Fiyst ooder Spter the fya otder ssitm becaye Thi epagsenes byfye Sysiem Can be Yder diffexentialeuaton Mexcuythemcm Lquid level Mixin tank eler syAtem spornxe from Tangient Osde oder Syaiem G1l&AOl F i l 35tance MeiCu thetmomeHI: SuxYoundin &m >Film eastane . 1fluid Fuid MevcuN Fim Gle unll estance mercu meYCu All xesigtane heat Actua the cluid Au pho w Allke ygsstonce the balb fusoandin at any (.e, the xexik tanCa ofoved by meYCuM nleced). l a nd i A the to eot tamzfer Koi des in the flm thevma capacity n the meYLu ipskant, the mercu oyhuns the Fsthe mo a unifsim mpeature thioh out i) The Gla walcontainin Cotiact duing the tay ent I t A aAuned that the Thi meagy that temPevatue besove t h time. mesCur dot not espand o) xes pone thcmomeje i a t iitially &are time Zeo, there A is no Change in time r o , the At Some Ome theYmbmeteY usill easubjeced Chovae n the usioundinj By aPplyin the 4emperatue z(t). usiedtole eneray balan@ Tnput Nate Out Put Taje = Rate of accumulokion hA Ca) dt wher A Ssoce oxea for af b heat tiaytfelf) Cheot Capacrty ofmeCu m1 = mas omeyuY in bulb t ime h = film Co-efictent fot heat bsaef *Aboe equakonStoes that the, Kate of flous f heot through the flm exstance fuvounding the bulb Cases the inBevna enexy of the mercury to ineyee at at the ame Boe The incvese in manifeAtd bj a change in nternal tmperatuNe and Coieponeling pangion o mercu, hch me cubey h Column ov) e deend heqy ene a cauzes the eading o on the flou Jate nd P1ofey ties of the&us5Ounding flm and the dimeryiops of the bulb Fosea&tae Cohditfon, eg"0 Can be vitth aA, hA (-s) =0 t<0 9 Noud esO - evO dt mc (9-1s ha-7--1 dt Cayant) ot Let y defne Vaviables, dev'otion Vayialbles: differ ende dy blus Vaviales and ther Steady Atoe yalue. X = -s hA X-Y) = d me mC X-Y = Time contan t X-Y= T dY dt trafstrya on both id, APPinaplo.ce L(X-Y) =TLdy At ime =o 4x)- YE)- * s Y)- yo) xs) - yC) out Put Value zeYd YCo) = TS Y$) TS YS) + yS) x ls)= (1+ TS) YtS) XS) Out Put inPut = Y) Xs) TFTS s xonfer +1 YIS) LLC-1) ( o )o XS L(7-1 fanckon of the Syem Phsica Syptem fo hich the xelaton aplace tsaafims of 7p and o/p deviation Any torm of The eg" called ato the a betoeen the fysi ovder 48km laplace tvanAfosm of the op to the YP calles taptfex fanc hon o y o.ye called deviaaon Vasiales. The eaßon foX the e Vovialles * ofdevaston Vaviae becae, thee 03 fee fom boundavy Condihong taled time Cortant for mecuTy thevmomek mC T A ()Resprye oYder procep for drfeent for Cn of unctgs S standard foY of 1 orde Y tragejer tEROun Cton X) O S input ult) = us) T4 CResFopge o t<0 Kp 14) S at20 . ( fist Fox ovdeY System, y) S+ X(S) yIs) XS) FS+ YS) = TS+ S YS)= A TS Y) ANous taking S+ inVeyse la.pla.ce tros fo Ye) =A -e t20 Theunchon yt) seachgn 3.2X itultmale value. hen t=T Se, time Cortont Can be defned a athe time sequived for the seGch 632 1Ponse, Yt) to oitultimae tme Value. Stvni lar y, Y t) t 2T t3T t = o: 632 A Y t ) = 0865 A Yt) = o:15A t)o.98 A o-632 impulpe poryse ResFane inpuleP X= A e X t) = e X S) = XS) A A (unit impule input to Ys) X) TSt YS) = XC) Y TSt = S bytaking inVese loplace tsamfosmation, Ytt)= tT| 0 Ko XS) and TS+1 TYe) x (s)= A YS) XS) K Ys)=KA ( YIS)YIS) = KAt/ T ) yC) + pA unit tmpulse rspore fos a oider Syatem Romp mput CRe Poyae