The equations are normalized to keep the numerical values in the same order of magnitude. Since Naphtali-Sandholm equations are grouped by stage and for a system with C components, the independent functions for any stage j are
The independent variables for the stage are
The functions and variables are solved together using a large •Jacobian of size N(2C + 1) x N(2C + 1). When originally presented, the Naphtali-Sandholm method used derivatives of if-values and enthalpies with respect to composition and temperature, but it was not stated whether these are analytical or numerical derivatives.
As in the 2N Newton methods, the condenser duty Qc and reboiler duty Qr are specified, while the reflux ratio and product rates are calculated. If some other specification is required such as product purity, the energy balance for that stage can be replaced with a specification function and the duty is calculated outside of the Newton-Raphson in a separate energy balance.
One characteristic of the global Newton methods is that large numbers of the derivatives are zero. In the Tomich method, all the component balances are solved in the tridiagonal matrix when evaluating every function and this gives the full Jacobian. In the global Newton methods, the component balances are left in native form and not rearranged using absorption and stripping factors. When one flow or temperature is varied, it usually has little effect on functions more than two stages away. The derivatives (except for the if-values and enthalpies) are easy to calculate. Those of the component balances, for example, will be either 1, -1, or 0. Because functions and variables are grouped by stage in the Naphtali-Sandholm method, its sparse Jacobian will be a block-banded matrix with the derivatives grouped as blocks along main, upper, and lower diagonals, giving fi
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