# Methodology Followed

## Estimation of equilibrium composition

Reforming reactions generally take place at high temperature and pressure conditions, typically in the range of 900-1100 °C and 15-25 bar, in order to achieve the maximum allowable equilibrium conversion. The equilibrium composition of gas phase reaction is estimated via the Gibbs free energy minimization method. Gibbs free energy (or the chemical potential of the system) is the total fr ee energy available in the system to do useful or external work, and is a function of temperature, pressure and composition of the system.

The Gibbs free energy of a multicomponent system is given by: where.

‘i^{-} represents the chemical component present in the system n represents the number of moles of component ‘i’ p_{1} represents chemical potential of component ‘i’

The chemical potential of each component is calculated using the following equation:

where.

AGg = Standard Gibbs free energy of formation for the component ‘i’

_ , /Ра х m^{3}

R = universal gas constant _{mo}} _{x} К/

Since the reforming reaction produces solid carbon, the presence of solid phase in the Gibbs free energy equation is accounted as follows:

where.

y_{i} = Equilibrium composition of the component ‘i’ in the system Ф[ = Fugacity coefficient of component ‘i’ n_{c} = Moles of carbon produced

AG°_{C} = Standard Gibbs free energy of formation of solid carbon

P = Total pressure of the system

P_{0} = Standard pressure of 1 atm

T = System temperature

T_{0} = Standard temperature of 298.15 К

To estimate gas phase composition, Peng Robinson (PR) equation of state was used, due to its relevance to reforming system (Challiwala et ah, 2016; Challiwala et ah, 2017; Challiwala et ah, 2015). Cubic equation model for compressibility factor calculation:

For each equation of state, different values of parameters p, e, о and q are available, and their incorporation in equation (12) will simplify the generic model to the pertinent equation of state. The parameters for the PR equation of state adopted from Chemical Engineering Thermodynamics textbook (Smith, 1950) are presented below:

The solution of the Peng Robinson equation of state enables the calculation of the fugacity coefficient based on the ‘vapor' like root of the compressibility factor. The fugacity coefficient is then incorporated into the Gibbs free energy equation as a collection factor for the pressure term.

There are two mathematical approaches to minimize the Gibbs free energy function: (1) Lagrange's Undetermined Multipliers method and (2) the Direct Minimization method.

*2.1.1 Lagrange's undetermined multipliers method*

The first method of Lagrange’s Undetermined Multiplier method is a teclmique to solve a non-linear equation by incorporating additional variables into the equation that artificially make the system of equations solvable. Details on the exact steps involved are provided below:

Step 1: Defining the Gibbs free energy equation

Let the total Gibbs free energy equation having “n” number of components be recognized as follows:

G^{t0Bl} is fixed at a specific temperature T and pressure P of the system. However, the problem is to identify the equilibrium moles of all the components, i.e., n,, n,.... n_{K}„

Step 2: Defining constraints:

As the total number of atoms remain constant for any reaction system, an atom balance provides element wise constraints that cannot be violated in search for equilibrium. These constraints can be written as follows:

If the equation above is multiplied with a constant, its overall solution remains unchanged, this mathematical manipulation, therefore, becomes:

Summation of the above equation for all the elements, therefore, becomes:

Step 3: Incorporation of constraints in G^{t0Bl}:

Mathematically, if a new function “F” is formulated, which is a summation of the total Gibbs free energy “G^{total}” and equation (24) of atom balance constraint, then the new function would theoretically be equal to “G'^{0Bl}”, since the total value of the atom balance constraint above is zero. This is shown below:

Although the value of the above equation is the same as “G^{t0Bl}”, its partial derivatives with respect to each component mole “n” are not. This is because function F now also incorporates the atom balance constraints.

Step 4: Gibbs free energy minimization

The minimum value of the modified Gibbs free energy equation “F” provided in equation (25), which is now also inclusive of atom balance, could be obtained by solving for partial derivative of function “F” with respect to molar composition “n” at any constant system temperature T and pressure P, as follows:

^{r} oG'^{oal} 'l

Since, - = p, or the chemical potential of component “i”. then the overall equation

l ^ Л.Р.П, '

above could be rewritten in terms of chemical potential, as follows:

Incorporation of equation (17) of chemical potential given earlier, the equation above becomes:

The equation above is for all the components “N” in the system, and therefore furnishes “N” equations. “N” equations, along with constraint equations for “in” number of elements as per the atom balance, furnishes “N+m” number of total equations. This will be used to solve for “N” number of unknown equilibrium molar composition.

*2.1.2 Direct gibbs free energy minimization*

The direct method of Gibbs free energy minimization is a relatively simple technique and does not involve the mathematical manipulation provided in step 2 of the Lagrange’s Undetermined multiplier method. However, the constraints of atom balance and mass balance remain the same.

The implementation of this method requires utilization of a built-in minimization tool box provided in MATLAB® called “fmincon”, which directly searches for minima of any given function subjected to a set of external constraints that bounds the solution within a desirable range of conditions.

Although “fmincon” provides a simple and robust technique, Lagrange’s undetermined multiplier’s method is a general procedure that needs to be followed if any other mathematical solver is desired to be used.

## Energy balance calculations

The total energy required or supplied from an endo/exothennic reforming reaction is estimated using energy balance calculations across the reformer block. This is done by using a simple equation that computes the difference between the enthalpies of the components that enter and leave the system and the energy liberated/consumed by the chemical reaction. The temperature and pressure of the components before entering and after leaving is of importance in this calculation, as per the following block diagram in Figure 2.

The following is the general equation to calculate the energy requirements in this process: where,

n = Molar flow of the exit gases n. _{feed} = Molar flow of the entering gases H _{f} = Enthalpy of gases in the feed

Figure 2. Reformer energy balance.

H ^{=} Enthalpy of gases in the exit

Е_{гхц} = Energy evolved'absorbed dining chemical change

Energy = Total energy difference between the inlet and the outlet streams

The contrasting features between the three reforming technologies are apparent in terms of their net energy requirements, hydrogen to carbon monoxide ratio (or Syngas ratio) and the amount of carbon formed in each process.

Based on the thermodynamic analysis using the procedure mentioned in the methodology section, the overall equilibrium product distribution of the different species was estimated, including their net energy requirements, and the syngas ratio for all the three reforming technologies (i.e., SRM, POX and DRM).