EngEtring
Interrelations between Distillation Curves
R O C W
development I
JOHN R. BOWMAN M E L L O N INSTITUTE, PITTSBURGH, PA,
F o r m u l a s for t r u e boiling point a n d volatility curves are dcrived from the composition distribution function treated i n a previous paper. These are alternative methods for characterizing mixtures of an indefinite n u m ber of components. The first m o m e n t of the composition distribution function is shown to be closely related t o t h o bubble point, Examples a r e presented for the case of simple batch distillation. By inversion of the formula for the t r u e boiling point curve, that function can be computed from the simple distillation curve.
mixtures for distillation study is the true boiling point curve. It is dwaya continuous-although, ideally, its derivatives are not-and can, to good approximation, be observed experimentally. With certain assumptions, the composition distribution function and the true boiling point curve function are simply related to each other, and either can be calculated from the other. Discussion of this relation forms the principal subject of this paper, but before it can be derived, another composition characterizing concept, the volatility curve function, must be introduced.
I
'
N A previous paper ( 1 ) the concept of the composition distribution function was introduced for characterizing the Composition of homogeneous mixtures, To construct it, each component of the mixture must first be assigned a single value of a component designating index which, for purposes of distillation theory, is most conveniently taken to be the relative volatility. In scanning the Components of a mixture this index takes on isolated discrete values for the components present in finite concentration with one to one correspondence, and varies over a continuous range corresponding to a continuum of components, each present in infinitesimal concentration, where such a continuum is present. With this method of designating components, the composition distribution function z ( a ) is defined as the coefficient of a differential interval in the relative volatility such that the: product is the fraction concentration of material in the mixture having relative volatility in that differential interval. The composition distribution function can be regarded physically as the density of material in the a direction where the concentrations are ranged along a linear a scale. Actually, however, it has more the properties of a conventional fraction concentration, because in nearly all equations the factor da divides out or is integrated. Where one or more components are present in finite concentration, the function x has infinite peaks a t the corresponding values of a. This difficulty can be overcome by introducing substitution operator functions, as was done in the previous paper, or more simply, by allowing the function x to equal the ordinary fraction concentrations a t the corresponding isolated values of a, and adding to all integrations with respect to a the summation of the integrand over these discrete values of a. These irregularities are wholly eliminated when the formulas are converted to volatility or true boiling point curve functions, as will be shown in the following sections. The composition distribution function is fundamental to the theory of performance of separation processes on complex mixtures becnuse it is the only aspect of composition that can be applied directly in kinetic, equilibrium, and material balance equations, hut it is inconvenient for practical work because it can be discontiriuous and is not directly observable. The most practical general characterization of the composition of complex
VOLATlIJTY CURVES
The volatility curve provides a means for specifying the composition of a complex mixture and is a compromise between the mathematically fundamental composition distribution function and the practically convenient true boiling point curve. Consider a conventional true boiling point batch distillation-i~., one in which each component distills over pure, and the components distill over in order of their boiling pointa. If constant relative volatility can be assumed, this order is the same as the order with respect to relative volatility, but in the opposite direction. The volatility curve ia defined to be the plot of the relative volatility of the component coming overhead a t any instant verRus the fraction of the original distilland charge remaining in the pot a t the same instant, The mathematical expression for a volatility curve will be termed a volatility curve function. The volatility curve is closely related to the true boiling point curve, the only difference being that the ordinate axia measures relative volatility instead of boilingpoint. For mathematical analysis, the inverse function for the volatility curve is usually more convenient to use than the function itself. Although this is unconventional in graphical representation, it introduces no complication. Accordingly the functions will be written a =
or
w
-
e(W)
(1)
W(a)
(2)
depending on which variable is explicitly dependent. Volatility curve functions corresponding to compositions of known distribution function are readily calculated by integration of the latter, becausc the amount of material remaining in the pot during an ideally sharp distillation when the distillate has a certain a is simply the total amount of material having an a less than the distillate value. Therefore the general formula is
2622
W ( I Y )=
(3)
INDUSTRIAL A N D ENGINEERING CHEMISTRY
November 1951
TRUE BOlLlNG POINT CURVES
