Analysis of Step Response, Impulse and Ramp Response in the Continuous Stirred Tank Reactor System

Transcription

1 ISSN: Volume 0 - Iue 05 May 07 PP Analyi of Step Repone, Impule and Ramp Repone in the ontinuou Stirred Tank Reactor Sytem * Zohreh Khohraftar, Pirouz Derakhhi, (Department of hemitry, Science and Reearch Branch, Ilamic Azad Univerity, Tehran, Iran) Abtract: The three main type reactor developed for batch, continuou-tirred tank, and plug-flow reactor are ueful for modeling many complex chemical reactor. In thi work, tep repone, impule and ramp repone i tudied theoretician in a continuou tirred tank reactor. The aim of thi tudy i to invetigate the mathematical analyi on function repone in a continuou tirred tank reactor. The parameter aumed for analyi are the flow rate i contant for the whole ytem, m 0.085, the volume of three thank i the equal : V = V = min V = V T =.05 m and reaction rate K= 0.04 min -. all three tranfer function howed favorable reult. Keyword: Step Repone, Ramp, Impule Repone, STR. I. INTRODUTION There are everal type of reactor ued in chemical or biochemical indutrie. ontinuou tirred tank reactor in the form of either ingle tank or (or more often) tank in erie, are ued widely and thee are particular uitable for liquid phae reaction (Danih et al, 05). The control of the operation of chemical reactor ha attracted the attention of reearcher for a long time (Adebekun,996). The three main reactor type developed thu far batch, continuou-tirred tank, and plug-flow reactor are ueful for modeling many complex chemical reactor. ontinuou tirred tank reactor ytem (STR) i a typical chemical reactor ytem. The STR conit of three tank a howed in igure. It i uppoe that the overflow tank are well-mixed, and the denity i the equal in three tank. So the aumption for the overflow tank, the volume in the two tank can be taken to be contant, and all flow are contant and ame (Saad et al, 0). In thi work, tep repone, impule and ramp repone i tudied theoretician in a continuou tirred tank reactor. The aim of thi tudy i to invetigate the mathematical analyi on function repone in a continuou tirred tank reactor. II. RESULT AND DISUSSION Modeling of the continuou tirred tank reactor ytem igure how the imple concentration proce control. It i aumed that the overflow tank are wellmixed iothermal reactor, and the denity i the ame in three tank. Due to the aumption for the overflow tank, the volume in the three tank can be taken to be contant, and all flow are contant and equal. It i aumed that the inlet flow i contant. igure how the block diagram of three tank of chemical reactor. ig.the imple concentration proce control 7 Page

2 ISSN: Volume 0 - Iue 05 May 07 PP ig. The block diagram of the three tank ytem The tranfer function of the firt tank can be expreed a follow: d () A 0 A V K V Where i the flow, A0 i the inlet concentration of the firt tank, A i the outlet concentration of the firt tank, A i the outlet concentration of the Second tank and it too the inlet concentration of third tank, A i the outlet concentration of third tank, V i the volume of the firt tank and K i the reaction rate. Equation can be rearranged to be after implify: d A V K V A 0 A A d KV V V A A0 A d KV V V d 0 V () A 0 By taking Laplace tranform from equation (): S ( ) ( ) 0( ) V ( ) A0 S ( ) V The tranfer function of the firt tank can be repreented a (): V ( ) V V KV KV K ( ) S A p 0 Where K p i the gain of the tranfer function of the firt tank. The tranfer function of the econd tank can be expreed a follow: d A - A -V K V (4) Equation 4 can be rearranged to be (5): () d A V A A (5) 7 Page

3 ISSN: Volume 0 - Iue 05 May 07 PP By taking Laplace tranform and rearranging equation 5, the tranfer function of the econd tank can be obtained: ( ) ( ) Where i the time contant for the econd tank. KV (6) The tranfer function of the third tank can be expreed a follow: d (7) A - A -V K V Equation 7 can be rearranged to be (8): d (8) V By taking Laplace tranform and rearranging equation8, the tranfer function of the third tank can be obtained: ( ) (9) ( ) Where KV The tranfer function of the combined three tank with the aumed parameter can be preented equation (0): ( ) G ( ) (0) 0( ) ( ) A igure how the block diagram of the open loop combined three tank ytem. ig. The block diagram of the open loop ytem The tranfer function of the whole ytem can be obtained according to the following aumption of parameter: m min - The flow rate i contant for the whole ytem. - The volume of the three thank i the equal: V V V V T.05m - Reaction rate K 0.04 min. Due to time contant ( V KV ( ) ) and the gain are ame for three tank, they can be calculated a follow: 74 Page

