Enggtring
Flash Distillation of an Indefinite Number of Components JOHN R. BOWMAN,
WAYNE
Process
development
MELLON INSTITUTE, PITTSBURGH, PA.
c. EDMISTER, CARNEOIE INSTITUTE OF TECHNOLOGY, PITTSBURGH, PA. however, are not suitable for computation in practical problem. The main purpose of thii paper is to transform them mathematically to explicit formulas useful for such computation. In most practical problems, the cornposition of the feed ia given, and either the fraction vaporized or the temperature is also fixed. For such cases, the dependent variables sought are the compositions of the products. The suitable formulas are presented in the following as Equations 4,7, and 8, and in equivalent form as Equations 11, 12,and 13; special cases for specific types of feed compositions are given by Equations 21, 22, 23, 31, 32, and 83. As in the previous papers, the equilibrium law is assumed to be
T h e previously developed general theoretical approach to distillation problems has been applied to equilibrium flash distillation. The results are explicit formulas relating the two product compositions to that of the feed, where either the fraction vaporized, or the temperature and vapor pressure-temperature characteristic, are given.
T
HE process of flash distillation is here conventionally idealized as that in which a liquid mixture feed is continuously subjected to a change in state such that liquid and vapor phases are formed concurrently and are removed as products in equilibrium with each other. The change of state may be brought about by alteration of the temperature or pressure or both. The following theoretical development of the performance of this process, however, permits immediate generalization to the case where the feed is gaseous, aa in equilibrium partial condensation, and to the case where both products are liquids, as in ooncurrent or batch extraction. The feed need not even be single phase. The only additional assumption to be made is that of constant relative volatility or, in the extraction caae, its analog.
which is the generalization of the classical law for multicomponent systems of constant relative volatility. The other baaic relation is the material balance vy
GENERAL THEORY
In a previous paper (I), distillation theory was approached from the standpoint of designating compositions by composition distribution functions x(a),so defined that the fraction of material in a mixture of composition x having relative volatilities between (I and a da is xda. Though z is by itself not a concentration, the conventional symbol is used because in nearly all of the equations d a divides out, leaving x with the formal mathematical properties of a concentration. This formulation has the advantage that many of the classical equations for binary systems readily generalize for multicomponent systems, even for the limiting case where the number of components is effectively infinite, as in most petroleum products. The relations between the cornposition distribution functions and the more nearly directly observable aspects of composition, such as the true boiling point curve, are discussed in another paper of this series (g). The formulas derived on this basis reduce to the usual ones where the concentrations are all finite; the only significant difference is the replacement of certain integrals by finite suwmtions. The variables comprising the operating conditions are the cornpositions of the single feed and the two product streams, the fraction vaporized, and the temperature. &om the phwe rule, there must exist 2n nontrivial relations between them variables, where n is the number of components of the system. Fund* mentally, these can be recognized as the material balances and equilibrium distributions of the components. These relations,
+ (1 - v)x = z/
(2)
which is essentially analogous to the operating line in continuow rectification. It stipulates that the sum of the amounts of any component in the products must equal the amount of the same component in the feed. I n this equation, as in Equation 1, do has been divided out of a11 terms. The integral in the denominator of Equation 1 is a3onstsnt for any given set of conditions. For simplicity, it is written
+
Equations 1 and 2 can be solved algebraically to give formal expressions for the product compositions, involving the constant, I, which remains to be determined. These are (4)
and
tr = ua
+ (1 -
U)Z
(5)
The constant, I , can be determined by making we of the relation
which, physically, ie the condition that the sum, over a11 componente of the fraction concentrations, must equal unity. A
2625
INDUSTRIAL AND ENGINEERING CHEMISTRY
2626
sjmilar relation applies to all composition distribution functions. Substituting the formal solutions 4 or 5 in identity 6 yields
which may be obtained by combining Equations 11 and 12, or more directly by integration of the operating line, Equation 2. FRACTION VAPORIZED GIVEN
(7) or
either of which provides an equation for the determination of I without recourse to its original definition by Equation 3. The two equations are equivalent but 8 is usually more convenient for computation purposes. Mathematically, since x / ( a ) ie monotonic, these equations have one and only one real positive root, and thia number is easily computable by graphical integre tion and interpolation. The limiting compositions are, for the liquid at the dew point of the feed
and for the vapor at the bubble point
For numerical computation, where the number of componenta la large, the relative volatility curves are more convenient than the composition distribution functions. The latter can be calculated from the former by integration, as has been shown previously (g), In these terms the foregoing resulto are
Vol. 43, No. 11
