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Process Control CHAPTER VI BLOCK DIAGRAMS AND LINEARIZATION Example: Consider the stirred tank blending process. X1, w1 X2, w2 I/P AT X, w1 AC xsp Control objective: regulate the tank composition x, by adjusting w2. Disturbance variable: inlet composition x1 Assumptions: w1 is constant System is initially at steady-state Both feed and output compositions are dilute Feed flow rate is constant Stream 2 is pure material Process dV 1 ( w1 w2 w) dt dx w w 1 ( x1 x ) 2 ( x2 x ) dt V V dx V w1 x1 w1 x w2 x2 w2 x dt w w 1.0 0 V dx wx1 wx w2 dt 0 w x1 w x w2 dx V w x1 w x w2 dt V dx 1 x1 x w2 w dt w K dx x1 x Kw2 dt ( sX ( s) X (0)) X 1( s) X ( s) KW2( s) 0 X ( s)(s 1) X 1( s) KW2( s) X ( s) 1 G1 ( s) X 1( s) s 1 X ( s) K G2 ( s) W2( s) s 1 X1(s) 1 s 1 W2(s) X (s ) K2 s 1 Measuring Element Assume that the dynamic behavior of the composition sensor-transmitter can be approximated by a first-order transfer function; X m ( s ) Km X ( s ) m s 1 when, m , m can be assumed to be equal to zero. X (s) Km X m Controller P( s ) KC E ( s) proportional P( s ) 1 KC 1 E ( s) s I P( s ) K C 1 D s E ( s) proportional-integral proportional-derivative P( s ) 1 KC 1 Ds E ( s) s I proportional-derivative-integral Current to pressure (I/P) transducer Assuming a linear transducer with a constant steady state gain KIP. Pt( s ) K IP P( s ) P(s ) K IP Pt(s) Control Valve Assuming a first-order behavior for the valve gives; Kv W2( s ) Pt( s ) vs 1 X d (s) Change in exit composition due to change in inlet composition X´1(s) X u (s) Change in exit composition due to a change in inlet composition W´2(s) X sp (s) ~ X sp ( s ) Set-point composition (mass fraction) Set-point composition as an equivalent electrical current signal Linearization A major difficulty in analyzing the dynamic response of many processes is that they are nonlinear, that is, they can not be represented by linear differential equations. The method of Laplace transforms allows us to relate the response characteristics of a wide variety of physical systems to the parameters of their transfer functions. Unfortunately, only linear systems can be analyzed by Laplace Transforms. Linearization is a technique used to approximate the response of non linear systems with linear differential equations that can than be analyzed by Laplace transforms. The linear approximation to the non linear equations is valid for a region near some base point around which the linearization is made. Some non linear equations are as follows; qT (t ) AT (t ) 4 k T (t ) k0 e E / RT ( t ) f p(t ) k p(t ) A linearized model can be developed by approximating each non linear term with its linear approximation. A non linear term can be approximated by a Taylor series expansion to the nth order about a point if derivatives up to nth order exist at the point. The Taylor series for a function of one variable about xs is given as, 2 dF 1d F F ( x ) F ( xs ) x s ( x xs ) 2 dx 2! dx ( x xs ) R 2 xs xs is the steady-state value. x-xs=x’ is the deviation variable. The linearization of function consists of only the first two terms; dF F ( x ) F ( xs ) dx xs ( x xs ) Examples: F ( x) x 1 F ( x ) xs 2 1 2 1 12 xs ( x xs ) 2 k T (t ) k 0 e E RT ( t ) dk k T (t ) k (Ts ) Ts T (t ) Ts dT E d RT ( t ) k T (t ) k (Ts ) k e 0 dt T Ts k T (t ) k (Ts ) k 0 e E RTs E 2 RTs E k T (t ) k (Ts ) k (Ts ) 2 RTs Example: Consider CSTR example with a second order reaction. r A kC A 2 Mathematical modelling for the tank gives; dCA 2 V F (C A 0 C A ) VkC A dt The non linear term can be linearized as; C A C As 2C As (C A C As ) 2 2 The linearized model equation is obtained as; dCA 2 V F (C A 0 C A ) VkC A 2VkC As (C A C As ) dt Example: Considering a liquid storage tank with non linear relation for valve in output flow rate from the system; dh qi Cv dt h h hs qi qi qi , s A h Cv d h A qi h dt 2 hs Cv 1 R 2 hs d h 1 A qi h dt R