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13.2: Objective Function and Newton-Raphson Procedure

We have seen that from a molar material balance applied to a two-phase system in equilibrium, and the definition of Ki, we can derive the Rachford and Rice Objective Function:

Contact your instructor if you are unable to see or interpret this graphic.(13.3)
 

Equation (13.3) is a non-linear equation in one variable, and the Newton Raphson procedure is usually implemented to solve it. In general, Newton Raphson is an iterative procedure with a fast rate of convergence. The method calculates a new estimate, αgnew, which is closer to the real answer than the previous guess, αgold, as follows:

Contact your instructor if you are unable to see or interpret this graphic.(13.6)

Substituting (13.3) and (13.4) into (13.6),

Contact your instructor if you are unable to see or interpret this graphic.(13.7)

In this iterative scheme, convergence is achieved when

Contact your instructor if you are unable to see or interpret this graphic.(13.8)

where ε is a small number (ε = 1.0 x 10– 9 is usually adequate). After solving for Contact your instructor if you are unable to see or interpret this graphic., the liquid molar fraction and composition of each of the phases can be calculated as follows:

Liquid Molar Fraction:     Contact your instructor if you are unable to see or interpret this graphic.(13.9a)
Percentage of Liquid:     Contact your instructor if you are unable to see or interpret this graphic.(13.9b)
Percentage of Vapor:     Contact your instructor if you are unable to see or interpret this graphic.(13.9c)
Vapor Phase Composition:     Contact your instructor if you are unable to see or interpret this graphic.(12.7)
Liquid Phase Composition:     Contact your instructor if you are unable to see or interpret this graphic.(12.11)

Contributors

  • Prof. Michael Adewumi (The Pennsylvania State University). Some or all of the content of this module was taken from Penn State's College of Earth and Mineral Sciences' OER Initiative.