CHE 322 Spring 2013 Term Test Solutions

# CHE 322 Spring 2013 Term Test Solutions - Name Student Q1...

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Unformatted text preview: Name: . Student #: Q1: - Q2: Q3: Q4: Total: CHE3ZZS — Process Dynamics and Control Term Test February 13, 2013 Closed Book All work to be marked must appear on front of page. Use back of page for rough ' work only. Given information: The unit step response of a ﬁrst order system in standard form is given by K — e‘t/T). T . So LoTIONS 1‘ onI 3 1. (25 marks) J Two tanks are connected together in the following unusual way in the Figure below: a) Write down the dynamic equations describing the mass balance in both the tanks in terms of the mass ﬂow rates, W1, wz, W3, the density of the ﬂuid p, and the cross sectional area of the tank A1 and A2. RATE : f" OUT lANK I d( V. -— ‘ ' ‘ 7F? ) = wY ~w;f,___iws \ﬁﬁxmawuo "T:qu 2. \$4er = wz. My M ' e.-m.>"Hms ct (MM — w. "Uh-“’3 (at) : 4—(00;’°J2.*w3) dl— J’A. 0‘i32. : U32. 6” fAL 20f21 b) If you assume that the mass ﬂow rates, wz, W3 are directly proportional to the pressure . drop across the valves with coefﬁcients of proportionality oz, and 03 respectively, simplify the. differential equations in part a. - (/32, = C7. Lfgk,~ 53352.] (/33 3 C3 H é\$f ié;‘:0,~ Cxtfﬁb,-fgk;)”csf3ki] 0H“ F 21 ﬂ (C,,ﬂ +653) L + 62.3%; Ar ’ 9A. A, A! N»; -'__._. '2 C2 3%, "" £27; '3 of21 c) If there is a two fold change in W1, calculate the relative change in the levels h] and h; with respect to their initial values. 6 w. ---—> 2w? all». - @ss \—)\=l7,, 37:0) d) After deﬁning the deviation variables and using Laplace Transforms, identify the transfer function models of h with respect to W1 (Hint: Derive hz with respect to h ﬁrst and then substitute into the differential equation for h) 0H»: ‘ c, r “a” : EL... ~ (62"? 5)3%‘ + CLﬂk; "' U) 1' fA' AI ~ A) ‘ ib- : 21L: — “’3‘” -— m CH” A). AL V 55 1 65 [\loi‘ci him):- l—, —‘hs:5 ; l); :‘h;""’z 3 L0,?— wu“ M) are olevlal-Coh .Variable e 4of21 LT o9 (a9 C; M ML 532(5) — ﬁk‘ A» L ( 1, 22% z 313 ‘w 6 A,“ A2.— L C;3/Ab I \ ﬂ: " M : A ' C23 ,3: +3 k, 5+ /AL LT of m 03‘ (Ca’ﬁ' C5) ﬂk‘ _r 2:2)”; sh“) = F “ *“f’” A (zﬂ/A\ (A), (C;+(5>SL) + 5“, = .———-« " A‘ Ala-r 9 3”“ cm ' w. L curs (at: ﬁis+m 5!», = H (Ls+‘)w I’M >9k’ (23.3 + Jr 623 3A» era—3 C a \Z _ A2 3.4%. ; “5‘” A. 13 (1+6; L-(z5+))'\-C>\$\nl 09! l V 3 ' Lt25+i)Sl->‘ : E' ((2.51‘ ) ( A. > (253%! L). (2 5+») ( ’____4 :. _~—« L raga“) _r [l+ (C%3)8ZLJSSB\ + A. fA' . ) ZL5+‘) :L\ —— G,\ W : WW —— (J C3119; ZL>SL *Cﬂil , I ’ {Lszk‘f A! \ A; 50f21 60f21 e) BONUS [5 extra points but is all or nothing]: Derive the transfer function of h2 with - respect to W1, 7of21 2. (15 marks) Figure below shows a system for heating a continuous ﬂow water kettle using a hot plate. ﬂ Assume that the hot plate temperature can be changed instantaneously by adjusting the rate of heat input into the hot plate, and assume that the temperature within the water kettle is uniform. The goal is to control the temperature of the water in the kettle using the electrical heat input Q. Here, F is the volumetric ﬂowrate, V is the volume of water in the kettle, To is the inlet temperature, T is the outlet temperature (and hence the temperature within the kettle), and Tp is the temperature of the surface of the hotplate. (5x5 = \BMARM) (i) Identify the output(s), input(s), state (s) and disturbance(s) in this process. MAR Ks ) Hot plate surface lNPUTs = To rise ‘) DnSTURBANCES: F To ‘ 3 MANIPULATED VAI'Z. -. Q OUTPUWS):'T STATES -. v, T' > 80f21 (ii) It is desired to implement a feedback control strategy to control the Temperature in the kettle T using the electrical heat input Q. Draw a feedback control strategy on this diagram below. Show the instrumentation required (sensors etc) and the controller as you would on a P & I D. Write down the variable (3) which you need to measure and which to manipulate. ‘ L 5 “A A R K s) Mme.sz : T V Hot plate suface ‘ m PEP—ATURE SENsoR Electricity, Q , ‘v l : PWAN,PULATE: Q a *6.