VIEWS: 5 PAGES: 10 POSTED ON: 4/5/2011
BSE 422 Midterm, 4 November, 2010 Take Home – Due Tuesday, 9 November; 4:00 PM PST This midterm is to be worked on by yourself with no assistance from anyone. Failure to do so will be met with severe consequences. This exam looks at lime mud filters used in the recovery area of a pulp mill. The model presented here comes from "The Modeling Of Rotary Lime Mud Filters", by Kent R. Davey, George Vachtsevanos, and Jim C. Cheng, Tappi Journal, Vol. 72, No.8, pp 150-156, August 1989. Lime mud consists of CaCO3 and residual alkali from the causticizing process. The lime kiln converts CaCO3 to CaO to be reused in the causticizing reaction to regenerate the kraft cooking liquor. Before the lime mud can be ran through the lime kiln two things need to be done. The lime mud is fed to the filter as slurry at an average solids content of 35%. The solids content must be increased to around 70% prior to the kiln (30% moisture). The alkali concentration in the mud also needs to be controlled. Too high an alkali concentration causes the drying lime to stick together and form large balls. Too low and it won’t stick together at all and high dusting losses occur. The job of the lime mud filter is to increase the solids and regulate the alkali concentration of the lime mud going to the lime kiln. A diagram of a drum filter is shown in Figure 1. Lime mud enters the vat and builds up on the vacuum drum. Wash water is applied to the lime mud cake to reduce the alkali concentration. This also has the effect of increasing the moisture content of the lime mud. A higher drum speed will give a lower moisture content and lower alkali concentration. For the problems presented here the production rate will be treated as a disturbance. Lime Mud Cake Wash Shower Scraper Blade Vacuum Drum Vat Slurry Mud Conveyor Figure 1. Drum mud filter. 1 Table I lists the nominal operating conditions for the mud filter. Table I. Nominal operating conditions for lime mud filter. Mud moisture content (%) 30 Alkali concentration (%) 1.5 Blade position (cm) 3.5 Drum vacuum (relative) 1 Wash shower flow (L/s) 3 Drum speed (rpm) 5 Production rate (m3/s) 1 Figures 2 and 3 represent block diagrams of the lime mud process. W P 9 36 S -8 M Figure 2. Lime mud filter process model; drum speed (S), wash flow (W), production (P), moisture content (M). S P -0.2 2 W -0.125 A Figure 3. Lime mud filter process model; wash flow (W), drum speed (S), production (P), alkali concentration (A). 2 Problem 1 The blade position (B) and the drum vacuum (V) affect the lime mud moisture content (M) and alkali concentration (A). The data in table I show operational data for the mud filter for these two variables. Note that the first row is the nominal operating point. Table 1. Lime mud filter operational data. Blade Position, cm (B) Vacuum, rel (V) Moisture, % (M) Alkali, % (A) 3.5 1 30 1.5 3.0 1 26 1.4 3.5 1.25 27 1.3 Derive a linear incremental model where the blade position (B) is the manipulated variable, drum vacuum (V) is a disturbance, and the mud moisture (M) is the output. Draw a block diagram for the model. M / B M / B M / B (26 30 ) /(3.0 3.5) M / B 8 %/cm M / V M / V M / V (27 30 ) /(1.25 1) M / V 12 %/rel M 8 B 12V V -12 B 8 M 3 Problem 2 Design a feed forward control strategy (using the block diagrams and process data from page 2) for the alkali concentration (A) that would result in perfect control. Use the wash water (W) as the manipulated variable. The drum speed (S) and production rate (P) are the disturbance variables. The solution should include a block diagram of the process with the control strategy. Show the numerical values of the feed forward controller gains in the block diagram. Calculate the change in alkali concentration (A) and the wash water (W) when