Instrumentation & Process Control

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1 Chemical Engineering (GTE & PSU) Poal Correpondence GTE & Public Secor Inrumenaion & Proce Conrol To Buy Poal Correpondence Package call a Poal Coure ( GTE & PSU) 5 ENGINEERS INSTITUTE OF INDI. ll Righ Reerved 8-B/7, Jia Sarai, Near IIT, Hauz Kha, New Delhi-6. Ph

2 Chemical Engineering (GTE & PSU) GTE 5 Cu-off Mark BRNCH GENERL SC/ST/PD OBC(Non-Creamy) Toal ppeared Chemical Engineering C O N T E N T. INTRODUCTION OF LPLCE TRNSFORM. 3-. TRNSFER FUNCTION PHYSICL EXMPLE OF FIRST ORDER SYSTEM RESPONSE OF FIRST ORDER SYSTEM IN SERIES HIGHER ORDER SYSTEM THE CONTROL SYSTEM CONTROLLERS CLOSED LOOP TRNSFER FUNCTION STBILITY INTRODUCTION TO FREQUENCY RESPONSE CONTROL SYSTEM DESIGN BY FREQUENCY RESPONSE DVNCE CONTROL STRTEGIES MESUREMENT OF PROCESS VRIBLES GTE QUESTION & SOLUTION PRCTICES SET-I PRCTICES SET-II.. -4 Poal Coure ( GTE & PSU) 5 ENGINEERS INSTITUTE OF INDI. ll Righ Reerved 8-B/7, Jia Sarai, Near IIT, Hauz Kha, New Delhi-6. Ph

3 Chemical Engineering (GTE & PSU) CHPTER- INTRODUCTION OF LPLCE TRNSFORM Need For Proce Conrol Why proce conrol ubjec in Chemical Engineering? When we run a kineic experimen, how do we mainain he emperaure and level a deired value? How do we manufacure produc wih conienly high qualiy when raw maerial properie change? How much ime do I have o repond o a dangerou iuaion? There i a need for coninuou monioring of he operaion of a chemical plan and exernal conrol o guaranee he aifacion of operaional objeciviie. conrol yem mu: Suppre he influence of exernal diurbance Enure he abiliy of a chemical proce. Opimize he performance of a chemical proce. Every engineer need baic knowledge abou conrol. Wih hi inigh, we will be able o deign plan ha can be conrolled afely and produce high qualiy produc Linear differenial equaion of order n n n d y d y Ln ()() y...() n n n y Linear differenial equaion arie from mahemaical modelling of chemical procee. Ue of Laplace ranform Laplace ranform offer a very imple mehod of olving linear differenial equaion. Uing Laplace ranform, a linear differenial equaion i reduced o an algebra problem. (which i impler han olving differenial equaion direcly). DEFINITION OF LPLCE TRNSFORM: The Laplace ranform of a funcion f() i defined a f() which can be find according o he equaion f()= - f()e Noaion of Laplace ranform of f() i {f()}=f() Example. Laplace ranform of funcion F() e 4 f()= 4e = = Poal Coure ( GTE & PSU) 5 ENGINEERS INSTITUTE OF INDI. ll Righ Reerved 8-B/7, Jia Sarai, Near IIT, Hauz Kha, New Delhi-6. Ph

4 Chemical Engineering (GTE & PSU) 4 {4}= FCTS BOUT LPLCE TRNSFORM. The Laplace ranform i no defined for he funcion f (), when he value of i le han zero.. The Laplace ranform i linear. Mahemaically af ()()()() bf a f b f a and b are conan. Where 3. Laplace ranform of he funcion f() exi if he inegral f () e ake a finie value (i.e. remain bounded) 4. Laplace ranform i a ranformaion of a funcion from ime domain (where ime i independen variable) o domain (where S i independen variable) S i a variable defined in complex plane (i.e. S = a + jb) LPLCE TRNSFORMS OF SIMPLE FUNCTIONS. The ep funcion f () [u()]=()e e L S 3: L (Sep funcion of ize ) = /S When funcion i uni ep. i.e. u(),, L(() u S. The exponenial funcion f () = o o a e o -a -(+a) - -(+a) {e }= e = e +a Poal Coure ( GTE & PSU) 5 ENGINEERS INSTITUTE OF INDI. ll Righ Reerved 8-B/7, Jia Sarai, Near IIT, Hauz Kha, New Delhi-6. Ph

