use as part of the mass and ene rgybalances class . Within a time-frame of approximate lytwo hours , s t ude nts ca n learn su fficient fu nda me n-tal s to use spreads heets as a tool for solving a va ri-ety of probl em s in the class. After solvi ng five toeight problems , most of them ha ve enough confi-den ce a nd exper ience to apply the tec hniques infuture engi neer ing classes.
Th e use of spre ads heets a lso encourages organi-zation in problem solving wh ich hopefull y will carrythrough to the st udent's non-computer work. Theflexibil ity and conveni ence of spre adsheets allowsstudents to solve more mea ningful problems a nd toexamine the solutions in detail by manipula t ing in-dependen t va r iables to determine the ir effect . Thebuilt-in gra ph ics capability a lso helps to ti e togethergr aphica l a nd algebraic solu tio n techniques wh ensuch alternate methods exist for a given problem.
These va lues, when substituted in to Eq. (22) yield9.91 plates below feed position .
CONCLUSIONS
Th e binary distillation problem considered byAmundson wa s re-examined , and a simpler methodinvolving powers of matrices has been give n a nd a nexplicit solution obtained . This approach is suitablefor us e in engineer ing mathematics or separatio nprocesses courses to illustrate the application ofmatrices to engineering problems.
ACKNOWLEDGEMENTS
Th e author is grateful to his colleague, KevinFree, for improvements in the clarity of this paper.
REFERENCES1. Amundson, N. , "Applica t ion of Matrices a nd Finite Differ-
ence Equations to Bina ry Dist ill ation," Tra ns. AIChE, 42 ,939(1946) 0
52
REFERENCES1. Rosen , E.M ., a nd R.N. Adams, "A Review of Spreads heet
Usage in Che mica l En gin eering Ca lcula t ions ," Computersand Ch ern, Engg ., 1U 6 ), 723 (1987 )
2. Grulke, E.A. , "Us ing S preads heets for Teach ing Design ,"Chern. Eng . Ed ., 20 , 128 (1986 )
3. Dijk stra, E.W., A Discipline of" Programming , Pre nt ice-Hall Inc., En glewood CIiITs, N.J (1976 )
4. Rosen , E.M., "The Use of Lot us 1-2-3 Macros in Engineer-ing Ca lcu la t ions," Chern. Eng . s. 24 , 100 (1990)
5. Asp enplus , Asp en Technology Cor porat ion, Ca mbridge,MA
6. H imme lhlau, D.M., Basic Princip les and Calculations inChemical Engineering , Pren ti ce-H all , Inc., Eng lewoo dcurn, NJ (198 2)
7. Felder, R.M. , and R.W. Roussea u , Elementary Prin ciplesof" Chemical Processes, .Joh n Wiley & So ns , New Yor k(1986 )
8. Wankat, P.C., "Wha t Will We Remov e From t he Cu rr icu-lum to Make Room for X?" Chem . Eng . Ed., 21, 72 (J 98 7)
9. Lotus 1-2-3, Lot us Developm e nt Corporation , Ca mbridge ,MA
10. S up ercalc, Com pute r Assoc ia tes In ternation al , San -Iose ,CA
11. Smartware, In formix Soft wa re Inc. , Len ex a , KS12. McKean , R.A., a nd E.L. Ha nzevack , "The Hea rt of t he
Matter: Th e Engineer's Essential On e-Page Memo," Chem .Eng. Ed. , 23 , 102 (1989) ,'"
Chemical Engineering Educati on
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