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Modeling Urea Processes: A New Thermodynamic Model and Software Integration Paradigm
(Special Shared Content with Virtual Materials Group)

High Pressure Equilibrium

Initially the high-pressure section was modeled using a full ionic model as described by Satyro (8). Albeit the model showed good performance when used to model industrial units, enhancements were possible in terms of computational speed and accuracy with respect to ammonia and carbon dioxide vapor compositions at the outlet of the urea synthesis reactor. The majority of the time spent in thermodynamic calculations was determined to be in the convergence of the ionic chemical equilibrium, and any simplification in that area would have significant impact in the calculation speed, and therefore would allow the use of the model not only for steady state calculations but also dynamic calculations necessary for safety studies and operator training.

The reactive system was simplified by considering all the chemical species in their molecular states. This is not true from a purely physical-chemical point of view, since the reactions happening in the liquid phase at high pressure are well represented by the following reaction system (8):

	(3)
	(4)
	(5)
	(6)
	(7)

The equilibrium constants for the equations above are functions of temperature, and the reaction equilibrium is supposed to be independent of pressure. Therefore, the equilibrium compositions for the several species (molecular and ionic) can be represented as in Equation 8:

(8)

Where the index i represents one of the chemical reactions defined by Equations 1 to 4, x is the composition vector in the liquid phase, T is the liquid phase temperature and the K's on the right of Equation 5 are defined as in Equations 9a and 9b.

(9a)

(9b)

Where is the activity coefficient and is the stoichiometric coefficient for each of the components present in reaction i. 

The calculation of ionic species activity coefficients is somewhat laborious and the details can be found in Satyro (8). Since the chemical equilibrium has to be evaluated at every iteration when calculating liquid phase fugacity coefficients, any reduction in computational load while keeping accuracy will translate into substantial time saving. Therefore, the reaction system defined by Equations 3 to 7 was replaced by the following simplified system:

	(10)
	(11)
	(12)

At equilibrium, the actual composition of the liquid phase will be denoted by z and the equilibrium expression is then given by:

(13)

For convenience we note that the fugacity coefficient in the liquid phase is given by the following:

(14)

Note that even if the solution was ideal from a physical point of view the fugacity coefficient is not unitary unless chemical reactions are not present. This is caused by the fact that the ratio z_i / x_i will be unitary only and only if the liquid phase does not present chemical reactions. The salts present in solution, ammonium carbamate, urea and ammonium bicarbonate are not present in the vapor phase and therefore have infinitesimal volatility. 

Careful analysis of the performance of different activity coefficient models on the representation of ammonia and water vapor-liquid equilibrium determined the final model used in this study and a 4 suffix Margules expression was determined optimal for our purposes as defined in the equations below:

	(15)
	(16)
	(17)
	(18)

Where d_ij is a symmetric, temperature independent interaction parameter and a_ij is defined as:

(19)

Standard state fugacities are determined based on vapor pressures for most components while specially determined standard state fugacities for ammonia and carbon dioxide are used, which are valid from 200 to 500 K. 

High Pressure Data Regression