the alkali concentration setpoint is increased to 1.7% and the drum speed is increased by 0.5 rpm. The block diagram for this system is given by P S KFP KFS -0.2 2 W ASP KSP -0.125 A The closed loop transfer function for the alkali concentration is given by A 0.125 K SP ASP (0.2 0.125 K FS ) S (2 0.125 K FP ) P For perfect control the closed loop gain for the alkali set point needs be one and the closed loop gain for the production rate and drum speed should be zero. G SP 0.125K SP 1 K SP 1 /(0.125) K SP 8 %/% G S (0.2 0.125K FS ) 0 K FS 0.2 / 0.125 K FS 1.6 L/s rpm G P (2 0.125K FP ) 0 K FP 2 / 0.125 K FP 16 L/m 3 The change in alkali concentration for a setpoint change of 0.2% (1.7%-1.5%) and drum speed of 0.5 rpm will be 0.2% by design. The wash flow can be calculated as W 8 ASP 1.6 S W 8(0.2) 1.6(0.5) W 2.4 L/s We can use the open loop process gains to double check to make sure we achieved the desired change of 0.2% for the alkali concentration. A 0.125W 0.2 S A 0.125 (2.4) 0.2(0.5) A 0.3 0.1 A 0.2 % 4 The block diagram with the proper values for the gains is shown below. P S 16 -1.6 -0.2 2 W ASP -8 -0.125 A 5 Problem 3 Design a feedback control strategy (using the block diagrams and process data from page 2) for the mud moisture content (M) using the drum speed (S) as the manipulated variable. The wash water (W) and production rate (P) are the disturbance variables. Use a loop gain of 4 for the controller. The solution should include a block diagram of the process with the controller. Calculate the change in mud moisture content (M) and the drum speed (S) when the mud moisture content setpoint is reduced to 25%. The block diagram for this system is given by W P 9 36 S MSP KC -8 M - The closed loop transfer function for mud moisture is 8K C 9 36 M MSP W P 1 ( 8 K C ) 1 ( 8 K C ) 1 ( 8 K C ) where loop gain 8 K C K C 4 / 8 K C 0.5 rpm/% so 9 36 M 0.8 MSP W P 5 5 For a -5% mud moisture setpoint change (25%-30%) M 0.8(5) M 4 % S K C ( MSP M ) S 0.5(5 (4)) S 0.5 rpm As was expected with a loop gain of 4 the process value came within 80% of the setpoint. 6 Problem 4 a) The mud moisture needs to remain in the range of ± 5%, but the wash flow varies by ± 2 L/s. For the feedback only control strategy in Problem 3, determine the minimum controller gain required to keep the mud moisture content in the specified range. What is the loop gain for this situation? With the wash flow varying by ± 2 L/s the mud moisture will vary by ± 18 %. We want the mud moisture to only vary by ± 5 %. 9 9 M W M (2) 5 1 LG 1 LG 1 LG 18 / 5 LG 2.6 LG 8 K C K C 2.6 / 8 K C 0.33 rpm/% Another way to look at this is we need to reduce the variation due to the disturbance to approximately 27% (5/18) of its open loop gain. This gives 1 R R(1 LG) 1 LG (1 R) / R 1 LG R reduction 5 / 18 LG (1 0.278) / 0.278 LG 2.6 Either way you come up with the same answer for the loop gain. The minimum controller gain to keep the mud moisture in the range of ± 5% with wash water variations of ± 2 L/s is -0.33. This gives a loop gain of 2.6. 7 BSE 422 Midterm, 4 November, 2010 Take Home – Due Tuesday, 9 November; 4:00 PM PST This midterm is to be worked on by yourself with no assistance from anyone. Failure to do so will be met with severe consequences. Take Home Problem 1 – (50 points) The following block diagram is a decoupled control system that uses the drum speed to control the mud moisture content and the wash flow to control the alkali concentration. The decouplers remove the influence of the drum speed controller output (SCO) on the alkali concentration (A) and the wash flow controller output (WCO) on the mud moisture content (M). The control engineer retuned the control system during a shutdown on a Friday afternoon right before he was leaving on vacation for a remote island in the