Lecture Note II Example 6 Continuou Stirred-Tank Reactor (CSTR) Chemical reactor together with ma tranfer procee contitute an important part of chemical technologie. From a control point of view, reactor belong to the mot difficult procee. Thi i epecially true for fat exothermal procee. We conider CSTR with a imple exothermal reaction A B. For the development of a mathematical model of the CSTR, the following aumption are made,. neglected heat capacity of inner wall of the reactor, contant denity and pecific heat capacity of liquid,. contant reactor volume, contant overall heat tranfer coefficient, and 3. contant and equal input and output volumetric flow rate. A the reactor i well-mixed, the outlet tream concentration and temperature are identical with thoe in the tank. Figure. A noniothermal CSTR. Ma balance of the component A can be expreed a dc A V qc Av qc A Vr( c A, ϑ) where t - time variable, c A - molar concentration of A (mole/volume) in the outlet tream, c Av - molar concentration of A (mole/volume) in the inlet tream, V - reactor volume, q - volumetric flow rate, r(c A, ϑ) - rate of reaction per unit volume, ϑ- temperature of reaction mixture. The rate of reaction i a trong function of concentration and temperature (Arrheniu law)
E R ϑ r( c A, ϑ) kca ke c A where k i the frequency factor, E i the activation energy, and R i the ga contant. Heat balance give dϑ Vρ c p qρ c pϑv qρ c pϑ αf( ϑ ϑc ) + V ( H ) r( c A, ϑ) 3 where ϑv - temperature in the inlet tream, ϑc - cooling temperature, ρ - liquid denity, c p - liquid pecific heat capacity, α - overall heat tranfer coefficient, F - heat tranfer area, (- H) - heat of reaction. Initial condition are c A () c A ϑ () ϑ The proce tate variable are concentration c A and temperature ϑ. The input variable are ϑ c, c Av, ϑ v and among them, the cooling temperature can be ued a a manipulated variable. The reactor i in the teady-tate if derivative with repect to time in equation (), (3) are zero. Conider the teady-tate input variable ϑ c, c Av, ϑv The teady-tate concentration and temperature can be calculated from the equation qc qc Vr( c, ϑ ) 4 Av A A qρ c ϑ qρ c ϑ αf( ϑ ϑ ) + V ( H ) r( c, ϑ ) 5 p Pleae finih the ret a homework. v p c A Example 7 Mathematical model of a thermocouple Figure. Control loop for the Stirred Heating Tank
θi θi θ θ θ a) bare thermocouple b) thermocouple with protect acket Figure 3. Temperature enor (thermocouple) a) Mathematical model of a bare thermocouple The energy balance for the bare thermocouple i, dθ C Qi Q 6 where C i the molar pecific heat capacity of the thermocouple, Q i i the heat flow from the media to the thermocouple, and Q i the heat lot by the thermocouple. And, θi θ Qi α A ( θi θ ) R 7 where R i the heat reitor, R α A, A urface area of the tip of the thermocouple, α i the heat tranfer coefficient of the heat tranfer between the media and the thermocouple. Aume Q, ubtitute Eq.7 into Eq.6, dθ R C + θ θ i 8 Follow the tep ued in the early example, dθ R C + θ θ i 9 Therefore it i a firt order ytem. b) Mathematical model of a thermocouple with protect acket. The energy balance for the thermocouple with protect acket i, dθ C Qi Q Q Q where C i the molar pecific heat capacity of the thermocouple protect acket, Q i i the heat flow from the media to the thermocouple protect acket, Q i the heat flow from the 3
thermocouple protect acket to the thermocouple, and Q, Q are the heat lot by the thermocouple acket and the thermocouple itelf repectively.. Aume Q, Q are equal zero, dθ C α A θ θ ) α A ( θ ) ( i θ where A i the heat tranfer urface area of the thermocouple protect acket, A i the heat tranfer urface area of the thermocouple tip, α i the heat tranfer coefficient of the heat tranfer between the media and the thermocouple protect acket, α i the heat tranfer coefficient of the heat tranfer between the thermocouple protect acket and the thermocouple. Since A >> A, dθ C α A ( θi θ ) we can alo arrive at d θ R C + θ θ i 3 where R. α A 3. The accumulation of the heat in the thermocouple tip i, dθ C ( ) αa θ θ 4 Similarly we can arrive, d θ R C + θ θ. 5 4. Now differentiate both ide of Eq.5 with repect to t, d θ d θ d θ R C +. 6 Subtitute Eq.3 and 5 into 6, and let τ R C, τ R C, d θ d θ τ τ + ( τ + τ ) + θ θ i. 7 It i a econd order ytem. 4
