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Catalytic Cracking of Ethane

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Catalytic Cracking of Ethane

Introduction
 

Ethylene and hydrogen are generated by a steam cracker using ethane as a feedstock.

 

C2H6 (g) # C2H4 (g) + H2 (g)

 

The feed is 4 mol of H2O to 1 mol of ethane, with the cracker operating at 1.01 bar and 1000 K. The products contain CH4, C2H4, C2H2, CO2, CO, O2, H2, H2O, and C2H6 in the molar amounts n1 to n9.

 

 

Species

In Feed

In Product

1

CH4

 

n1

2

C2H4

 

n2

3

C2H2

 

n3

4

CO2

 

n4

5

CO

 

n5

6

O2

 

n6

7

H2

 

n7

8

H2O

4

n8

9

C2H6

1

n9
 

 

This application calculates the composition of the reaction products by

 

• 

calculating the Gibbs Energy of Formation of the individual species in the products, employing data from the ThermophysicalData:-Chemicals  package
• 

constructing a function that describes Gibbs Energy of the products as a function of product composition
• 

and minimizing the Gibbs Energy of the products, subject to constraints on the amount of carbon and hydrogen in the system.

 

> 

restart:
with(ThermophysicalData:-Chemicals):
with(Optimization):
> 

temp := 1000 * Unit(K):
R    := 8.314 * Unit(J/mol/K):

Physical Properties
 

Enthalpies

> 

h_CH4  := Property("Hmolar", "CH4",            temperature = temp):
h_C    := Property("Hmolar", "C(gr)",          temperature = temp):
h_H2   := Property("Hmolar", "H2",             temperature = temp):
h_H2O  := Property("Hmolar", "H2O",            temperature = temp):
h_O2   := Property("Hmolar", "O2",             temperature = temp):
h_CO   := Property("Hmolar", "CO",             temperature = temp):
h_CO2  := Property("Hmolar", "CO2",            temperature = temp):
h_C2H4 := Property("Hmolar", "C2H4",           temperature = temp):
h_C2H2 := Property("Hmolar", "C2H2,acetylene", temperature = temp):
h_C2H6 := Property("Hmolar", "C2H6",           temperature = temp):

 

Entropies

> 

s_CH4  := Property("Smolar", "CH4",            temperature = temp):
s_C    := Property("Smolar", "C(gr)",          temperature = temp):
s_H2   := Property("Smolar", "H2",             temperature = temp):
s_H2O  := Property("Smolar", "H2O",            temperature = temp):
s_O2   := Property("Smolar", "O2",             temperature = temp):
s_CO   := Property("Smolar", "CO",             temperature = temp):
s_C    := Property("Smolar", "C(gr)",          temperature = temp):
s_CO2  := Property("Smolar", "CO2",            temperature = temp):
s_C2H4 := Property("Smolar", "C2H4",           temperature = temp):
s_C2H2 := Property("Smolar", "C2H2,acetylene", temperature = temp):
s_C2H6 := Property("Smolar", "C2H6",           temperature = temp):

Gibbs Energy of Formation
 

CH4
 

C (gr) + 2 H2 (g) → CH4 (g)

 

Enthalpy change

> 

DeltaH := 1*Unit(mol)*h_CH4 - (1*Unit(mol)*h_C + 2*Unit(mol)*h_H2)

-89068.55636*Units:-Unit(mol)*Units:-Unit(J/mol)

 (3.1.1)

Entropy change

> 

DeltaS := 1*Unit(mol)*s_CH4 - (1*Unit(mol)*s_C + 2*Unit(mol)*s_H2)

-108.5548332*Units:-Unit(mol)*Units:-Unit(J/(mol*K))

 (3.1.2)

Gibbs Energy of Formation

> 

DeltaG_CH4 := DeltaH - DeltaS*temp

108554.8332*Units:-Unit(mol)*Units:-Unit(J/(mol*K))*Units:-Unit(K)-89068.55636*Units:-Unit(mol)*Units:-Unit(J/mol)

 (3.1.3)

H2O
 

H2 (g) + 0.5 O2 (g)  → H2O (l)

 

