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Chapter 2 Mathematical Modeling of Chemical Processes Mathematical Model (Eykhoff, 1974) “a representation of the essential aspects of an existing system (or a system to be constructed) which represents knowledge of that system in a usable form” Everything should be made as simple as possible, but no simpler. General Modeling Principles • The model equations are at best an approximation to the real process. Chapter 2 • Adage: “All models are wrong, but some are useful.” • Modeling inherently involves a compromise between model accuracy and complexity on one hand, and the cost and effort required to develop the model, on the other hand. • Process modeling is both an art and a science. Creativity is required to make simplifying assumptions that result in an appropriate model. • Dynamic models of chemical processes consist of ordinary differential equations (ODE) and/or partial differential equations (PDE), plus related algebraic equations. Chapter 2 Table 2.1. A Systematic Approach for Developing Dynamic Models 1. State the modeling objectives and the end use of the model. They determine the required levels of model detail and model accuracy. 2. Draw a schematic diagram of the process and label all process variables. 3. List all of the assumptions that are involved in developing the model. Try for parsimony; the model should be no more complicated than necessary to meet the modeling objectives. 4. Determine whether spatial variations of process variables are important. If so, a partial differential equation model will be required. 5. Write appropriate conservation equations (mass, component, energy, and so forth). Table 2.1. (continued) Chapter 2 6. Introduce equilibrium relations and other algebraic equations (from thermodynamics, transport phenomena, chemical kinetics, equipment geometry, etc.). 7. Perform a degrees of freedom analysis (Section 2.3) to ensure that the model equations can be solved. 8. Simplify the model. It is often possible to arrange the equations so that the dependent variables (outputs) appear on the left side and the independent variables (inputs) appear on the right side. This model form is convenient for computer simulation and subsequent analysis. 9. Classify inputs as disturbance variables or as manipulated variables. Chapter 2 Modeling Approaches Physical/chemical (fundamental, global) • Model structure by theoretical analysis Material/energy balances Heat, mass, and momentum transfer Thermodynamics, chemical kinetics Physical property relationships • Model complexity must be determined (assumptions) • Can be computationally expensive (not realtime) • May be expensive/time-consuming to obtain • Good for extrapolation, scale-up • Does not require experimental data to obtain (data required for validation and fitting) • Conservation Laws Theoretical models of chemical processes are based on conservation laws. Chapter 2 Conservation of Mass rate of m ass rate of m ass rate of m ass in out accum ulation (2-6) Conservation of Component i rate of com ponent i rate of com ponent i accum ulation in rate of com ponent out i rate of com ponent produced i (2-7) Conservation of Energy Chapter 2 The general law of energy conservation is also called the First Law of Thermodynamics. It can be expressed as: rate of en ergy rate of en ergy in rate of en ergy o u t accu m u lation b y con vection b y con vection n et rate of h eat ad d ition to th e sy stem from th e su rrou n d in gs n et rate of w ork p erform ed on th e sy s tem b y th e su rrou n d in gs (2-8) The total energy of a thermodynamic system, Utot, is the sum of its internal energy, kinetic energy, and potential energy: U tot U int U K E U PE (2-9) Black box (empirical) • Large number of unknown parameters • Can be obtained quickly (e.g., linear regression) Chapter 2 • Model structure is subjective • Dangerous to extrapolate Semi-empirical • Compromise of first two approaches • Model structure may be simpler • Typically 2 to 10 physical parameters estimated (nonlinear regression) • Good versatility, can be extrapolated • Can be run in real-time • linear regression y c 0 c1 x c 2 x 2 • nonlinear regression Chapter 2 y K 1 e t / • number of parameters affects accuracy of model, but confidence limits on the parameters fitted must be evaluated • objective function for data fitting – minimize sum of squares of errors between data points and model predictions (use optimization code to fit parameters) • nonlinear models such as neural nets are becoming