5 -a {e }= +a Chemical Engineering (GTE & PSU) 4: Similarly, a L e S a 3. The ramp funcion o o f () = a o - {a()}= ae - a a = -e + a= 4. The ine funcion o o { } F ink o (ink)= - ink e - -e =(.in k+kco k) +k K {in k}= +K Funcion Graph Tranform u() u() n u() n! n e -a u() a Poal Coure ( GTE & PSU) 5 ENGINEERS INSTITUTE OF INDI. ll Righ Reerved 8-B/7, Jia Sarai, Near IIT, Hauz Kha, New Delhi-6. Ph

6 n a e u() Chemical Engineering (GTE & PSU) n! () n a in k u() k k co k u() k inh k u() k k e e a a in k?() u r co() k u k () a k a () a k (), uni impule (), uni pule () e S S LPLCE TRNSFORMS OF DERIVTIVES a) Fir order derivaive df() =f()-f(o) b) Second order derivaive d f d df df df() = = - = =f()-f(o) -f(o) = f()-f(o)-f(o) c) n h order derivaive Poal Coure ( GTE & PSU) 5 ENGINEERS INSTITUTE OF INDI. ll Righ Reerved 8-B/7, Jia Sarai, Near IIT, Hauz Kha, New Delhi-6. Ph

7 n d f n Chemical Engineering (GTE & PSU) = f()- f()- f()...f()-f() n n- n-() n- n- Example. Find he Laplace ranform of he funcion dx dx + +x=, x(o)=x(o)= Soluion : dx = x()-x(o)-x(o) dx =x()-x(o) x =x() We ge S x()-sx(o)-x(o)+sx()-x(o)+x()= S (S +S+)x()= S x()= ( ++) Laplace ranform of an inegral if f() =f(),hen o f() f() = Inverion by parial fracion In he following example, he echnique of parial fracion inverion for oluion of differenial equaion i repreened. Example.3 olve he following equaion for x() Soluion : dx o x o 3 x() Taking Laplace ranform of above equaion dx = x() - Poal Coure ( GTE & PSU) 5 ENGINEERS INSTITUTE OF INDI. ll Righ Reerved 8-B/7, Jia Sarai, Near IIT, Hauz Kha, New Delhi-6. Ph

8 Chemical Engineering (GTE & PSU) x() x()-x(o)= - x() x()-3= x()= (+)(-) Expanding i by parial fracion mehod 3 - B C = + + (+)(-) (+)(-) ( -)+B{(-)}+C{(+)} = (+)(-) 3 - (+B+C)+(C-B)+(-) = (+)(-) (+)(-) Comparing he coefficien on boh ide +B+C = 3, C B =, - = - We ge =, B =, C= X()= X()=+e +e PROPERTIES OF TRNSFORMS Final value heorem If f() i he Laplace ranform of f(), hen lim f = lim f Provided ha f() doe no become infiniy for any value of aifying Re () o. he limi of f() i found o be correc only if f() i bounded a approache infiniy 6 The final value heorem allow u o compue he value ha a funcion approache a when i laplace ranform i known. Example.4 Find he final value of he funcion x() for which he Laplace ranform i x()= ( ) Soluion apply final value heorem Lim[x( )] lim f = 8 Poal Coure ( GTE & PSU) 5 ENGINEERS INSTITUTE OF INDI. ll Righ Reerved 8-B/7, Jia Sarai, Near IIT, Hauz Kha, New Delhi-6. Ph

9 Lim [x()]= 8 Chemical Engineering (GTE & PSU) The condiion of he heorem aified unle = or (+) Iniial value heorem lim f = lim f Tranlaion of ranform If f = f hen a () a {e f()}=f(+a)=() f e Tranlaion of funcion if f = f hen o f - =e f for > 7: Uni Pule Funcion Uni pule funcion of duraion i defined by., (),, Uni pule funcion i he difference of wo ep funcion of equal ize /. Fir ep funcion occur a =, f(), Second ep funcion occur a =, f (), Poal Coure ( GTE & PSU) 5 ENGINEERS INSTITUTE OF INDI. ll Righ Reerved 8-B/7, Jia Sarai, Near IIT, Hauz Kha, New Delhi-6. Ph