Enthalpy Change

> 

DeltaH := 1*Unit(mol)*h_H2O - (1*Unit(mol)*h_H2 + 0.5*Unit(mol)*h_O2)

-247855.2767*Units:-Unit(mol)*Units:-Unit(J/mol)

 (3.2.1)

Entropy change

> 

DeltaS := 1*Unit(mol)*s_H2O - (1*Unit(mol)*s_H2 + 1/2*Unit(mol)*s_O2)

-55.2736411*Units:-Unit(mol)*Units:-Unit(J/(mol*K))

 (3.2.2)

Gibbs Free Energy of Formation

> 

DeltaG_H2O := DeltaH - DeltaS*temp

55273.64110*Units:-Unit(mol)*Units:-Unit(J/(mol*K))*Units:-Unit(K)-247855.2767*Units:-Unit(mol)*Units:-Unit(J/mol)

 (3.2.3)

CO
 

C (gr) + 0.5 O2 (g)  → CO (g)

 

Enthalpy change

> 

DeltaH := 1*Unit(mol)*h_CO - (1*Unit(mol)*h_C + 0.5*Unit(mol)*h_O2)

-111996.5341*Units:-Unit(mol)*Units:-Unit(J/mol)

 (3.3.1)

Entropy change

> 

DeltaS := 1*Unit(mol)*s_CO - (1*Unit(mol)*s_C + 0.5*Unit(mol)*s_O2)

88.2948423*Units:-Unit(mol)*Units:-Unit(J/(mol*K))

 (3.3.2)

Gibbs Energy of Formation

> 

DeltaG_CO := DeltaH - DeltaS*temp

-88294.84230*Units:-Unit(mol)*Units:-Unit(J/(mol*K))*Units:-Unit(K)-111996.5341*Units:-Unit(mol)*Units:-Unit(J/mol)

 (3.3.3)

CO2
 

C (gr) + O2 (g) → CO2 (g)

 

Enthalpy change

> 

DeltaH := 1*Unit(mol)*h_CO2 - (1*Unit(mol)*h_C + 1*Unit(mol)*h_O2)

-394612.5062*Units:-Unit(mol)*Units:-Unit(J/mol)

 (3.4.1)

Entropy change

> 

DeltaS := 1*Unit(mol)*s_CO2 - (1*Unit(mol)*s_C + 1*Unit(mol)*s_O2)

1.2572526*Units:-Unit(mol)*Units:-Unit(J/(mol*K))

 (3.4.2)

Gibbs  Energy of Formation

> 

DeltaG_CO2 := DeltaH - DeltaS*temp

-1257.252600*Units:-Unit(mol)*Units:-Unit(J/(mol*K))*Units:-Unit(K)-394612.5062*Units:-Unit(mol)*Units:-Unit(J/mol)

 (3.4.3)

C2H4
 

2 C (gr) + 2 H2 (g) → C2H4 (g)

 

Enthalpy change

> 

DeltaH := Unit(mol)*h_C2H4 - (2*Unit(mol)*h_C + 2*Unit(mol)*h_H2)

38223.73474*Units:-Unit(mol)*Units:-Unit(J/mol)

 (3.5.1)

Entropy change

> 

DeltaS := Unit(mol)*s_C2H4 - (2*Unit(mol)*s_C + 2*Unit(mol)*s_H2)

-80.9312581*Units:-Unit(mol)*Units:-Unit(J/(mol*K))

 (3.5.2)

Gibbs Energy of Formation

> 

DeltaG_C2H4 := DeltaH - DeltaS*temp

80931.25810*Units:-Unit(mol)*Units:-Unit(J/(mol*K))*Units:-Unit(K)+38223.73474*Units:-Unit(mol)*Units:-Unit(J/mol)

 (3.5.3)

C2H2
 

2 C (gr) + H2 (g) → C2H2 (g)

 

Enthalpy change

> 

DeltaH := Unit(mol)*h_C2H2 - (2*Unit(mol)*h_C + 1*Unit(mol)*h_H2)

224981.2831*Units:-Unit(mol)*Units:-Unit(J/mol)

 (3.6.1)

Entropy change

> 

DeltaS := Unit(mol)*s_C2H2 - (2*Unit(mol)*s_C + 1*Unit(mol)*s_H2)