popular (automatic modeling) Chapter 2 Number of births (West Germany) Number of sightings of storks Uses of Mathematical Modeling • to improve understanding of the process • to optimize process design/operating conditions • to design a control strategy for the process • to train operating personnel •Development of Dynamic Models Chapter 2 •Illustrative Example: A Blending Process An unsteady-state mass balance for the blending system: rate of accum ulation rate of of m ass in the tank m ass in rate of m ass out (2-1) or d V ρ dt w1 w 2 w (2-2) Chapter 2 where w1, w2, and w are mass flow rates. • The unsteady-state component balance is: d V ρ x dt w1 x1 w 2 x 2 w x (2-3) The corresponding steady-state model was derived in Ch. 1 (cf. Eqs. 1-1 and 1-2). 0 w1 w 2 w (2-4) 0 w1 x1 w 2 x 2 w x (2-5) Chapter 2 The Blending Process Revisited For constant , Eqs. 2-2 and 2-3 become: dV dt w1 w 2 w d V x dt w1 x1 w 2 x 2 w x (2-12) (2-13) Equation 2-13 can be simplified by expanding the accumulation term using the “chain rule” for differentiation of a product: d V x V dt dx x dt dV (2-14) dt Chapter 2 Substitution of (2-14) into (2-13) gives: V dx x dV dt dt w1 x1 w 2 x 2 w x (2-15) Substitution of the mass balance in (2-12) for dV / dt in (2-15) gives: V dx dt x w1 w 2 w w1 x1 w 2 x 2 w x (2-16) After canceling common terms and rearranging (2-12) and (2-16), a more convenient model form is obtained: dV 1 dt dx w1 dt w1 w 2 w V x1 x w2 V (2-17) x2 x (2-18) Chapter 2 Chapter 2 Stirred-Tank Heating Process Figure 2.3 Stirred-tank heating process with constant holdup, V. Stirred-Tank Heating Process (cont’d.) Chapter 2 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 and heat capacity C of the liquid are assumed to be constant. Thus, their temperature dependence is neglected. 4. Heat losses are negligible. Chapter 2 For the processes and examples considered in this book, it is appropriate to make two assumptions: 1. Changes in potential energy and kinetic energy can be neglected because they are small in comparison with changes in internal energy. 2. The net rate of work can be neglected because it is small compared to the rates of heat transfer and convection. For these reasonable assumptions, the energy balance in Eq. 2-8 can be written as dU int w H Q dt denotes the difference betw een outlet and inlet conditions of the flow ing stream s; therefore U int the internal energy of the system H enthalpy per unit m ass w m ass flow rate (2-10) -Δ w H = rate of enthalpy of the inlet Q rate of heat transfer to the system stream (s) - the enthalpy of the outlet stream (s) Chapter 2 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 Uˆ int Hˆ 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, dUˆ int , dUˆ int dHˆ C dT (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 V Uˆ 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): Chapter 2 dU int V C dt dT (2-31) dt Note that this term appears in the general energy balance of Eq. 210. 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, Hˆ Hˆ ref C T Tref (2-32) 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 Tref (2-33) Model Development - III For the inlet stream Chapter 2 Hˆ i C Ti Tref (2-34) Substituting (2-33) and (2-34) into the convection term of (2-10) gives: w Hˆ w C Ti Tref w C T Tref (2-35) Finally, substitution of (2-31) and (2-35) into (2-10) V C dT dt w C Ti T Q (2-36) steam-heating: V C dT dt Q ws H v w C (Ti T ) w s H v (1) 0 w C (Ti T ) w s H v (2) subtract (2) from (1) V C dT dt w C (T T ) ( w s w s ) H v divide by wC V dT w dt T T H v wC ( ws w s ) Define deviation variables (from set point) y T T T is desired operating point Chapter 2 u ws w s V dy w s (T ) from steady state y w dt note w hen H v 1 dt note that wC dy 0 dt dy u H v wC K p and V w 1 y K pu y K pu G eneral linear ordinary differential equ ation solution: sum of exponential(s) S uppose u 1 (unit step response) t y (t ) K p 1 e 1 Chapter 2 Example 1: Ti = 40 C, T = 90 C, Ti = 0 C o o s.s. balance: o w C ( T - Ti ) = w s H v w s = 0.83 10 g hr Chapter 2 6 H v = 600 cal g o C = l cal g C 4 w = 10 kg hr = 10 kg m 3 V = 20 m 3 3 V 2 10 kg 4 V w 2 dy dt 2 10 kg 4 4 2hr 10 kg hr = -y + 6 10 u y T T u ws ws -5 dynamic model Step 1: t=0 double ws T (0) = T y(0) = 0 u = +0.83 10 g hr 6 Chapter 2 2 dy dt = -y + 50 y = 5 0 l - e final -0 .5 t o T = y ss + T = 50 + 90 = 140 C Step 2: maintain Step 3: then set 2 dy dt o T = 140 C / 24 hr u = 0, w s = 833 kg hr = -y + 6 10 u, y(0) = 50 -5 Solve for u = 0 y = 50e -0.5t t y 0 (self-regulating, but