Page 529 - Modelling in Transport Phenomena A Conceptual Approach

A.7. THE ROOT OF AN EQUATION 509

Application of the sign rule indicates that the maximum number of real positive
roots is equal to three. The terms M and N are
M=- 3q - P"
9
- (3)(0.8618) - (3.1923)2
- = -0.845 (3)
9

9pq - 27r - 2p3
N=
54
- (9)(- 3.1923)(0.8618) - (27)(- 0.0567) + (2)(3.1923)3
-
54
= 0.775 (4)
The discriminant, A, is
A=M3+N2
= (- 0.845)3 + (0.775)2 = - 0.003 (5)

Therefore, all the roots of Eq. (2) are real and unequal. Before calculating the roots
by wing Eqs. (A.7-12)-(A.7-14), 6 must be determined. From Eq. (A.7-15)

Hence, the roots are
3.1923
+
VI = (2)mcos (y) - 2.902
=
3
3.1923
+
= (2)dmZcos (y 120) + -j- = 0.109
3.1923
= (2)acos (y +240) + 7
0.181
=
The molar volume of saturated vapor, Vg, corresponds to the largest root, i.e.,
2.902 m3/ kmol. Since, the molecular weight, M, of methanol is 32, the density of
saturated vapor, pg, is given by

- (32) = 0.01103 g/ cm3
-
(2.902)(1 x lo3)