Page 84 - Separation process engineering

and

(2-41)

th
Either of these equations can be used to solve for V/F. If we clear fractions, these are C -order
polynomials. Thus, if C is greater than 3, a trial-and-error procedure or root-finding technique must be
used to find V/F. Although Eqs. (2-40) and (2-41) are both valid, they do not have good convergence
properties. That is, if the wrong V/F is chosen, the V/F that is chosen next may not be better.

Fortunately, an equation that does have good convergence properties is easy to derive. To do this,
subtract Eq. (2-40) from (2-41).

Subtracting the sums term by term, we have

(2-42)

Equation (2-42), which is known as the Rachford-Rice equation, has excellent convergence properties. It
can also be modified for three-phase (liquid-liquid-vapor) flash systems (Chien, 1994).

Since the feed compositions, z, are specified and K can be calculated when T drum and p drum are given, the
i
i
only variable in Eq. (2-42) is the fraction vaporized, V/F. This equation can be solved by many different
convergence procedures or root finding methods. The Newtonian convergence procedure will converge
quickly. Since f(V/F) in Eq. (2-42) is a function of V/F that should have a zero value, the equation for the
Newtonian convergence procedure is

(2-43)

where f is the value of the function for trial k and df /d(V/F) is the value of the derivative of the function
k
k
for trial k. We desire to have f k + 1 equal zero, so we set f k + 1 = 0 and solve for Δ (V/F):

(2-44)
This equation gives us the best next guess for the fraction vaporized. To use it, however, we need
equations for both the function and the derivative. For f , use the Rachford-Rice equation, (2-42). Then
k
the derivative is