Ramp input Th uncton incxean% ines depcribed b ith time ond euatio the t0 t20 X = bt Slope-b Puse t input X(S) =X()edt A t Ae Jt t0 tw -S Ae 5 A X()=A - -3) Sts tt Stnugoidal ReAponAe w Vevy ThyYery impovtont the Stabiltyf eful ond in find in Control Aystem t40 At) At) Asin t X(S) A 20 A S 1 cyele A Amplitude 6 Fe guency, sad/s Ditonce Peviod blu tao peaks 21T TS+ XS) Y) YS) = XS) TStIJ =A TS+ bCommonepesaHon method TS4 P A (S+)(Ts+1) A (s Tst) TSt S+R st P(S)+ (0$tR) (Yst )(rst) the P(s++(9$+R) Es*) CYS+) AG A = Ps'+Pw+ TS+ 95t RTS R A = s(Pt9T)+ s(p+TR)+ P+R P+t=O P=-9T P-9T -TR +TR=o A IT A= P+R R A P P=A T AW - RT R A R It A wTY Y S) = - A9T S ltus (Sh) ts Yt) Yt) ) A As By tkn A nveyae lapla.ce tafoYmahon, AT A c9s t 4A Sin t I+ AwT-t/7ATcoA A ut t A i n 9t in tono medric eloon sin (P+) + Sin (+ 8) uhere tan (4) dentity above eq" above Yt) SinLt+ T wheve, Find A A 9T wrteh4 be can Total out Put = to(-sT) fo inysoida) APonse Value af y (tt) StRo.dy Atat hen A i n in ( - t t ) The out Put a egual to that f Awplite Raho Ae aVe sith frequerc s input Agnal The xato of out Put toinput anplitude. AR = (3 Phot oad pheaa a f g>o if 2o Cmpitude Lag Lead Liuid ( Level Rotape Syskm):- (1 order Tank q9- Th AA4em Corn f a tank afUnioím C9s-Bechon al area A hich to attached 3etance ' . Such a pipe, () a 9 a eit M e Aume Out let E 9,t) tho+ the though the vAana s bthe E Ro ffat)-S It) = byvidin t) lin2o R bolan de Inle flo Valve Volume sic flous Ya4eVoltime seaeloled to the heod If ht) ee accunu lahan AL) t th bo-th Aide . ) =A dh dt elatonakie. above eqh SubstituR. 9 A dh dt Som Stodj Aate Roe of aCuiulation=o] R btta Su fom e e O hh-hs (1-3)-(-) - A Ch-hs) d +A =-hs) dt 9-4) 9 Ad tokin loplace trorfom, c0 wV ) +ALsH)- R 96) =LH)+AS H() tARS R HS) (S) HS) R () SAR+) R TSt) he -AR M * R coled Find ultimate Nolue of Stead Value H+), im t ate ain. o HE). e., {ead Zto lims H$)> So H t) fos unitA4ep input, t0 9t) t20 PS)= Sulstiue e n Ht) t Subtitute Qs) to a pply ti im Ht) mit S es lin S S->0 in te R (TS+-1 o) aboe e im S>o (TS +1) S S-So, Ht)=R when, ) = 5 This Show thot ulote change in H chonge in 9 inply R. for un't Cote 2 Fsom e - Fo Steodyy Stoye then HS) *-9%s G) St where , if Ch-hs) R H S) = . 0 R cone O. From e T'St1 R TSt1 s) Minin R HS s)TSt1 (1* otdes 8m): Tank W the mixing PyoCeM Conider of Solution flos Contaning diolved alt in bich at a stream Volumetic floa eV vate Corytant atonk af ConAtat into Volume v . hold *the Conentyo4tion moyf Salt Volum Volume o Salt a in enterinStream, rth ime. Aimi desxed It elon out to to detetmine ConCentration let tsasfev func on y tothe inlet Con Cen a i o n Ma balance eq 9-93= Fo dt Sttady t o , sSbtract oth the quos* dt x 9Y =v dt x-y -Y NO dy dt takin lapla.ce rard fosm, Xs)-Ys) = T |S Ye)-o Xs)=T'S Y(S) -+ Ys) xS) = (rs+1) Ys) x S+| Y) 3whee, T= Sumnayy o mc T= 0 hA T the momeRY, level fov liguid AR to mixin T izinq fo RC tonX ene RC poce, pyoCC% CiCut balane Sttam (o") Elecici Heatn Yoce A Stream at tmpeY akureT T fd a dded t» the tonk. Heot to the tank by meoa T electic keae The af the tonk ellmixed etit S+Se am Flous KoR an the tem pexahure T bh Yat of of ener flo enegs flb in out Teey flow in f%om heatr Ra of aCcumu latoh oeneyn ton k + 1 T 2 1 1) s+) whee, TS