53.7969897*Units:-Unit(mol)*Units:-Unit(J/(mol*K))

 (3.6.2)

Gibbs Energy of Formation

> 

DeltaG_C2H2 := DeltaH - DeltaS*temp

-53796.98970*Units:-Unit(mol)*Units:-Unit(J/(mol*K))*Units:-Unit(K)+224981.2831*Units:-Unit(mol)*Units:-Unit(J/mol)

 (3.6.3)

C2H6
 

2 C (gr) + 3 H2 (g) → C2H6 (g)

 

Enthalpy change

> 

DeltaH := Unit(mol)*h_C2H6 - (2*Unit(mol)*h_C + 3*Unit(mol)*h_H2)

-105040.1991*Units:-Unit(mol)*Units:-Unit(J/mol)

 (3.7.1)

Entropy change

> 

DeltaS := Unit(mol)*s_C2H6 - (2*Unit(mol)*s_C + 3*Unit(mol)*s_H2)

-215.8750025*Units:-Unit(mol)*Units:-Unit(J/(mol*K))

 (3.7.2)

Gibbs Free Energy of Formation

> 

DeltaG_C2H6 := DeltaH - DeltaS*temp

215875.0025*Units:-Unit(mol)*Units:-Unit(J/(mol*K))*Units:-Unit(K)-105040.1991*Units:-Unit(mol)*Units:-Unit(J/mol)

 (3.7.3)

H2
 
> 

DeltaG_H2:= 0:

O2
 
> 

DeltaG_O2:= 0:

Constraints
 

A mole balance on the carbon, hydrogen and oxygen gives these constraints

 

Carbon

 n1 +  2 n2 + 2 n3 + n4 + n5 + 2 n9 = 2      

Hydrogen

4 n1 + 4 n2 + 2 n3 + 2 n7 + 2 n8 + 6 n9 = 14

Oxygen

2 n4 + n5 + 2 n6 + n8 = 4
 

 

> 

cons :=    
 n1 +  2*n2 + 2*n3 + n4 + n5 + 2*n9 = 2 * Unit(mol)
,4*n1 + 4*n2 + 2*n3 + 2*n7 + 2*n8 + 6*n9 = 14 * Unit(mol)
,2*n4 + n5 + 2*n6 + n8 = 4 * Unit(mol):

Gibbs Energy of the Products
 
> 

DeltaG := [DeltaG_CH4, DeltaG_C2H4, DeltaG_C2H2, DeltaG_CO2, DeltaG_CO, DeltaG_O2, DeltaG_H2, DeltaG_H2O, DeltaG_C2H6] /~ Unit(mol):

 

Gibbs Energy of the product

> 

Gt := n -> add(n[i] * DeltaG[i], i=1..9) + R * temp * add(n[i] * ln(((n[i] + 1e-10*Unit(mol))/ add(n[i], i=1..9))), i=1..9):

Optimization
 
> 

Digits := 20:
> 

composition := Minimize(Gt([n1, n2, n3, n4, n5, n6, n7, n8, n9]), {cons}, initialpoint = [n1 = 1 * Unit(mol), n2 = 1 * Unit(mol), n3 = 1 * Unit(mol), n4 = 1 * Unit(mol), n5 = 1 * Unit(mol), n6 = 1 * Unit(mol), n7 = 1 * Unit(mol), n8 = 1 * Unit(mol), n9 = 1 * Unit(mol)], assume = nonnegative, iterationlimit = 300):

 

Hence the product composition is

> 

composition[2]

[n1 = 0.67518947817769323367e-1*Units:-Unit(mol), n2 = 0.91347569119127277945e-7*Units:-Unit(mol), n3 = 0.27748906056945665557e-9*Units:-Unit(mol), n4 = .55669335375188894522*Units:-Unit(mol), n5 = 1.3757872150171095145*Units:-Unit(mol), n6 = 0., n7 = 5.3541353936680476732*Units:-Unit(mol), n8 = 1.5108260774791125946*Units:-Unit(mol), n9 = 0.15008155792897638429e-6*Units:-Unit(mol)]

 (6.1)
> 

 

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