Figure 4. Blending ytem and Control Method, meaure x and adut w Example 8 The pneumatic control valve p p q Figure 5. The chematic diagram of a pneumatic control valve Ma balance of compre air (ignal), dp F F i C 8 where C i the capacitor, p i the preure in the diaphragm chamber, F i, F, are the in and out compre air flow rate. p p Since the diaphragm chamber i ealed, F F, i, R i the reitance of the R compre air line, therefore, 5
dp RC + p p 9 and d( p + p ) RC + ( p + p ) ( p + p ) when there i no action, p p. Let T v RC, d p T v + p p Aume the effective area of the diaphragm i A, and let c be the rigidity coefficient of the pring, according the Hooke' law, A p c l 3 where l i the diplacement of the diaphragm caued by the force acted on by p. Subtitute into Eq., d l A Tv + l p 4 c Aume a linear relationhip between the change of the fluid flow rate q and the diplacement of the diaphragm l, q K l 5 Subtitute Eq.5 into 4, d q A Tv + q K p 6 c or, d q Tv + q K v p 7 where A K v K i the gain of the pneumatic control valve and, c T v RC i the time contant of the pneumatic control valve. Thi i a firt order ytem. If the time contant of the pneumatic control valve i much maller than the time contant of the proce it i controlling, T v, q K v p. 8 It i a proportional ytem. Tranfer-Function Repreentation and time domain repone of Control-Sytem Element. The general form of the firt order element, dy( τ + y( Kx( 9 It tranfer function, 6
( ) G ( ) Y K X ( ) τ + Step input x( MU(, K M Y ( ) G( ) X ( ) τ + K M y( L τ + KM e When M (unit tep inpu, t ( ) τ y t K e.63. t τ τ t 3 3 3 33 Figure 6. Step repone of a firt order ytem. The general form of a econd order element, d y( dy( τ m + ζτ + y( x( 3 m 4 Tranfer function, Y ( ) G ( ) 35 X ( ) τ m + ζτ + m ω + ζω + ω 36 where ω i the undamped natural frequency and ζ i the damping ratio of the τ m ytem. Given a tep input x( MU(, 7
M Y ( ) G( ) X ( ) 37 τ + ζτ + M y( L τ + ζτ + W e will dicu thi ytem in great detail in later chapter. 38 Figure 7. Step repone of a econd order ytem 3. The general form of the proportional element, y( Kx(. 39 It tranfer function i, Y ( ) G ( ) K 4 X ( ) Given a tep input x( MU(, M Y() G()X() K 4 M y ( L K KMU ( 4 It i a tep with KM a it magnitude. 4. The general form of the integral element, dy( τ i Kx( 43 8
It tranfer function i, Y ( ) K G ( ) X ( ) τ i Given a tep input x( MU(, K M Y() G()X() τ i KM KM y ( L t τ i τ i It i a ramp at the lop of KM/τ i. 44 45 46 5. The general form of the differential element, dy( y( τ d 47 It tranfer function i, Y ( ) G( ) τ d 48 X ( ) Given a tep input x( MU(, M Y() G()X() τ d τ d M 49 y( L [ τ d M ] τ d Mδ ( 5 It i a impule. 6. The general form of delay element y( x( t τ ) It tranfer function i, Y ( ) τ G ( ) e X ( ) Given a tep input x( MU(, τ M Y ( ) G( ) X ( ) e y( L e τ M M ( t τ ) It i a tep after a time delay of τ. 5 5 53 54 Development of Empirical Dynamic Model from Step Repone Data Higher order ytem and dead time Connecting many tank make the ytem correpondingly higher order. Thu by a erie of firt order ytem, we can get an infinite number of higher order ytem. Suppoe we have a well-mixed overflow tank of time contant τ. If we introduce a tep increae in the inlet temperature or concentration, we will (by the well-mixed aumption) 9
immediately detect a rie in the outlet tream the familiar firt-order lag repone a in the Figure 6. If we have intead two tank in erie, each half the volume of the original, we will detect a econd-order, igmoid repone at the outlet a in the Figure 7. If we continue to increae the number of tank in the erie, alway maintaining the total volume, we oberve a lower initial repone with a fater rie around the time contant. Thi behavior i hown in Figure 9. Figure 8. Well mixed tank in erie. Figure 9. Time repone of well mixed tank in erie. If taken to the limit of an infinite number of tank, we finally obtain a pure delay, in which the full tep diturbance i not een at the outlet until time τ ha paed. Thi i the dead time, or tranmiion delay; it i familiar to anyone who ha waited at the faucet for the hot water to arrive. That lead u to conider a imple model of the firt-order-plu-time-delay.
