ut d(s) = 2/5. ******** and U(S) = Y(S) Чртах Eman woman 7. £} стан __y(s) = # F B (max. ST+1 + B - чтам в P (²-5) 27. (
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The Linearised equation given as v a { c(t)} (CA_C² + q* cut) = q (t) (C₁-C*) + q (t) (x + ²). Ⓒ scaled variables y(t) = C(t); d (t) = PACE) ; u(A)=2B(t) (1) Omax Aman Imax. multiply едно о with. 1 on both sides. Cmax ✓ & (CU)) + q* cut) dt (max Act) (C) x Vinax + 9B²6) (²-2) d Brod <maz. <max- Cmax. автах Amar. va [d] + q* cu) c(t) dt Стал ACH) ar Aman gamax (AC) (man Cmar + 98(t) & Bran •Bmax (CB-C) . ✓ &/12 (y(E)) + 2 + y (4) Cman V₁_y(t) + ₁ + y(t) = d(t). & Amax ((-²) + uct) 23max (C₁3²_C²) Gmax Cmax Appy Laplace transfam. VS Ycs) + 2 YCS) = Amon (CA²_c²) dis) + Upman (₁)U(S) Eman cmar Vmax A*-*) des) + V/max ()ucs) . . Y(S) = [√₁² +₁+] [(CA+= VS Cman Cmax ) (CA-C*) чамак d(s) + (CB NBman, Jes (Y/₁+ s +₁1) [ (C₁² .U(S)] 2 Cmax 27 Cmaa (CÃ_(*) Tamax des) + ((8²_(²) BMax [CH q* Cmax q* Conax Y(5)= as &> v/q* 2 TS+1 Y(S)= _.UIS)]
VẫC«()} = 4/()C%A+ ()Cu-g()Ch+rV V (Con (t)) = Qa (1) Coma + Qu (1) Conx-q(1) Con+rV where r(moles/second.m³) is the rate for the reaction H₂0 un H + OH- which for completely dissociated ("strong") acids and bases is the only reaction in which Hand OH participate. We may eliminater from the equations by taking the difference to get a differential equation in terms of the excess of acid C(t) = £x(t) - Cow(t) Hence V(C(t)) = qa(t)C₁ + Qu(t)Cu - q(t)C(t) (1) where C₁=CHA-CONA and Chinhome This is the material balance for mixing tank without reaction. The overall model is bilinear due to the product of flow rate and concentration q(e)C(t). Note that C(t) will take on negative values when pH is above 7. The acid and base feed concentrations C, and Ca for both Hand OH-are assumed to be constants. Linearising equation (1) around a steady-state nominal point (denoted with an asterisk) V(C(t))+q°C (t) =q₁(1)(CC) + (1)(CC) is used to denote steady-state values, and q' = C₁+C₂ C= 10-10-144p C₁=CHA-CONA and С;= Ска-Соня Scaled variables for the input are introduced for the input, output and the disturbance as follows C(1) 9 (1) y(t) d(t)=9₁(1) u(t) 7) Assuming a unity negative feedback loop, derive the following transfer functions a. Gry(S) b. Gdy (s) c. Gre(s) d. Gde(s)