Select thermodynamic models for process simulation
A Practical Guide to a Three Steps Methodology

Example 2.11: Effect of temperature on the reaction constant

Calculate the equilibrium of the dodecane dehydrogenation reaction as a function of temperature, at a total pressure of 3 bar and a molar ratio of H2/HC of 6:

image

Analysis:

Required properties:

Pressure is known. The system composition must be calculated as a function of temperature (which is therefore known). In order to apply the chemical equilibrium, two types of properties must be computed: fugacities and Gibbs energy of reaction at the reference state.

Mixture:

Phase condition:

In this example, the main argument for the choice of the fugacity model is based on the observation that the pressure is low and that in the reaction condition the mixture is in its vapour phase. Under these conditions, the ideal gas approximation can be used and the fugacities written as partial pressures (hence ideal mixture behaviour).

Solution:

See complete results in file (xls):

Some help on nomenclature and tips to use this file can be found here.

In order to calculate the equilibrium constant, the component formation properties are required. Table 1 provides these values as found in the DIPPR database.

Table 1: Data of the example
Component image (kJ/mol) image (kJ/mol) image (kJ/(mol.K) image (kJ/(mol.K)
n-dodecane -290.7 49.81 -1.1443 0.62415
1-dodecene -165.4 136.1 -1.0139 0.6185
Hydrogen 0 0 0 0.13057

Note that the entropy of formation is not given by the database, but that it is here calculated using the relationship with enthalpy and Gibbs energy:

image

These values are different from the absolute entropy that is also provided in the database (here given as S0 but not used).

Since gaseous hydrogen is considered as the component that represents the element H, its formation properties are zero by definition.

Using the Gibbs energies of formation:

image

we can now calculate the Gibbs free energy of reaction at 298.15 K: image = 86.74 kJ/mol, and therefore the equilibrium constant: image

ln K = -34.99. This value is very small, indicating that at room temperature, the reaction equilibrium is such that the alkane is much more stable than the alkene. To observe dehydrogenation, the temperature must be raised, as demonstrated by the positive heat of reaction image = 125.26 kJ/mol (indicating an endothermic reaction: the higher the temperature, the larger the conversion).

The temperature-dependence of the equilibrium constant is written according to

image
where image itself depends on temperature according to image. This is why the ideal gas heat capacity of the components must be taken into account. Using the model of Passut & Danner (see chapter 3):

image

the parameters are found in the database:

Table 2-1: Heat capacity parameters for the Passut & Danner expression
Component image image image image image
n-dodecane 6.35E+01 8.51E-01 -2.09E-04 -1.20E-07 5.50E-11
1-dodecene 5.45E+01 8.04E-01 -1.59E-04 -1.65E-07 6.58E-11
Hydrogen 2.54E+01 1.94E-02 -3.65E-05 3.00E-08 -8.18E-12

We can now integrate

image

Which is equivalent to

image

to find the value of the equilibrium constant as a function of temperature. The result is shown in figure 1. We see that a high temperature is required in order to have a visible conversion.

image Figure 1: Equilibrium constant of the dehydrogenation of n-dodecane as a function of temperature.

The mixture equilibrium composition as a function of temperature can be calculated using

image

and the information provided in the problem statement (the pressure is 3 bar and the molar ratio H2/HC = 6). This will yield three equations with three unknowns:

image

It is for example possible to calculate the conversion as:

image

Doing so, figure 2 is found:

image Figure 2: Effect of pressure and temperature on the n-dodecane dehydrogenation reaction.

It is visible that increasing pressure decreases the conversion because new moles are created (the sum of the stoechiometric coefficients, image is positive). Not shown on the figure, but easy to simulate with the excel sheet, is when the hydrogen content (alpha) increases, the conversion also decreases.