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l or tory SIMPLE

PRO ESS DYN MI S EXPERIMETS RINIVAS P AL AN KI , SHAK S AMPATHFAMU FS U College of Engineering Tallahassee FL 323 10 6 6

Impulse input

_ ..J

. . . .... -- >- To d rain

h ,Tank 1

t Main InletI II I- - - - : , T - - ~ Overflow pipeI to drainIIIII

r ...L.1. _ _ _ J

Copyright ChE Division of ASEE 1997Chemical Engineering Education

Tank 2 - ...--1

Figure 1. Sch em atic rep resentation of theexperimenta l set up

Tank withconstant head.

Srinivas Palanki is an assistant professor ofchemical engineering at the FAMUFSU Col-lege of Engineering. He received his BTech de-gree in chemical engineering from the IndianInstitute of Technology Delhi in 1986. and hisMS and PhD degrees from the University ofMichigan in 1987 and 1992, respectively. Hiscurrent research interests are in batch processoptimization and nonlinear control.

Vishak Sampath is a doctoral candidate inthe Department of Chemical Engineering atFlorida State University. He received his BTechdegree in chemical engineering from the Insti-tute of Technology Banares Hindu Universityin 1994. His research interests include chemi-cal process simulation optimization and con-trol of dynamic nonlinear processes.

EXPERIMENTAL SET-UPA sc he matic of the experimental se tup is show n in FigureI . The re are two tan ks co nnect ed in se ries. with eac h tank

having a cro ss-sec tional area of 4 8 .65 cm2. Water is fed toTank I fro m an overhead res ervo ir. T he res ervoir has anoverflow pipe n r the top and is co ntinuously supp lied withwater. T he ove rflow pipe e nsure s that a constant flow rate ismainta in ed at the inlet of the first tank. A strip of masking

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Process dyna mics is co ncerned with analyzing the timedepe nde nt beh avior of a process in res ponse to aninput c hange . Unde rstanding the dynamic behav io r isessential fo r process design. for se lection of opt imal operating co nditions, and for impl ementing process contro l strategies. but a majorit y of ex periments in a typica l unit operation s laborator y focu s on the steady state behavior of che mica l processes. In this paper . we will descr ibe a simple. lowcos t experi ment fo r ana lyzing the dynami c beha vior of asec ond-order system. Whil e the ex pe riment is ea sy to perfo rm, it requi res the stude nt to co mbine analytica l as we ll asco mputational ski lls to ana lyz e ex perimental data.

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tape is attached to the side of eac h tank so that the water level at any given timec n be marked of f on the tape. The flow rates to the first tank and the secondtank can be adjusted by ball valves.

Initially, the water levels in the two tanks are at steady state. The objective ofthe experiment is to pred ic t ho w the w ter level in the two tanks ch nges withtim e when a beaker full of water is suddenly adde d to the first tank. and to verifythis predict ion experimentally .

simp le low costdynamics experiment

THEORY is described inAssuming that the cross -sectional area of both tank s is uni form and equal to A.

and that the density of water is constant. an unsteady-state mass balance aroundeach indi vidu al tan k res ults in the followi ng equations:

where qin and ql are the vo lumetric flow rates o f the inle t and outle t strea ms of nk I. and g2 is the volumetric flow rate of the outlet stream of Tank 2. Theflow rates in the ex it lines of the two tanks depend on the pressure drop. which inturn depends on the water levels h i and h2 in the two tanks.

less than

this paper Theapparatus costs

1 to build The

Inti

LINE R MO EL experiment introduces

whe re R I and R2 are the flow res istance terms of the pipes exi ting from nk Iand Tank 2. respectively.Substituting Eq. 2 into Eq. I) . and putting in mtr ix form. we get

If a linea r head-flow relationship is ass umed. we ge tanti 2) the concept ofprocess

dynamics andillustrates the

3 differences betweenlinear and

At steady state tIll l tit tit nonlinear model

hu s, from Eq. 3) predictionanti 4) The students are

where his and h2s are the steady-s tate values of the wa ter le vel in Tank I andTank 2, respectively. and qins is the steady-state flow rate.It can be easi ly show nl l l that when qin is subjected to an imp ulse change of

strength M. and R I F R2, the impulse response is given bytested on their

analytical

5) well their

computational skillsEq. 5) pro vides analytical ex press ions that predi ct the wate r levels h i and h2with time .w nt r 99 65

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NONLINEAR MODELTh e valve disc harge rate in eac h ta nk ca n be model ed by

the followi ng square-roo t Imv:121

6)whe re C 1 and C2 are the valve coefficients of va lves I and 2.respectively. and dep end on the val ve openings .

u stituti ng Eq. 6) into Eq . I). the follow ing nonlinea rmodel is obtai ned:

5 . T ake a beaker of water about 400 ml) andmeasur e the vo lum e o f wate r in a graduatedcy linder. Ad d this beaker of wa te r to the first tan k.At the sa me time , start the stop wa tch and ma rkoff the level of wa ter in bo th tan ks.

6. Mark off the le vel of water in bo th tanks in 10second inter vals for 120 seco nds.

7. Measure the water le vel recorded with time on themask ing tape attached to eac h tan k.

S. Plot the wa te r level in eac h tan k versus time.

7)

At steady state,

dt dt

9 . Using the steady-state values of the water le vel inea ch tan k and the steady-s ta te flow rate, calculatethe flow res istance R, and R2 in the linear modelusing Eq . 4) .

