H83PDCH83PDCH83PDCH83PDC ProcessProcessProcessProcess Dynamic & Control Dynamic & Control Dynamic & Control Dynamic & ControlLecture 2 Mathematical Modelling of Chemical ProcessesH83PDC Process Dynamic & Control Lecture 2 - 2Lecture Outline Derivation of dynamic models of chemical processesH83PDC Process Dynamic & Control Lecture 2 - 3Mathematical Model Mathematical equations that describe/ represent the unsteady-state behaviour of a systemTheoretical ModelTheoretical ModelSemi-empirical ModelSemi-empirical ModelEmpirical ModelEmpirical ModelH83PDC Process Dynamic & Control Lecture 2 - 4Dynamic Process Models Derive from physical & chemical principles Mass & energy balances (conservation laws) Heat, mass, momentum transfer Thermodynamics, chemical kinetics Physical property relationships Consist of ordinary differential equations (ODE), partial differential equations (PDE), algebraic relations Can be solved numerically using computer simulationH83PDC Process Dynamic & Control Lecture 2 - 5Process ModelsProcess ModelsImprove understanding of the processImprove understanding of the processTrain plant operating personalTrain plant operating personalDevelop a control strategy for a new processDevelop a control strategy for a new processOptimize process operating conditionsOptimize process operating conditionsH83PDC Process Dynamic & Control Lecture 2 - 6Conservation LawsTheoretical models of chemical processes are based on conservation laws Conservation of Mass Conservation of Component irate of component i rate of component iaccumulation inrate of component i rate of component i(2-7)out produced = ` ` ) ) + ` ` ) )rate of mass rate of mass rate of mass(2-6)accumulation in out = ` ` ` ) ) )H83PDC Process Dynamic & Control Lecture 2 - 7 Conservation of EnergyThe general law of energy conservation is also called the First Law of Thermodynamics. It can be expressed as:The total energy of a thermodynamic system, Utot, is the sum of its internal energy, kinetic energy, and potential energy: = ` ` ` ) ) ) + + ` )rate of energy rate of energy in rate of energy outaccumulation by convection by convectionnet rate of heat addition net rate of workto the system from performed on the systhe surroundings ` )tem (2-8)by the surroundingsint(2-9)tot KE PEU U U U = + +H83PDC Process Dynamic & Control Lecture 2 - 8Development of Dynamic ModelsExample (1):A Blending ProcessAn unsteady-state mass balance for the blending system:rate of accumulation rate of rate of(2-1)of mass in the tank mass in mass out = ` ` ` ) ) )H83PDC Process Dynamic & Control Lecture 2 - 9where w1, w2, and w are mass flow rates.The unsteady-state component balance is:Steady-state material balance: Overall balance Component balance( )1 2(2-2)d Vw w wdt= + ( )1 1 2 2(2-3)d Vxw x wx wxdt= + H83PDC Process Dynamic & Control Lecture 2 - 10Simplify the dynamic model by assuming perfect mixing and that the density of the liquid () is a constant.For constant , , , ,Equation 2-13 can be simplified by expanding the accumulation term using the chain rule for differentiation of a product:1 2(2-12)dVw w wdt = + ( )1 1 2 2(2-13)d Vxwx wx wxdt= + ( )(2-14)d Vxdx dVV xdt dt dt = +H83PDC Process Dynamic & Control Lecture 2 - 11Substitution of (2-14) into (2-13) gives:Substitution of the mass balance in (2-12) for dV/dtin (2-15) gives:After canceling common terms and rearranging (2-12) &(2-16), a more convenient model form is obtained:1 1 2 2(2-15)dx dVV x wx wx wxdt dt + = + ( )1 2 1 1 2 2(2-16)dxV x w w w w x wx wxdt + + = + ( )( ) ( )1 21 21 21(2-17)(2-18)dVw w wdtw w dxx x x xdt V V = + = + H83PDC Process Dynamic & Control Lecture 2 - 12Example (2):Stirred-Tank Heating Process (Constant Holdup)Assumptions:1. Perfect mixing The exit T is also the temperature of the tank contents.2. Liquid holdup V is constant because the inlet & outlet flow rates are equal.3. The density and heat capacity C of the liquid are assumed to be constant. Their temperature dependence is neglected.4. Heat losses are negligible.Stirred-tank heating process with constant holdup, VH83PDC Process Dynamic & Control Lecture 2 - 13Assuming: Changes in potential energy and kinetic energy can be neglected because they are small in comparison with changes in internal energy. The net rate of work can be neglected because it is small compared to the rates of heat transfer and convection.Energy balance:( )int(2-10)dUwH Qdt= +)intthe internal energy of the systementhalpy per unit massmass flow raterate of heat transfer to the systemUHwQ====)( )denotes the differencebetween outlet and inletconditions of the flowingstreams; therefore- wH = rate of enthalpy of the inletstream(s) - the enthalpyof the outlet stream(s) =)H83PDC Process Dynamic & Control Lecture 2 - 14 For a pure liquid at low or moderate pressures, the internal energy is approximately equal to the enthalpy, Uint H H depends only on temperature We can assume that Uint= H and int = where the caret (^) means per unit mass A differential change in temperature, dT, produces a corresponding change in the internal energy per unit mass, where C is the constant pressure heat capacity (assumed to be constant). The total internal energy of the liquid in the tank is:int (2-29) dU dH CdT = =int int(2-30) U VU =H83PDC Process Dynamic & Control Lecture 2 - 15 Expression for the rate of internal energy accumulation can be derived from Eqs. (2-29) and (2-30): The liquid in the tank is at a temperature T and has an enthalpy, . Integrating Eq. 2-29 from a reference temperature Trefto Tgives,where refis the value of at Tref. Without loss of generality, we assume that ref= 0. Thus,int(2-31)dU dTVCdt dt =( ) (2-32)ref refH H CT T = ( )(2-33)refH CT T = H83PDC Process Dynamic & Control Lecture 2 - 16 For the inlet stream Substituting (2-33) and (2-34) into the convection term of (2-10) gives: Finally, substitution of (2-31) and (2-35) into (2-10)( ) ( ) ( )(2-35)i ref refwH wCT T wCT T (( = ( )(2-36)idTV C wCT T Qdt = +( )(2-34)i i refH CT T = H83PDC Process Dynamic & Control Lecture 2 - 17Learning OutcomesAbility to model process dynamics of simple unit operationTypes of modelConservation lawsODE, PDE, algebraic relation