Approximate uing firt-order-plu-time-delay model The tranfer function of the firt-order-plu-time-delay, Ke θ G( ) τ + Step repone, t θ y( KM e τ 55 56 θ Figure. The time repone of firt-order-plu-time-delay ytem For a firt-order-plu-time-delay order model, we note the following characteritic (tep repone) a. The repone attain 63.% of it final repone at one time contant (t τ + θ ). b. The line drawn tangent to the repone at maximum lope (t θ) interect the % line at (t τ +θ ). [ee Fig. ] c. K i found from the teady tate repone for an input change magnitude M. The tep repone i eentially complete at t 5τ. In other word, the ettling time i t 5τ. There are two generally accepted parameter τ, θ, and K. graphical technique for determining model Method. Slope-intercept method: Firt, a lope i drawn through the inflection point of the proce reaction curve in Figure. Then t and θ are determined by inpection. Alternatively, τ can be found from the time that the normalized repone i 63.% complete or from determination of the ettling time, τ. Then et τ τ / 5. Method. Sundarean and Krihnawamy Method: Thi method avoid ue of the point of inflection contruction entirely to etimate the time delay. They propoed that two time, t and t, be etimated from a tep repone curve, correponding to the 35.3%
and 85.3% repone time, repectively. etimated from the following equation, The time delay and time contant are then θ.3t. 9t 57 ( t ) τ.67 t 58 Thee value of θ and τ approximately minimize the difference between the meaured repone and the model, baed on a correlation for many data et. Example F-6XL Roll Mode Time Contant In the early 98, two F-6 airplane were modified to extend the fuelage length and incorporate a large area delta wing planform. Thee two airplane, deignated the F-6XL, were deigned by the General Dynamic Corporation (now Lockheed Martin Tactical Aircraft Sytem) (Fort Worth, Texa) and were prototype for a derivative fighter evaluation program conducted by the United State Air Force. In thi method hown in figure, t i defined a the time when the lateral tick input reache 5 percent of maximum value. A line repreenting the maximum lope of the roll rate i plotted; the time at which thi line interect the x-axi i denoted a t. The roll rate reache 63 percent of it maximum value at t 3. The τ eff i the time difference between t and t. The τ r i the difference between t and t 3.
A ample comparion i hown a figure. Although the flight data how a higher orde r roll rate repone, the model accurately reproduce the initial delay and roll rate onet. Figure. Time hitory method for τ eff and τ r calculation. Figure. Sample reult of comparion between model and F-6XL flight data. 3
Etimating Second-order Model Parameter Uing Graphical Analyi In general, a better approximation to an experimental tep repone can be obtained by fitting a econd-order model to the data. Figure 3 how the range of hape that can occur for the tep repone model, K G( ) 59 τ + τ ( )( ) + Figure 3 include two limiting cae: τ /τ, where the ytem become firt order, and, τ /τ, the critically damped cae. The larger of the two time contant, τ, i called the dominant time contant. The aumed model i, θ Ke G( ) 6 ( τ + )( τ + ) Parameter are etimated uing o called Smith Method,. Determine t and t 6 from the tep repone.. Find ζ and t 6 /τ from Figure 4. 3. Find t 6 /τ from Figure 4 and then calculate τ (ince t 6 i known). Figure 3. Step repone for everal overdamped econd-order ytem. 4
Figure 4. Relationhip of ζ, τ, t, and t 6 in Smith method. 5