L4-1 Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign. Ideal CSTR Design Eq with X A : Review: Design Eq & Conversion BATCH SYSTEM: FLOW SYSTEM: Ideal Batch Reactor Design Eq with X A : Ideal SS PFR Design Eq with X A : Ideal SS PBR Design Eq with X A : j ≡ stoichiometric coefficient; positive for products, negative for reactants

L4-2 Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign. Review: Sizing CSTRs We can determine the volume of the CSTR required to achieve a specific conversion if we know how the reaction rate r j depends on the conversion X j Ideal SS CSTR design eq. Volume is product of F A0 /-r A and X A Plot F A0 /-r A vs X A (Levenspiel plot) V CSTR is the rectangle with a base of X A,exit and a height of F A0 /-r A at X A,exit

L4-3 Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign. Area = V PFR or W catalyst, PBR Review: Sizing PFRs & PBRs We can determine the volume (catalyst weight) of a PFR (PBR) required to achieve a specific X j if we know how the reaction rate r j depends on X j Ideal PFR design eq. Plot F A0 /-r A vs X A (Experimentally determined numerical values) V PFR (W PBR ) is the area under the curve F A0 /-r A vs X A,exit Ideal PBR design eq.

L4-4 Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign. Numerical Evaluation of Integrals (A.4) Simpson’s one-third rule (3-point):Trapezoidal rule (2-point): Simpson’s three-eights rule (4-point): Simpson’s five-point quadrature :

L4-5 Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign. Review: Reactors in Series 2 CSTRs2 PFRs CSTR→PFR V CSTR1 V PFR2 V CSTR1 V CSTR2 V PFR1 V CSTR2 V CSTR1 + V PFR2 ≠ V PFR1 + C CSTR2 PFR→CSTR If is monotonically increasing then:

L4-6 Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign. L4: Rate Laws & Stoichiometry Reaction Rates (–r A ) 1.Concentration 2.Temperature 3.Reversible reactions How to derive an equation for –r A [–r A = f(X A )] 1.Relate all r j to C j 2.Relate all C j to V or  3.Relate V or  to X A 4.Put together

L4-7 Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign. Concentration and Temperature Molecular collision frequency  concentration Rate of reaction  concentration Reaction rate is a function of temperature and concentration C A : Concentration of A C B : Concentration of B As temperature increases, collision frequency increases Rate of reaction = f [( C A, C B, ……), (T)] At constant temperature : r = f(C A, C B, …….) Specific rate of reaction, or rate constant, for species A is a function of temperature

L4-8 Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign. Elementary Reactions & Rate Laws Dependence of reaction rate –r A on concentration of chemical species in the reaction is experimentally determined Elementary reaction: involves 1 step (only) Stoichiometric coefficients in an elementary reaction are identical to the powers in the rate law: Reaction order:  order with respect to A  order with respect to B Overall reaction order n =  Zero order: -r A = k A k is in units mol/(volume∙time) 1st order: -r A = k A C A k is in units time -1 2nd order: -r A = k A C A 2 k is in units volume/(mol∙time) 3rd order: -r A = k A C A 3 k is in units volume 2 /(mol 2 ∙time)

L4-9 Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign. Examples: This reaction is not elementary, but under some conditions it follows an elementary rate law Forward reaction is 2nd order with respect to NO and 1 st order with respect to O 2 (3nd order overall) Overall Stoichiometric Equations Overall equations describe the overall reaction stoichiometry Reaction order cannot be deduced from overall equations Compare the above reaction with the nonelementary reaction between CO and Cl 2 Forward reaction is 1 st order with respect to CO and 3/2 order with respect to Cl 2 (5/2 order overall)

L4-10 Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign. Specific Rate Constant, k A k A is strongly dependent on temperature Where : A = Pre-exponential factor or frequency factor (1/time) E = Activation energy, J/mol or cal/mol R = Gas constant, J/mol K (or cal/mol K) T = Absolute temperature, K Arrhenius Equation To determine activation energy E, run the reaction at several temperatures, and plot ln k vs 1/T. Slope is –E/R Taking ln of both sides: 1/T ln k -E/R

