N Newton Methods

In both the BP and SR methods (Secs. 4.2.5 and 4.2.7) the temperatures and the total flow rates are calculated separate of each other. An alternative is to calculate the temperatures and total flow rates together in a Newton-Raphson technique. The name 2N Newton comes from that there are two equations per stage for a total of 2 x N functions and variables per column for the Newton-Raphson. In all three of these approaches, the component flow rates are still calculated in an intermediate step. 

The best-known presentations are by Tomich (32), Holland (8), and Orbach et al. (33). These vary in their choice of Newton-Raphson equations and independent variables and each may solve a different range of columns, These methods have been shown to work well for wide-boiling mixtures including refinery fractionators, absorber-stripper columns, and reboiled absorbers. 

The Tomich method. This method has the summation equations for the vapor and liquid compositions are merged for the first independent equation  

Then s andy s are found from the component balances and the equilibrium equation. 

The other independent function is the stage energy balance and is the same as Eq. (4.67) of the SR method above. 

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