A new approach to separation calculations has been developed which has many practical advantages. The equations of conservation of mass and energy and of equilibrium are grouped by stage and then linearized. The resulting set of equations has a block-tridiagonal structure which permits solution by a simple technique. Thus a new way to apply the Newton-Raphson technique to separation calculations is devised. The method has been tried on a number of problems chosen to exhibit characteristics which cause other solution technique to fail. Most problems are solved with ten iterations or less. * Functions (l), ( 2 ) , and ( 3 ) are discrepancy functions, that is, they are a quantitative measure of the failure of the values of LAJ, V t j , and T , to satisfy the physical relationships. The value of Ei is the number of heat unitdtime by which the enthalpy balance is unsatisfied, whereas in M i j and Qii the discrepancy is measured in moledtime. A solution to the problem has been obtained when one has values of the vnriahles which make Ei, Mi,, and Otj zero. (I), ( 2 ) , and ( 3 ) can be considered equations if we replaced Ei, M i j , and Q i i by zero. We would then say we are looking for the values of the variables which satisfy the equations. L1 -V1 (b> V n -L, ( 5 ) Temperature Ti -T D 7, -T B Overall product rate V1 -D Ln -B Key component rate V1, -D