Where the feed composition and the fraction vaporized are the known independent variables, the general formulas obtained in the preceding section are immediately applicable to solution of all types of problems. If the number of components in the system is small, the compositions are best designated by sets of fraction concentrations. These may be on any consistent basis-such aa mole, mass, or volume fractions-provided they can be assumed to be additive. The relative volatilities must of course be on the same basis. For this case, Equations 4 and 6, where Z is the one real positive root of Equations 7 or 8, present the compositions of the products explicitly, because the integrations reduce to summations over the components and the composition distribution functions reduce to fraction concentrations. 1x1 effecting the summations, care must be taken to assure that, in each term, the concentration of a component is associated with the relative volatility of the same component. The following sample calculation on a hypothetical ternary mixture will illuetrate the method. Let the relative volatilitie~ of the components be 3.000,1.000,and 0.600,respectively. With this notation, the intermediate component is selected as the key component in classical terminoloa, but this is unimportant in the present theory because of the symmetry of the formulas. Let the respective mole fraction concentrations in the feed be 0.500, 0.200, and 0.300. Finally, let half of the feed be vaporized. This renders the process exactly determinate, and the compositicns of the products can be computed from Equations 4 and 7. The first step of the computation is the determination of the constant, I. Of the alternative equivalent equations 7 and 8 for the purpose, Equation 8 is the more convenient. Numerically, it yields
3.000 X 0.500 0.5 X 3.000 0.5 X
+
1.Ooo
Z
x 0.200
0.5 X LOO0
+ 0.51
+
0.500 x 3.000 0.61
0.5 X 0.500
and
+
a
This simplifies to a cubic equation, and the one real positive root is easily found to be Z = 1.50, approximately. Subst.itution of this value for Z together with the given operating data in Equations 4 and 5, gives explirit numerical evaluation of the mole fraction concentrations of each of the three components in each of the two product stream. These results are given in Table I.
where I is the one real positive root of
TABLE I. PRODUCT COMPOSITIONS FOR HYPOTRBTICAL EXAMPLP~ The limiting volatility curves are
Produot Liquid
Vapor
Lightest 0.333 0.667
Intermediate 0.240
0.160
Heavieet 0.428 0.172
Additional numerical examples have been presented recently (5,
and
Where computation of both product compositions is required, the simplest procedure is usually to compute one of them from Equations 11 or 12 and then calculate the other from the first and the equation VWV,
+ (1 - v)Wt
- w/
(16)
4).
If the number of components is large or effectively infinite, designation of compositions by relative volatility curves is more convenient or, in the infinite case, necessary. The products are then computed as integral transforms of the relative volatility curve of the feed by Equations 11, 12,and 13. The integrations are readily performed graphically. True boiling point curves are closely allied to the relative volatility curves and are easily calculated from them, when the functional relation between the normal boiling point and the relative volatility of the components is known, by a transformation of the relative volatility axis scale to a temperature scale. Conversely, if the feed.composition is known only from ita true boiling point curve, its relative volatility curve must be calculated before the formulas can be applied.
INDUSTRIAL AND ENGINEERING CHEMISTRY
November 1951
TEMPERATURE GIVEN
Where the feed composition and the temperature at which the products are brought to equilibrium and separated are the known independent variables, application of the general formulas is less direct, but can be effected as follows. In a paper on this subject (Z), the assumption of constant relative volatility was shown to be equivalent to
2627
CASE 11: LOGARITHMIC VOLATILITY DISTRIBUTION IN THB FEED. For petroleum products, the feed composition can usually be approximated by a logarithmic volatility curve-Le., a function of the form W j ( a ) = 0 for a log a = log
LYE
+ (log - log aa)Wj ai
W / ( a ) = 1 for For a given value of the temperature, therefore, I can be determined numerically if the vapor pressure-temperature function for the kev component is known. Equation 7,8, or 13 then provides means for the determination of v, after which the general formulas can be applied directly. Though the three alternate equations relating Z and v always have one real root in one variable when the other is fixed, the root for u does not always fall in the significant range zero to unity. Such cases arise when the value of I corresponds to a temperature below the bubble point or above the dew point of the feed composition. CASE I: LINEARVOLATILITY DISTRIBUTION OF THE FEED. Specific formulas will be derived for the important special case wherein the feed is assumed to have an infinite number of components uniformly distributed with respect to relative volatility. The relative volatility curve for such a mixture is a straight line and the composition distribution function is a constant for a certain range of a and zero outside of that range. This function will be written x l - 0
f o r a < l - a
(Y
>
Expressing Equation 27 explicitly in
Wj
log
a a.
w/(a)= ai log -
Application of the general formulas to this casc is facilitated by calculation of the composition distribution function. It is
T
w/(a)= - =
(
a
log ::)-I
in the range of Equation 27 and aero outside this range. The limits of the range are the initial and end-point values for a. With this form for q,equation 8 can be integrated to yield the relation between v and I. After solving the result for I,
(18)
x/ = 2a -f o r l - a < a < l + - a x / = O