10 S ()()() ()()() f f ()()() f f () L L f e L f L Chemical Engineering (GTE & PSU) S e S S () e S S Uni Impule Funcion, uni pule funcion become uni impule or Dirac funcion. Repreened by () () L () L lim() lim() lim() e e S e lim S Uing L Hopial rule () e S e lim lim S S L () Example.5 Solve he following equaion for y () o d y y=,yo = Soluion aking Laplace ranform S S d(y()) y() = o y() y = y y o Poal Coure ( GTE & PSU) 5 ENGINEERS INSTITUTE OF INDI. ll Righ Reerved 8-B/7, Jia Sarai, Near IIT, Hauz Kha, New Delhi-6. Ph

11 Chemical Engineering (GTE & PSU) - y()= [y()]= =coh. Laplace ranform of a funcion f () - f()= f()e o. < o ; Laplace ranform i no defined KEY POINTS 3. {af()=bf()}=a {f()}+b {f()} 4. df() =f()-f(o) 5. d f() 6. For n h order = f()-f()-f() n d f() 7. o n n(n-)(n-) = f()- f(o)- f(o)... f f = 8. Final value heorem ; lim f = lim f 9. Iniial value heorem lim() f lim() Sf S (n-)(n-) f(o)-f(o). If f = f (-a) () a Then {e f()}=f(+a)= f()e [f(-)]=e o -o f() Table of Laplace Tranform Table of Laplace Tranform ) L () = /S ) L() e a S a Poal Coure ( GTE & PSU) 5 ENGINEERS INSTITUTE OF INDI. ll Righ Reerved 8-B/7, Jia Sarai, Near IIT, Hauz Kha, New Delhi-6. Ph

12 3) 4) 5) 6) 7) 8) n S Chemical Engineering (GTE & PSU) n n! L() when > & n N () n n L where n N n i funcion n S L(in) a a S a L(co) a S S a L(inh) a a S a L(coh) a S S a 9) L() e a n n! () S a n Poal Coure ( GTE & PSU) 5 ENGINEERS INSTITUTE OF INDI. ll Righ Reerved 8-B/7, Jia Sarai, Near IIT, Hauz Kha, New Delhi-6. Ph

13 Chemical Engineering (GTE & PSU) CHPTER- TRNSFER FUNCTION MERCURY THERMOMETER We will develop he ranfer funcion for a fir order yem by conidering he uneady ae behavior of an ordinary mercury gla hermomeer Figure-.: cro ecional view of hermomeer. Conider he hermomeer which i locaed in a flowing eam of fluid for which he emperaure x varie wih ime. We need o calculae he ime variaion of hermomeer reading y for a paricular change in x. SSUMPTIONS. The film urrounding he bulb only govern he reiance o hea ranfer. Therefore he reiance offered by gla and he mercury i negleced.. There i no expanion or conracion of gla during he ranien repone. 3. The emperaure of mercury i uniform hroughou Iniially hermo meer i a eady ae. Time he zero he hermomeer will be ubjeced o ome change in he urrounding emperaure x () By pplying he energy balance Inpu rae oupu rae = rae of accumulaion dy h () x y mc (.) Where = urface area of bulb for hea ranfer C = hea capaciy of mercury m = ma of mercury in bulb = ime h = hea ranfer coefficien of film eady ae h() x -y = (.) The ubcrip i ued o indicae he variable in he eady ae value Subracing equaion (.) from equaion (.) give d () y y h[()()] x x y y mc Poal Coure ( GTE & PSU) 5 ENGINEERS INSTITUTE OF INDI. ll Righ Reerved 8-B/7, Jia Sarai, Near IIT, Hauz Kha, New Delhi-6. Ph

14 Noice d(y-y) Pu X = x-x, Y= y-y Chemical Engineering (GTE & PSU) dy = becaue y i conan dy h X-Y mc dy X-Y = τ mc dy X-Y = h mc pu ha Taking Laplace ranform X Y () SY Y() = X() τ+ The parameer τ i called he ime conan of he yem and ha he uni of ime Y() = G = ranfer funcion X() Where G () i he ymbol for he ranfer funcion of he yem PROPERTIES OF TRNSFER FUNCTION. I decribe he dynamic behavior of he yem. I i raio of he Laplace ranform of he deviaion in oupu variable o he Laplace ranform of he deviaion in he inpu variable 3. In mercury hermomeer oupu variable i he hermomeer reading and inpu variable i he urrounding emperaure Block diagram repreenaion Sample Sudy Maerial Poal Coure ( GTE & PSU) 5 ENGINEERS INSTITUTE OF INDI. ll Righ Reerved 8-B/7, Jia Sarai, Near IIT, Hauz Kha, New Delhi-6. Ph