4 ISSN: Volume 0 - Iue 05 May 07 PP KV Tranient repone The tranfer function of a firt- order ytem ha been tudied. Due to the tudy of repone to different common forcing function: tep, impule, and ramp. Step repone The tranfer function, which i written by equation (), i G ( ) ombining equation () and (.) give: Thi can be expanded by partial fraction to give: A B D (8.6 ) (8.6 ) (8.6 ) (8.6 ) ombining equation () and (.) give: (.) Thi can be expanded by partial fraction to give: A B D (8.6 ) (8.6 ) (8.6 ) (8.6 ) (8.6 ) A (8.6 ) B(8.6 ) (8.6 ) D (8.6 ) (8.6 ) (8.6 ) (8.6 ) (8.6 ) A(8.6 ) B(8.6 ) (8.6 ) D Solve the unknown parameter by plugging the real root of the denominator: 0, 8.6 or the denominator root 0 and : 8.6 A 0.994, D.47 Plug in the olution to the known parameter: (8.6 ) B(8.6 ) (8.6 ) (.47) Expand: A A ( ) ( ) (8.6 ) ( t ) U ( t ) ( ) ( ) (8.6 ) (8.6 ) A (8.6 ) B(8.6 ) (8.6 ) D (8.6 ) (8.6 ) (8.6 ) (8.6 ) (8.6 ) () (8.6 ) (8.6 ) (8.6 ) A0 A0 A B D ( ) (8.6 ) (.) (.) 75 Page

5 ISSN: Volume 0 - Iue 05 May 07 PP B B B Group element according to power of : (68.76B 68.74) ( B 6.86) ( B ) Equate the coefficient of imilar term on both ide to create a lit of equation: B B B Solve ytem of equation: B.47,.47 Plug the olution to the partial f partial fraction parameter to obtain the final reult: (.47) (.47) (.47) ( ) (8.6) (8.6) (8.6) Invere Laplace :Taking A ( ) e t ( e t t )( e t t ) Solve the unknown parameter by plugging the real root of the denominator: Ramp Repone: The tranfer function, which i written by equation () i G ( ) ( ) 0 0 t t 0 ombining equation () and (.) give: A Thi can be expanded by partial fraction to give: ( ) ( ) ( ) ( ) ( ) ( ) (8.6 ) (8.6 ) A (8.6 ) B (8.6 ) (8.6 ) D (8.6 ) E (8.6 ) (8.6 ) (8.6 ) (8.6 ) (8.6 ) () A B D E (8.6 ) (8.6 ) (8.6 ) (8.6 ) A(8.6 ) B (8.6 ) (8.6 ) D (8.6 ) E Solve the unknown parameter by plugging the real root of the denominator: 0, 8.6 or the denominator root 0 and : 8.6 B E Plug in the olution to the known parameter: A(8.6 ) 0.994(8.6 ) (8.6 ) D (8.6 ) Expand: 76 Page

6 ISSN: Volume 0 - Iue 05 May 07 PP A A 8.6D A D A Group element according to power of : 4 A D A A D A Equate the coefficient of imilar term on both ide to create a lit of equation: A A D A 8.6D A 0 A 7.496, 6.86, D Plug the olution to the partial fraction parameter to obtain the final reult: ( ) 8.6 (8.6 ) (8.6 ) Invere Laplace :Taking Impule repone The tranfer function, which i written by equation () i 0( t ) ( t ) 0( ) ombining equation () and (.) give: Thi can be expanded by partial fraction to give: Solve the unknown parameter by plugging the real root of the denominator: Plug in the olution to the known parameter: t t ( t ) t 7.496e te e t G ( ) ( ) A ( ) Y ( ) ( ) X ( ) 0( ) (0.669) ( ) (8.6) A B ( ) ( ) ( ) ( ) A B ( ) 8.6 ( ) 0.994( ) ( ) ( ) ( ) ( ) ( ) ( ) A B 77 Page

7 ISSN: Volume 0 - Iue 05 May 07 PP A(8.6 ) B (8.6 ) Expand: A 6.5A A 8.6B B Group element according to power of : Equate the coefficient of imilar term on both ide to create a lit of equation: Plug the olution to the partial fraction parameter to obtain the final reult: Invere Laplace: A A B B A 6.5A 8.6B B A A 0 A 0, B 0, A ( ) (8.6 ) 8.6 (8.6 ) (8.6 ) ( ) ( ) (8.6 ) A ( t ) e t t III. ONLUSION The mathematical analyi on tranform of the repone i invetigate in a continuou tirred tank reactor. All three tranfer function howed favorable reult. Reference []. Adebekun. (996). Output feedback control of a tirred tank reactor, omputer and hemical Engineering, 0 (8), []. Danih and et al, (05). Effect of Operating ondition on STR performance: an Experimental Study. Journal of Engineering Reearch and Application. 5 () []. Saad and et al. (0), Modeling and ontrol Deign of ontinuou Stirred Tank Reactor Sytem.Proceeding of the 5th International onference on Automatic ontrol, Modelling & Simulation (AMOS '). 78 Page