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**A Selection of Chemical Engineering Problems Solved using Mathematica**

Housam BINOUS National Institute of Applied Sciences and Technology 1- Chemical Kinetics and Catalysis 2- Applied Thermodynamics

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**Successive First-Order Reversible Reactions**

We consider successive first-order reversible reactions : Governing equations are :

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**Forward and Backward rate constants:**

Steady state solution [Ai] i Forward and Backward rate constants: ki,i+1=1 and ki+1,i=0.9

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**Transient solution A100, A200, A300, A400, A500,**

A600, A700, A800 and A900 A1 A1000

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**Eley-Rideal Mechanism**

Rate expressions for Reaction A + B ’ C 1/ equilibrium constant KA and rate constants are k1 and k2 2/ rate limiting step, rate constant is kp, equilibrium constant KAB 3/ equilibrium constant 1/KC and rate constants are k5 and k6

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**Adsorption competition with an inert component**

Rate expressions for Reaction A + B ’ C Adsorption competition with an inert component 1/ equilibrium constant KA and rate constants are k1 and k2 2/ equilibrium constant KB and rate constants are k3 and k4 3/ equilibrium constant KD and rate constants are k7 and k8 4/ rate limiting step, rate constant is kp, equilibrium constant KAB 5/ equilibrium constant 1/KC and rate constants are k5 and k6

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**Rate expression when H2 follows dissociative adsorption**

Reaction A + H2 ’ AH2 (for example: hydrogenation reactions) Rate expression when H2 follows dissociative adsorption 1/ equilibrium constant KA and rate constants are k1 and k2 2/ equilibrium constant KH2 and rate constants are k3 and k4 3/ rate limiting step, rate constant is kp 4/ equilibrium constant KAH2 and rate constants are k5 and k6 H 2 s catalyst A AH2 s catalyst

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**Liquid-liquid Equilibrium of Ternary mixture**

Liquid phase activity coefficients from NRTL model :

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**So far we have 5 equations and 6 unknowns, we need one more equation.**

We choose values for X1 and X2 than we solve the nonlinear system of 8 equations with 8 unknowns.

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**Liquid-liquid equilibrium for Water-Benzene-Ethanol at 25 °C**

Tie line Plait point

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**Liquid Liquid Extraction**

Liquid Liquid Equilibrium of ternary system Isopropyl ether-water-acetic acid at 20 °C and 1 atm : wt % water wt % acetic acid water rich phase isopropyl ether rich phase tie line

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**Hunter and Nash Graphical Equilibrium Stage Method**

F E1 RN Mixing Point M = F + S= E1 + RN Operating Point P = Ri-1 - Ei = F - E1 = RN - S E1 E2 EN S 1 2 N-1 N F R1 RN-1 RN

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**Stepping off Equilibrium Stages**

wt % water wt % acetic acid P S F E1 RN 5.35 equilibrium stages are needed to achive raffinate specifications

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**McCabe and Thiele Diagram**

wt % Acetic Acid in Raffinate wt % Acetic Acid in Extract Equilibrium Curve Operating Line 5.35 equilibrium stages are needed to achive raffinate specifications

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**Two Feed Extraction Column**

Total Feed FT = F1 + F2 F1 M S EN R1 OP2 OP1 F2 FT Mixing Point M = FT + S= R1 + EN Operating Points OP1 = Ri+1 - Ei = R1 - E0 OP2 = Ek - Rk+1= EN - RN+1 OP1 + OP2 = F2 F2 S=E0 E1 EN-1 EN 1 2 N-1 N R1 R2 RN F1=RN+1

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**Stepping off Equilibrium Stages**

wt % water wt % acetic acid F1 S EN R1 OP2 OP1 F2 FT 2.89 equilibrium stages are needed to achive raffinate specifications

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Residue Curve Map vapor y x liquid

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**Obtaining the boiling temperature :**

Liquid phase activity coefficients from Wilson model : Obtaining equilibrium vapor phase mole fractions :

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**Residue curve map for the ternary system **

acetone-methanol-chloroform at P=760 mmHg Azeotrope Residue Curve UN SN SP

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**y x Simple reactive distillation Chemical reaction Phase equilibrium**

Reaction equilibrium

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**Transformed compositions**

Equation for simple distillation with reaction equilibrium

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**XB XA Residue curve map for the isopropyl acetate chemistry at P=1 atm**

water Acetic Acid Isopropanol Isopropyl acetate XA XB Reactive azeotrope Need to take into account acetic acid dimerization

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**XB XA Residue curve map for the methyl acetate chemistry at P=1 atm**

Methanol Methyl acetate XB water Acetic Acid XA Need to take into account acetic acid dimerization

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**Flash Distillation P and T Vapor V yi Feed F zi Liquid L xi**

Rachford and Rice :

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**Equilibrium constants using virial equation of state:**

Equilibrium constants using the equations that fit the DePriester Charts : Equilibrium constants using virial equation of state: Phase Equilibrium :

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**Hydrocarbon Mixture P=3.5 bars and T=300 K Feed z1=0.2 z2=0.3 z3=0.4**

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**Vapor and Liquid Compositions**

Mass Balance equations give vapor and liquid compositions : P=3.5 bars T=300 K Feed F=1 z1=0.2 z2=0.3 z3=0.4 z4=0.05 z5=0.05 Liquid L=0.4330 x1=0.1066 x2=0.3425 x3=0.3790 x4=0.0886 x5=0.0830 Vapor V=0.5770 y1=0.2683 y2=0.2688 y3=0.4153 y4=0.0216 y5=0.0257

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**Binary ideal mixture with**

McCabe and Thiele Method for Distillation of Binary Ideal Mixture feed Zf=0.5 bottom xb=0.05 distillate xd=0.9 Binary ideal mixture with constant relative volatility = a

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**Pinch Point and Minimum Reflux Ratio**

y Feed line : Rectifying operating line :

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**9 equilibrium stages are needed to achieve the separation**

McCabe and Thiele Diagram R=1.5 Rmin x y 9 equilibrium stages are needed to achieve the separation

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**19 equilibrium stages are needed to achieve the separation**

Murphree Liquid Stage Efficiency EML=0.5 y x 19 equilibrium stages are needed to achieve the separation

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**Multicomponent Distillation**

feed Z1=0.3 Z2=0.3 Z3=0.4 bottom xb1=0.05 distillate xd1=0.95 xd2=0.049 xd3=0.001 Ternary ideal mixture of Pentane, Hexane and Heptane with constant relative volatilities = 6.35, 2.47 and 1

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**8.3 equilibrium stages are needed to achieve the separation**

Liquid Compositions Pentane mole fraction Hexane mole fraction Stripping Section Rectifying Section D F B Rectifying operating line : Stripping operating line : R=2.5 and S=1.35 8.3 equilibrium stages are needed to achieve the separation

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**Enthalpy-Composition Diagram for hexane-octane system at 760 mmHg**

Hexane mole fraction Enthalpy of saturated liquid and vapor tie line H(y) h(x) Conjugate line

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**Ponchon and Savarit method**

Enthalpy of saturated liquid and vapor Hexane mole fraction F P1 P2 D B L V feed q=0.41 Z1=0.5 bottom xb1=0.05 distillate xd1=0.95 7 stages are needed to achieve the separation

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Conclusion Mathematica’s algebraic, numerical and graphical capabilities can be put into advantage to solve several chemical engineering and chemistry problems including equilibrium-staged separations with McCabe-Thiele, Hunter-Nash and Ponchon-savarit Methods.

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