slow) how long to reach y = 0.5 ? Chapter 2 Step 4: How can we speed up the return from 140°C to 90°C? ws = 0 vs. ws = 0.83106 g/hr at s.s ws =0 y -50°C T 40°C Process Dynamics Process control is inherently concerned with unsteady state behavior (i.e., "transient response", "process dynamics") Stirred tank heater: assume a "lag" between heating element temperature Te, and process fluid temp, T. Chapter 2 heat transfer limitation = heA(Te – T) Energy balances w C Ti +h e A (Te -T )-w C T =m C Tank: Chest: Q - h e A(T dT At s.s. dt 0, e dT e - T) = m e C e dT dt dT e dt 0 dt Specify Q calc. T, Te 2 first order equations 1 second order equation in T Relate T to Q (Te is an intermediate variable) y=T-T u=Q-Q mm e C e d 2 y m eC e m eC e m dy 1 y u wC w dt wC h eA e Chapter 2 wh e A e dt 2 T i fixed Note Ce 0 yields 1st order ODE (simpler model) Fig. 2.2 Rv q = 1 R h v Rv: line resistance A dh Chapter 2 dt qi 1 h P p gh (2 - 57) Rv linear ODE P p gh If q = Cv A dh dt * P - Pa qi C v Pa : ambient * nonlinear ODE ρgh q i C v h pressure (2-61) Chapter 2 Chapter 2 Table 2.2. Degrees of Freedom Analysis 1. List all quantities in the model that are known constants (or parameters that can be specified) on the basis of equipment dimensions, known physical properties, etc. 2. Determine the number of equations NE and the number of process variables, NV. Note that time t is not considered to be a process variable because it is neither a process input nor a process output. 3. Calculate the number of degrees of freedom, NF = NV - NE. 4. Identify the NE output variables that will be obtained by solving the process model. 5. Identify the NF input variables that must be specified as either disturbance variables or manipulated variables, in order to utilize the NF degrees of freedom. Chapter 2 Degrees of Freedom Analysis for the Stirred-Tank Model: 3 parameters: V , ,C 4 variables: T , Ti , w , Q 1 equation: Eq. 2-36 Thus the degrees of freedom are NF = 4 – 1 = 3. The process variables are classified as: 1 output variable: T 3 input variables: Ti, w, Q For temperature control purposes, it is reasonable to classify the three inputs as: 2 disturbance variables: Ti, w 1 manipulated variable: Q Biological Reactions Chapter 2 • Biological reactions that involve micro-organisms and enzyme catalysts are pervasive and play a crucial role in the natural world. • Without such bioreactions, plant and animal life, as we know it, simply could not exist. • Bioreactions also provide the basis for production of a wide variety of pharmaceuticals and healthcare and food products. • Important industrial processes that involve bioreactions include fermentation and wastewater treatment. • Chemical engineers are heavily involved with biochemical and biomedical processes. Bioreactions • Are typically performed in a batch or fed-batch reactor. Chapter 2 • Fed-batch is a synonym for semi-batch. • Fed-batch reactors are widely used in the pharmaceutical and other process industries. • Bioreactions: cells substrate m ore cells + products (2-90) • Yield Coefficients: YX /S YP / S m ass of new cells form ed (2-91) m ass of substrate consum ed to form new cells m ass of product form ed m ass of substrate consum ed to form product (2-92) Chapter 2 Fed-Batch Bioreactor Monod Equation rg X (2-93) Specific Growth Rate m ax Figure 2.11. Fed-batch reactor for a bioreaction. S Ks S (2-94) Chapter 2 • Modeling Assumptions 1. The exponential cell growth stage is of interest. 2. The fed-batch reactor is perfectly mixed. 3. Heat effects are small so that isothermal reactor operation can be assumed. 4. The liquid density is constant. 5. The broth in the bioreactor consists of liquid plus solid material, the mass of cells. This heterogenous mixture can be approximated as a homogenous liquid. 6. The rate of cell growth rg is given by the Monod equation in (293) and (2-94). • Modeling Assumptions (continued) 7. The rate of product formation per unit volume rp can be expressed as Chapter 2 r p Y P / X rg (2-95) where the product yield coefficient YP/X is defined as: YP / X m ass of product form ed (2-96) m ass of new cells form ed 8. The feed stream is sterile and thus contains no cells. • General Form of Each Balance R ate of accum ulation rate in rate of form ation (2-97) • Individual Component Balances • Cells: d ( XV ) dt Chapter 2 • Product: V rg d PV dt • Substrate: d ( SV ) dt F Sf • Overall Mass Balance (2-98) V rp 1 YX /S (2-99) V rg 1 YP / S V rP (2-100) • Mass: d (V ) dt F (2-101) Chapter 2 Previous chapter Next chapter