— ..” .. TEMPE )IATURE/Com-rrzoLLE ,1 9of21 (iii) It is desired to implement a FEEDFORWARD control strategy to control the Temperature in the kettle T using the electrical heat input Q. Draw a FEEDFORWARD control strategy on this diagram below. Show the instrumentation required (sensors etc) and the controller as you would on a P & I D. Write down the variable (5) which you need to measure and which to manipulate. (5 M A RK s > Hot plate surface 10 onI 3. (15 marks) . . i) The following plot is the proﬁle of the temperature change (T(t)) for a batch reactor that includes a heating step. Identify T(s). L 5 M A R K s > 20 15 0 5 10 ‘ 15 20 110f21 ii) Given the transfer functions, qualitatively sketch their response to a unit step input. Indicate the ﬁnal value of output and the approximate time taken to reach it where applicable. Justify your answer where possible. 3 ' 3/7. a)'G"10s+'2 T ’ 5 ’ C" W (5 MARKS) _ 2(5S —— 1) 3 b) G ‘ (15s +1)(2s +1) 3 (5 MARK s RHP Zara Re. frat-mt Haemﬁm inverscr responsb i6 FV¢~\$¢“\: 12 of21 4. (35 marks) Consider the following ﬂow diagram for a series of three tanks with a recycle stream. If you assume that the upstream ﬂow rate is F and the recycle stream ﬂow rate is PR and the volume of the tanks are V1, V2, and V3 respectively. Assume that composition of component A in the tank is cm The output stream composition is CAe Feed Composition Proﬁle Mlxmg POlnt Exit Composition (desired: dashed line) l3 0f21 21). Consider the single tank (Tank 1) for the case without the recycle. Assume that input (11) is the composition of component A in the feed (CAf ) and the output is the composition of component A in the stream leaving the tank (CA1)- Assuming that the volume in the tank is constant (V1), derive the transfer function that relates the input to the output in terms of the volume and the ﬂow rates. Using this equation, write down the transfer function of the other two tanks in terms of their volumes V1 and V2,( Hint: You should write the mass balance and derive a ﬁrst order transfer function) I M AR Ks ) VI §1F[CA\) 2‘ "‘ FCAb Aft“ LJI‘I‘iZifj C): ’ 52> \zolzi—nj Lapin“ “Transicrm in l'cnws ol- Oieviai'iOh Variables : CAI, 3 CA! i C/‘l' is + t F m i" \v. :5 Sirmiafia KL! lami-t 2. l or \am C” l c’ l ,TAL : V AL : CAL .13.. 5 '1' l till 242:. 8 “l' l i- F 14 of21 b) If there is a recycle stream, ﬁrst derive the transfer function for tank 1 alone based on your answer in part a. Assume that volumes in the tanks remain the same and that the fraction of ﬂow being recyled is alpha (alpha = FR/(FR + F). In the second part, extend your answer to tanks 2 and 3. (5 MARKs) dc» "' M Male: ‘ai 33030 c “T our : FTCAm” FTC“ r M \l a V) c ———-\$ “3" FT 2: {:31 + F c’ l —— Ar = M ETAMM \ i" -‘ WP ::. “"7" F For Toma 2.. Far “Enh— 3 ) \ V ’ CAL ‘ CA5 ~ 3 ‘ ~ V2. I C‘ V5 ) QM [email protected]+l A» .ﬂJkos+ F F 15¢f21 i 0) Draw the block diagram representation of the entire process that relates input C Af to the output CAe including the three tanks, the recycle stream and the mixing transfer function. Assume that the transfer function for the tanks is given by G1, G2, and G3, L5 MARKS) 16 of21 l d) Consider the case Where there is no recycle stream. Qevelop the overall transfer function that relates input CAf to the output CAe in terms of G1 , G2 and G3. (3 MA RK 6) O‘Jcraﬂ OPcn LOO)? 1 gig" I thizé’li CAI; l7 of'21 i e) If we now consider the recycle stream, develop the olverall transfer function that relates input CAf to the output CAe in terms of G1, 2 and G3 and the recycle fraction alpha = FR/(FR+ F). (5 mums) Cn ' CA3 f 6“ G); 613 5 : Cloecok LOOP 0!” 9-164 - (l’d)6n (“2/673 , CPI: l "" 0‘67) 601,613 2 18 of21 f) If the volumes of the tanks in m3 and the ﬂow rates in m3/hour are speciﬁed by the following equations: ' V1 =20 V2 =10 V3 =2 F = 2 Derive the overall transfer function between the input GM to the output CAe for the case Without the recycle. Identify the poles and zeros of this transfer function and sketch them on the complex plane. <5 M» RKS Cm: l 19 onI g) Assuming the same volumes and ﬂow rates in part f) derive the overall transfer function between the input CAf to the output CAe for the case with the recycle assuming that alpha=0.5. Write down the equations for calculating the poles and zeros of this transfer function. MARKS > CM: 0- 5 : WWW Ch? (ros+\)(5\$ﬂ) (5+0 —-Oi"3 P Eq- 3 (IO\$+))(5s+w)(e-+l) :0-5 20 of21 E h) Identify the type of interconnection for the case with the recycle and comment on the impact of this interconnection on the poles and zeros of the overall transfer function. (lmmzms) ’ H o", \nhrconnccl-iéh‘. ‘r’c:¢.ol)3aclz 3:) Pairs Cam almost bul- Zcr‘ée’ came)» 3.5 a \:cC°‘-1>0LC}2 . BYE. inlerﬁonbc CHDD . I E l l E 21 onl l . ...
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