Chemical Process Control ChE 462 Theoretical Models of Chemical Processes Develop understanding of process a mathematical hypothesis of process mechanisms Dynamic model is used to predict how process responds to given input Modeling Types Dynamic vs. Steady-state Steady-state Variables not a function of time Dynamic Variables are a function of time Empirical vs. Mechanistic models Empirical Models only limited representation of the process. Simpler model forms. Mechanistic Models Rely on understanding of a process. Derived from fundamental laws of physics and chemistry. Useful for exploration of new conditions. Linear vs Nonlinear Linear: simpler model form, solution has the form ezx (useful for empirical model) ; Nonlinear: more complex and difficult to identify. Modeling procedure 1. Decide modeling variables and fundamental laws (Mass, Energy and/or Momentum) 2. Make appropriate assumptions (Simplify, e.g. isothermal, no friction, incompressible flow, etc,…) 3. Develop the model equations and Check model degree of freedom do we have more unknowns than equations? 4. Determine unknown constants e.g. fluid density , temperature, viscosity 5. Solve model equations 6. Validity the model compare to experimental data/sensitivity analysis Process Dynamic Model The balance equations to model chemical engineering processes. Mass Conservation Equations Thermal Energy Balance Equation Rate of accumulation  =  of thermal energy  + Rate of convective  heat transfer    entering the system  Net rate of  heat generation    by reaction  Mass, Moles, or Energy Balances + − Rate of convective  heat transfer    leaving the system  Net rate of heat transfer   through the boundaries    of the system  Rate of  Rate Entering   Accumulation =  the  System     Rate Leaving −   the System   Rate of Generation by  +   Reaction within the System   Part 1: Process Model Examples 1. Blending systems 2. Series Reaction 3. Stirred Tank and Bioreactor Part 2: Process optimization Example 1: Control A Blending Process X1 and W1 V, ρ X 2, W2 Product A X, W ???: X=Xsp Overall balance: Component A balance: Model Equations X2, W2 Mixture of A and B X1 and W1 Product X, W ???: X=Xsp Feed forward control Model AC Method 1 Mixture of A and B X1 and W1 AT X2=1 W2 X1 Product X, W t ???: X=Xsp A simple feedback control algorithm (proportional control) Method 2 Mixture of A and B X1 and W1 X1 AC X2 W2 AT Product X, W t ???: X=Xsp Unsteady State Blending Process Considering the density of the liquid, ρ, is a constant Overall balance: 0≠W1+W2-W Component A balance: 0≠W1X1 +W2X2-WX dV ρ = w1 + w2 − w dt ρd (Vx) = w1 x1 + w2 x2 − wx dt How to find dx/dt??? Equation below can be simplified by expanding the accumulation term using the “chain rule” for differentiation of a product: Degree of freedom NF=NV-NE NF: Degree of freedom NV: Number of Process Variables (time is not process variable) NE: Number of Independent Equation NFC: the control degree of freedom (independent material/energy streams), e.g., Temp, Flow, Comp… Considering the density of the liquid, ρ, is a constant For example: freedom = ? dV 1 = ( w1 + w2 − w) dt ρ dx w1 w2 = ( x1 − x) + ( x2 − x ) ρV dt ρV Calculation (Example 2.1) A constant liquid holdup: a) Steady state value of x V=2 m3 , ρ = 900kg/ m3 b) W1 change to 400kg/min suddenly Expression of x(t); V is still constant. Pure A X2=0.75 W2 =200 kg/min Mixture of A and B X1 =0.4 W1 =500 kg/min AT Product X, W Steady state: 0 = w1 + w2 − w 0 = w1 x1 + w2 x2 − wx Solve: x(0)=0.5 W1 change to 400kg/min suddenly, V is constant 0 = w1 + w2 − w dx Vρ = w1 x1 + w2 x2 − wx dt Initial condition: x(0)=0.5 Plug w1, x1, w2, x2, w into the equation above. 600 Example Plot 500 400 X(t)=0.517-0.017exp (-t/3) W 300 W1 200 100 Control 1: FF, good model Control 2: FF, bad model Control 3: FB, large Kc Control 4: FB, small Kc 0.52 0 0 10 15 20 25 30 35 T no control Feedforward 0.516 ??? 