The previous series was titled "Multi Reaction Equilibria" but reactions were never discussed. Maybe I should have titled it "Reactionless Equilibria". Today's post begins an alternative method of modeling equilibria where reactions are defined.
The Example: Benzene hydrogenation
A single reaction, hydrogen of benzene to cyclohexane, will be used as an example. The method can also be extended to multiple reactions.
The 7/2/2015 post showed how to obtain the thermodynamic equilibrium constants, K, as a function of temperature. We also can define a function, KX, involving the activities, a, of each compound.
The above equation is written for the j reaction. The powers are the elements of the stoichiometric matrix. Recall that these elements are negative for reactants and positive for products.
Assuming ideal gas, we obtain the following for ln(KX) as a function of the fractional extent of reaction:
Note that for this single reaction example, the fractional extent of reaction is a scalar. For multiple reactions, a vector will be needed. Also, the lr index refers to the limiting reactant and nf is the molar feed vector.
At equilibrium the composition KX will equal the thermodynamic K. The equilibrium fractional extent of reaction is found using a root function.
Adiabatic operating curves
A heat balance is defined:
A root function is then used to find the equilibrium temperature given the bed inlet feed fractional extent of reaction and temperature.
When interstage cooling is used, I assume that the feed streams are all cooled to the same temperature. This assumption would allow all streams to be cooled by a single source such as a boiler. This assumption removes a degree of freedom so that we can't divide an overall conversion into a given number of beds as I did with the reforming example. Therefore, I determine how many beds are needed to exceed a desired conversion.
For the bed curves in the following plot, I chose 400 K for the feed temperatures. To develop the curve, the feed temperatures were first assumed as 400 K plus the 25 C approach. After finding the equilibrium outlet conditions, the final operating curve was obtained by subtracting the approach temperature.
This method can be modified easily to accommodate multiple reactions. One lnK = lnKX equation is needed for each reaction. The system of equations are then solved for the fractional extents of each reaction. With Mathcad, a Find solve block would replace the root operation for this step. To obtain a single equilibrium curve, use the fractional extents of reaction to find the conversion of a main reactant.
Which is better for multiple reactions...minimization of free energy of the equilibrium constant method? I don't have an answer for that question because it depends upon the chemical system. The two methods involve different mathematical operations, i.e. minimization and simultaneous equations. Both methods can have problems obtaining a solution and both methods may find only one of several solutions. If you obtain an equilibrium curve that has a severe outlier or jog, that is probably an indication that the method found an alternative solution for that point. In that case, first try a different initial guess. If that doesn't work, then try the other method.
The free energy method is very useful for systems involving numerous compounds and reactions. You may need to add constraints to prevent the formation of some compounds by slow although thermodynamically favorable reactions. I will show an example in the future.