L4-11 Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign. Reversible Reactions K C : concentration equilibrium constant (capital K) At equilibrium, the reaction rate is zero, r A =0 Rate of disappearance of A (forward rxn): Rate of generation of A (reverse reaction): Thermodynamic equilibrium relationship K C is temperature dependent (no change in moles or  C P ):  H RX : heat of reaction If K C is known for temperature T 1, K C for temperature T can be calculated

L4-12 Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign. L4: Rate Laws & Stoichiometry Reaction Rates (–r A )  1.Concentration  2.Temperature  3.Reversible reactions  How to derive an equation for –r A [–r A = f(X A )] 1.Relate all r j to C j 2.Relate all C j to V or  3.Relate V or  to X A (Wednesday) 4.Put together (Wednesday)

L4-13 Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign. 1. Relate all r j to C j r A as a function of C j is given by the rate law The rate relative to other species (r j ) is determined by stoichiometry “A” is the limiting reagent r j is negative for reactants, positive for products In general: j ≡ stoichiometric coefficient positive for products, negative for reactants

L4-14 Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign. For the reaction the rate of O 2 disappearance is 2 mol/dm 3 s (-r O2 = 2 mol/dm 3 s). What is the rate of formation of NO 2 ? r NO2 = 4 mol/dm 3 s

L4-15 Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign. 2a. Relate all C j to V (Batch System) Reaction rate is a function of C j : How is C j related to V and X A ? Batch: Put N A in terms of X A : Do the same for species B, C, and D: C j is in terms of X A and V. But what if V varies with X A ? That’s step 3a!

L4-16 Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign. 2a. Additional Variables Used in Textbook Book uses term Θ i : So species N i0 can be removed from the equation for C i Multiply numerator by N A0 /N A0 :

L4-17 Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign. 3a. Relate V to X A (Batch System) Volume is constant (V = V 0 ) for: Most liquid phase reactions Gas phase reactions if moles reactants = moles products If the volume varies with time, assume the equation of state for the gas phase: At time t: PV = ZN T RT and at t=0: P 0 V 0 = Z 0 N T0 RT 0 P: total pressure, atmZ: compressibility factor N T : total molesT: temperature, K R: ideal gas constant, dm 3 ∙atm/mol∙K Want V in terms of X A. First find and expression for V at time t: N T at time t is: What is N T at t?

L4-18 Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign. 3a. Relate V to X A (continued) Can we use the eq. for N T above to find an expression for N T /N T0 ?

L4-19 Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign. What is the meaning of ε? When conversion is complete (X A =1): The expansion factor  is the fraction of change in V per mol A reacted that is caused by a change in the total number of moles in the system If we put the following equation in terms of ε: where

L4-20 Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign. 4a. Put it all together (batch reactor) Batch: For a given X A, we can calculate C j and plug the C j into –r A =kC j n What about flow systems?

L4-21 Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign. 2b. Relate all C j to  (Flow System) How is C j related to aa nd X j ? Flow: Reaction rate is a function of C j : Put F A in terms of X A : Do the same for species B, C, and D: We have C j in terms of X A and , but what if  varies with X A ? That’s step 3b!

L4-22 Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign. 3b. Relate  to X A (Flow System) Start with the equation of state for the gas phase: What is C T0 at the entrance of the reactor? Rearrange to put in terms of C T, where C T = N T /V: Can we relate C T to  ? Rearrange to put in terms of  : Put in terms of  0 : Use these 2 equations to put  in terms of known or measurable quantities

L4-23 Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign. 3b. Relate  to X A (continued) When conversion is complete (X A =1):

L4-24 Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign. 4b. Put it all together (flow reactor) Flow: For a given X A, we can calculate C j and plug the C j into –r A =kC j n This is the same equation as that for the batch reactor!

L4-25 Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign. 4. Summary: C j in terms of X j Batch: Flow: This is the same equation as that for the batch reactor! For a given X A, we can calculate C j and plug the C j into –r A =kC j n