10. Plot the analytical solutio n give n by Eq . 5) on thesame graph as the ex perime ntal dat a.

I I. Using the stea dy-s tate values o f the wa ter le vel inThu s. fro m Eq . 7)C - l in\ - - -h is le Steady State Measurements

8)W he n a be ak er o f water o f vo lu me M is suddenly add ed toTa nk I, the initial co nditions of the system represent ed byEq . 7) are give n by

A 48.65 emhi 24.5 emh,. 16.6 emM 400 em

Th e nonlinear model rep resent ed by Eq. 7) canbe integra ted numerica lly usin g the initial co nd ition s give n by Eq . 9) to predict how the wate rleve l in the two tanks changes wit h time.

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eac h tank and the steady-state flow rate. calculatethe va lve coefficients C 1 and in the non linearmodel usi ng Eq. (8) .

12. Integrate the nonlinear mod el numeric ally usingthe initial conditions given in Eq . (9).

13. Plot the numerical solution given of the nonlinearmodel on the same graph as the experimental data .

14. Comment on the acc uracy of the linear model andthe nonlinear model.

PI L RESULTS ND DIS USSIONTab le I shows the experimental values obtained in a typil experimental run . Figure 2 shows the prediction of theear model and the nonlinear model along with the experient al data point s. A program in MTLAB \I [31to genera tegure 2 is shown in Tabl e 2. We note that the nonlinearod el prediction of the experimental dat a is better than the

inear model prediction .A compa rison of the initial volume of wate r in Tank I andank 2 with the vol ume of water added at the initia l timehows that there is a 33 .6% deviation from the initial steadytate of Tank I. Th is deviation may be too large fo r the linea rodel to predict dynam ic behavior acc urately.It is ass umed that the beak er of wate r is added instanta-

neou sly. Th is add ition takes finite time, however , and so theuse of an impu lse response is an idealization of the actua lpulse response that mig ht lead to some disc repancy betweenpredicted and experimental values. Errors in the experimentcould also resu lt due to lack of coordina tion between thestudent who is keeping time and the students who are marking of f the water level in the two tanks.

HOMEWORK EXER ISE - - - - - - - I I. Derive Eq. (5) . II 2. The derivatio n of Eq. (5 ) ass umes that R) : II Under what physical conditions would R1 = II Derive the impulse response when R 1 = II 3. For what magnitude of the impulse input M is the II linear model predi ct ion close to the nonlinear II model predicti on? II 4 . If the sys tem is subjected to a ste p input of II magnitud e M. what are the predictions of the II linear and nonlinear models? IL ON LUSIONS

T LE 2MATL Program

Numerical ln tegration of Nonlinear Modelto= 0: f = 120 :xO = 132.7 16.6]:II.xl = ode23 ode . 10. if: xO): Analytica l So lution of Linear ModelI I = lillSpl/ce O. 120. 100):1 1= 24.5 + S.222 * exp -111l 2.6):12 = 16.6 + 17.435 * exp -111l 2.6) - exp - I I/ S.562) ): Experimental Data12 = 10: 10 : 20: 30: 40: 50: 60: 70: SO 90: 100: 110: 120]: I = [32.7: 30.S: 29.5: 2S.2: 27.4: 26.S: 26.3: 26: 25.7: 25.3: 25.2: 25: 24.9]2 = [16.6: IS.6: 19.2: 19.4: 19.2: IS.9: IS.5: IS.2: 17.S: 17.6: 17.4: 17.2: 17]

% Comparison between sol ution s1 10/ /. .r, II. 1 1, - I J. 1 2. -- , /2 . J.'+ ' .12. 2, function xd ot = ode I..r)xdott I) = 1.942 - 0.392 ' sqrl x I:xdo l 2 )= 0.392 * sq rl x I)) - 0.477 * sqrt x 2 :

illIeI / 997

A simple low-cost dy namics experiment is describedin this paper. The apparatus cos ts less than 100 tobuild. The experime nt introduces the concept of pro cess dynamics and illu strates the differences betweenlinea r and nonlinear model pred iction . The studen ts aretested on thei r analytica l as well as their computationalskills .NOMEN L TURE

A area of cross section of Tank I and Tank 2. cnr'valve coefficient of Valve Ivalve coefficient of Valve 2water level in Tank I. ernwater level in Tank 2. ernvolumetric flow rate into Tank I. cmvs ecvolumetric flow rate out of Tank I. cnr'zsecvolumetric flow rate out of Tank 2. cnr'zsecflow resistance of Pipe Iflow resis tance of Pipe 2time. sec

REFEREN ES1. Seborg, D.E., T.F. Edgar , and D.A. Mellicharnp, P rocess

Dyn mics nd Con trol, John Wiley an d Sons, New York,NY (1989 )2. Perry, R.H ., and C.H. Chilton, Ch em ic l n g in eersHa n d book , 5th ed., McGraw-Hill, New York , NY (1973 )3. MATLABT , The Mat hwork s, Inc Cochitua te Place,Sout h Na tick, MA 0

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