Adiabatic Reactor Analysis for Methanol Synthesis

An important industrial reaction is the combination of carbon monoxide with hydrogen to produce methanol. Methanol is quite useful for a variety of chemical synthesis reactions, including the transesterification of triglycerides in vegetable oils for biodiesel production. The gaseous mixture of carbon monoxide and hydrogen can be used to synthesize a wide array of hydrocarbons, including synthetic fuels, and is therefore often referred to as “syn-gas”. Syngas can be obtained from coal, as discussed in this paper by Octave Levenspiel (Professor, Oregon State University).

The overall reaction for methanol synthesis from syngas is written as:

CO + 2 H₂ à CH₃OH

And can be approximated as an elementary reaction, such that the rate expression (assuming irreversible reaction, as written above) is:

Where , k_o = 5.7x10¹⁰ L².mol^-2.min^-1, E_A = 100.5 kJ/mol

I. Energy Balance and the Conversion-Temperature Relationship

The Thermodynamics of reaction species are (From Felder & Rousseau)

Species	Cp(J/mol.K)	Hf (kJ/mol) at 298K
CO	29	-110.5
H2	28.8	0
CH3OH	43	-201.2

We can then calculate the change in thermodynamic properties (Enthalpy and heat capacity) upon reaction, using stoichiometry.

DCp_rxn = 1*43 – 2*28.8 – 1*29 = -43.6 J/mol.K

DH_rxn = -201.2 – 2*0 – 1*(-110.5) = -90.7 kJ/mol.K

For an adiabatic reactor (either a CSTR or PFR w/o significant heat dispersion), the energy balance yields the following temperature vs. conversion dependency (Fogler, eqn 8-30),

II. Mass Balance and Rate Expression

We can write a general stoichiometric table for this reaction system, accounting for the presence of inert diluent, I.

Species	Initial	Change	Final
CO
H2
CH3OH
Inert		0
Total

If we assume that everything is ideal gas, then concentration = moles/volume, and accounting for changing volume with temperature and conversion, (Pressure is constant, or pressure drop defined by momentum balance, e.g. Ergun equation)

Our concentrations for each species can then be calculated from stoichiometric table, in terms of conversion, pressure and temperature

We can substitute these terms into the rate expression, as follows:

III. Inlet Conditions

• Feed Temperature T_o = 25^oC or 298.15 Kelvin
• Feed Pressure P_o = 1 atm
• Molar Feed Rate F_TO = 1 mol/min
• Stoichiometric Feed, No diluent , ,

We further assume that there is no pressure drop associated with gas flow through the continuous reactor, i.e. P = P_o.

IV.A. Calculation 1: Solve for 5% conversion (X = 0.05)

Using Equation (8-30) from Fogler,

350.97 Kelvin

For our rate expression, substituting T = 350.97 and X = 0.05,

= 2.8x10-11 mol.min-1.L-1

For obtaining a Levenspiel plot (for sizing either a CSTR or PFR), we want to calculate ,

Liters.

IV.B. Calculation 2: Solve for 10% conversion (X = 0.10)

404.7 Kelvin

For our rate expression, substituting T = 404.7 and X = 0.1,

= 3.4x10^-9 mol.min^-1.L^-1.

For obtaining a Levenspiel plot (for sizing either a CSTR or PFR), we want to calculate ,

Liters.

IV.C. Calculation 3: Solve for 50% Conversion (X = 0.50)

Using Equation (8-30) from Fogler,

870.8 Kelvin

For our rate expression, substituting T = 870.8 and X = 0.50,

=9.1x10^-3 mol.min^-1.L^-1

For obtaining a Levenspiel plot (for sizing either a CSTR or PFR), we want to calculate ,

Liters.

IV.D. Generate Data for Levenspiel Plot

X	T	k	-rCO	Fao/-rCO
0.05	351.0	6.28 x 10^-5	2.82x10^-11	1.18x10^10
0.10	404.7	6.08 x 10^-3	3.43x10^-9	9.73x10^7
0.15	459.4	2.13 x 10^-1	1.18x10^-7	2.83x10^6
0.20	515.0	3.65 x 10^0	1.83x10^-6	1.82x10^5
0.25	571.7	3.74 x 10^1	1.63x10^-5	2.05x10^4
0.30	629.3	2.59 x 10^2	9.60x10^-5	3.47x10^3
0.35	688.0	1.34 x 10^3	4.14x10^-4	8.04x10^2
0.40	747.8	5.44 x 10^3	1.30x10^-3	2.38x10^2
0.45	808.7	1.84 x 10^4	3.88x10^-3	8.60x10^1
0.5	870.5	5.33 x 10^4	9.12x10^-3	3.65x10^1
0.55	934.0	1.36 x 10^5	1.87x10^-2	1.79x10^1
0.60	998.4	3.14 x 10^5	3.38x10^-2	9.85x10^0
0.65	1064.0	6.64 x 10^5	5.49x10^-2	6.08x10^0
0.70	1130.9	1.30 x 10^6	7.99x10^-2	4.17x10^0
0.75	1199.1	2.39 x 10^6	1.04x10^-1	3.20x10^0
0.80	1268.7	4.15 x 10^6	1.20x10^-1	2.78x10^0
0.85	1339.7	6.87 x 10^6	1.17x10^-1	2.85x10^0
0.90	1412.1	1.09 x 10^7	8.76x10^-2	3.81x10^0
0.95	1486.0	1.67 x 10^7	3.61x10^-2	9.23x10^0
0.96	1501.0	1.81 x 10^7	2.55x10^-2	1.31x10^1
0.97	1516.0	1.96 x 10^7	1.58x10^-2	2.11x10^1
0.98	1531.0	2.12 x 10^7	7.75x10^-3	4.30x10^1
0.99	1546.2	2.29 x 10^7	2.13x10^-3	1.56x10^2

We can then plot this data to see how to best perform this reaction.

We can see from the plot that a CSTR will get us to a conversion of ~ 85% - after that we would prefer a PFR to keep reactor volume to a minimum.

Caveat

Sometimes people write for a gas-phase reaction the rate expression in terms of partial pressures

Our true form of the rate expression, which is in terms of concentrations, is

Comparing the two, we see that

Which when linearized does not fit an Arrhenius relationship.