Federated States of Micronesia. After the start up the system is very unstable and the operators have to run the process in manual control. You get called in at midnight that night to fix the problem. There appears to be some problems with the controller gains. What changes would you make, if any, to the feedback controller gains (KCM and KCA) and the decoupler gains (KDM and KDA) to make the process stable? Note that the 4 gains on the right are the process gains and SCO and WCO are the feedback controller outputs for the drum speed and wash flow, respectively. - SCO S MSP -5.7 -7 M KCM KDA -2.14 7 -10 -0.3 KCA KDM ASP 28.6 -0.14 A WCO W - Feedback Decouplers Process The first thing to check is the decouplers. The values for these are only based on the process gains. First let’s look at the decoupler that removes the effect of drum speed controller output on alkali concentration. The influence coefficient between the alkali concentration and the drum speed controller output is given by A (0.14 K DA 0.3) S CO where K DA 2.14 L/s rpm 8 This influence coefficient should be zero. Let’s check this by calculating our own value for KDA 0.14K DA 0.3 0 K DA 0.3 / 0.14 K DA 2.14 L/s rpm This decoupler gain is good. Now we look at the decoupler that removes the effect of wash flow controller output on mud moisture. The influence coefficient between the mud moisture and the wash flow controller output is given by M (7 K DM 7)WCO where K DM 10 s rpm/L Again the influence coefficient should be zero. 7 K DM 7 0 K DM 7 / 7 K DM 1 s rpm/L Not only is the magnitude wrong it has the wrong sign as well. It made the influence worse by going in the wrong direction and too much. Dealing with the feedback controller gains is a little trickier since there are two paths between the controller output and the controlled variable. You can see from the block diagram the engineer ignored this. It looked like he was trying to tune the loops for a loop gain of 4 (admittedly I did that making up the problem without thinking about it). Right off you should at least see the sign of the feedback controller gain on the alkali loop is wrong. It should be negative. If we write the closed loop transfer function for the mud moisture loop we get 7 K CM K DA 7 K CM K CM (7 K DA 7) M MSP M MSP 1 (7 K CM K DA 7 K CM ) 1 K CM (7 K DA 7) The decoupler eliminates any interaction of the wash flow on the mud moisture so there are no terms from the alkali loop (i.e., ASP). First lets look at the current loop gain. LGM K CM (7 K DA 7) LGM 5.7(7(2.14) 7) LGM 5.7(21.98) LGM 125.3 We can see this is way too large. For a feedback controller a loop gain of 4 should be reasonable so LG M K CM (7 K DA 7) 4 K CM 4 /(7 K DA 7) K CM 4 /(7(2.14 ) 7) K CM 4 /(21 .98 ) K CM 0.18 %/% 9 Similarly for the alkali loop we write 0.3 K CA K DM 0.14 K CA K CA (0.3 K DM 0.14 ) A ASP A ASP 1 (0.3 K CA K DM 0.14 K CA ) 1 K CA (0.3 K DM 0.14 ) Calculating the loop gain for this loop (using the proper value for the decoupler gain and the proper sign for KCA) gives LG A 28.6(0.3(1) 0.14) LG A 28.6(0.44) LG A 12.6 Even with the correction on the decoupler (LGA=81.8 without correction) the loop gain is still too high. Again specifying a loop gain of 4 and using the value we calculated for KDM gives LG K CA (0.3 K DM 0.14 ) 4 K CA 4 /(0.3 K DM 0.14 ) K CA 4 /(( 0.3)(1) 0.14 ) K CA 4 /(0.44 ) K CA 9.1 %/% This confirms the negative sign for the feedback controller gain. This makes sense physically. When the alkali concentration is above the setpoint the error will be negative. With a negative controller gain this will cause the wash flow to increase thus having the desired effect of lowering the alkali concentration towards the setpoint. To summarize Gain Before After KCM -5.7 -0.18 KCA 28.6 -9.1 KDM -10 1 KDA -2.14 -2.14 10