0.512 X 5 0.508 0.504 Feedback 0.5 0.496 0 5 10 15 Xsp=0.5 20 25 30 35 Example 2: Model for Product Composition for CSTR with a Series Reaction r1 r2 → B  → C A  F: mass/time CAO , F, ρ, Vr r1 = k1 ⋅ CA2 r2 = k 2 ⋅ CB FC Component A: FT Feed CA, CB, CC Product Component B: AT Equations for Component Compositions. Vr is constant. Component C: Example 3: Stirred tank heater Chapter 2: 2-3 Stirred-Tank Heating Process Assumptions: 1. Perfect mixing; thus, the exit temperature T is also the temperature of the tank contents. 2. The liquid holdup V is constant because the inlet and outlet flow rates are equal. 3. The density ρ and heat capacity C of the liquid are assumed to be constant. Thus, their temperature dependence is neglected. 4. Heat losses are negligible. Figure 2.3 Model Development - I For a pure liquid at low or moderate pressures, the internal energy is approximately equal to the enthalpy, Uint ≈ H , and H depends only on temperature. Consequently, in the subsequent development, ^ ^ we assume that Uint = H and Uint = H where the caret (^) means per unit mass. As shown in Appendix B, a differential change in temperature, dT, produces a corresponding change in the internal energy per unit mass, ^ ^ d U int = d H = CdT (2-29) where C is the constant pressure heat capacity (assumed to be constant). The total internal energy of the liquid in the tank is: U int = ρVUˆ int (2-30) Model Development - II An expression for the rate of internal energy accumulation can be derived from Eqs. (2-29) and (2-30): dU int dT (2-31) = ρVC dt dt Note that this term appears in the general energy balance Suppose that the liquid in the tank is at a temperature T and has an enthalpy, Hˆ . Integrating Eq. 2-29 from a reference temperature Tref to T gives, ^ ^ (2-32) H − H ref = C (T − T ) ref where Hˆ ref is the value of Hˆ at Tref. Without loss of generality, we assume that Hˆ ref = 0 (see Appendix B). Thus, (2-32) can be written as: = Hˆ C T − T (2-33) ( ref ) Model Development - III For the inlet stream Substituting (2-33) and (2-34) into the convection term of (2-10) gives: (2-10) Finally, substitution of (2-31) and (2-35) into (2-10) Degrees of Freedom Analysis for the Constant Volume Stirred-Tank Model parameters: dT V ρ C = wC (Ti − T ) + Q dt (2-36) variables: equation: The process variables are classified as: w Models of Bioreactor Operations Model of Fed-Batch Bioreactor Growth Rate rg = µ X (2-93) Monod Equation µ = µmax S Ks + S µ is Specific growth rate Fed-batch reactor for a bioreaction. (2-94) Monod Equation Reaction rate 0 order 1/2 umax µ = µmax S Ks + S (2-94) Ks 1st order Substrate Concentration Modeling Assumptions 1. 2. 3. 4. The fed-batch reactor is perfectly mixed. Isothermal reactor operation can be assumed. The liquid density is constant. The broth in the bioreactor can be approximated as a homogenous liquid. 5. The rate of product formation per unit volume rp can be expressed as rp = YP / X rg (2-95) where the product yield coefficient YP/X is defined as: YP / X = mass of product formed mass of new cells formed (2-96) • General Form of Each Balance {Rate of accumulation = } {rate in} + {rate of formation} (2-97) Model of Fed-Batch • Individual Component Balances for Fed Batch • Cells: • Product: • Substrate: • Overall Mass Balance • Mass: Part 2: Process Control and Optimization • Control has to do with adjusting flow rates to maintain the controlled variables of the process at specified set-points. • Optimization chooses the values for key set-points such that the process operates at the “best” economic conditions. Optimization and Control of a CSTR Optimizer RSP What is optimal operation temperature? TC RSP Feed FC FV CA0 FT Steam A→B→C TT Product CA,CB, CC Optimization Example (steady state) Economic Objective Function Product Substrate Φ = Q C A VA + Q C B VB + Q CC VC − Q C A0 VAF • • • • VB > VC, VA, or VAF V is the chemical values. At low T, little formation of B At high T, too much of B reacts to form C Therefore, the exits an optimum reactor temperature, T* Graphical Solution of Optimum Reactor Temperature, T* Process Design and Economic Analysis 1. Develop a process model 2. Convert process model to financial model 3. Optimization Algorithm 1. Select initial guess (within certain ranges) for reactor temperature 2. Evaluate CA, CB, and CC 3. Evaluate Φ 4. Choose new reactor temperature and return to 2 until T* identified.