Calculation of true boiling point curve functions for compositions of known volatility functions requires, in general, bowledge of the relation between the relative volatility and the normal boiling point of the compounds of the system concerned. The calculation then becomes a mere transformation of the scale of the a axis to a temperature scale. Development of the relation between the boiling points and the relative volatilities of the components of the system baaically requires experimentally determined boiling points and equilibrium data, but analytic expressions can be obtained if an additional idealizing assumption is made. Let the vapor pressuretemperature function for the key component be aasumed to be
2623
Because the bubble point depends only on the first moment of the composition distribution function, it does not fully characterize the composition. This is obvious from physical considerations; if the system contains more than two components, an infinite class of compositions can have the same bubble point. However, if another parameter, such as the fraction remaining in the pot during a batch distillation, is introduced, the composition associated with a definite bubble point becomes uniquely determined, and can be calculated by inversion of an integral transform. This is illustrated by the example of the next section. EXAMPLE:SIMPLEDISTILLATION CURVES.In the previous paper (1), parametric formulas were derived for several limiting types of distillation processes. The one for simple .distillationLe., batch distillation without a rectifying column-reduces to
(4)
The previously made assumption of constant redative volatility gives P(U,
T ) = UPO(T)
(5)
Therefore
is the general vapor pressure-temperature function for the components of this ideal system. Setting the pressure equal to the total preasure yields the general relation between the boiling temperature and the relative volatility’ which is required to effect the transformation of the volatility curve into the true boiling point curve. log
1
-R
(7)
BUBBLE POINT CURVES
Ordinary distillation curves, where the separations are not ideally sharp, can be computed if the assumptions of the preceding section can be made and the distillate compositions are calculated by the methods of the previous paper (1). The temperatures observed are ideally the bubble pointa of the instantaneous distillate compositions and, for ideal system, the bubble pbint of the composition ~ ( uis)the root in T of the equation
P = J.’a,
-
T)x(u)da
p
PO(T)Z
and (9)
This is a remarkable result. The variable or the functional I , which is the first moment of the composition distribution, arises frequently aa a purely mathematical device in dietillation theory where only material balances and equilibrium laws are taken into account. By indirect means, it now appears to be closely asclociated with the temperature. The result in Equation 9 depends only on the assumption of constant relative volatility and does not require any particular form for the vapor pressuTtemperature law for the key component. Such a law must be aasumed or determined, however, before an explicit formula for the bubble point can be derived. Taking Equation 4 gives
A Plog I
-B
where the fraction remaining in the pot, W , depends upon the parameter J aa before by Equation 13. For purposes of mathematical analysis, or where the charge composition is known only by its volatility curve, infinite quantities and graphical differentiation can be avoided by using the Stieltjes integrals
(8)
Uging Equation 5 to eliminate p(u, T)gives the formula
Tb
In accordance with Equation 3, the first equation is readily transformed to express the volatility eurve function for the instantaneous distillate composition,
A
T-
where
and
True boiling point curves characterizing the instantaneoua distillate compositions can be obtained from the last results by change of variable, converting u to T by Equation 7 or a similar relation. Compositions of dietillate cuts over a b i t e range of W can be obtained in t e r m of volatility functions by integration with appropriate limite, according to the usual formula for the mean, and the corresponding true boiling point curves derived by change of variable as before. For calculation of the ordinary distillation curve-i.e., distillate bubble point versus fraction remaining in the pot-the first moment, I , is required. From Equations 10 and 12
(17)
(11)
Vd. 43, No. 11
INDUSTRIAL AND ENGINEERING CHEMISTRY
2624
Putting this value for Z in Equation 11 yields, together with Equation 13, the distillation curve in parametric form. The inverse problem, where calculation of the charge composition from the simple distillation curve is required, is somewhat more difficult, but of considerable practical importance. It has been treated by Pease (6)but the following analysis leads to more convenient numerical computation. Equations 13 and 17 can be combined to give
where the subscripts designate components, as in the classical notation. This can obviously be accomplished by assigning J different arbitrary values, 2n in number, and solving the resulting system of simultaneous equations. This, however, is difficult if n is even moderately large. I n that case, the following method can be used if the relative volatilities of the components are sufficiently well separated in value. Multiplying both sides of Equation 22 by an exponential factor gives the critically discontinuous relation lim Oifj3 ma
(24)
where a,,is the largest of the a’s. Determination of the relative volatility and concentration of the most volatile component is therefore easily effected by plotting W ( J )on iemilog coordinate paper. For sufficiently large values of J the curve must approach a straight line. The slope and intercept of this line then give the relative volatility and concentration, respectively, of the lightest component in the charge. The term corresponding to that component can then be subtracted from W ( J ) and the process repeated. In this way the entire curve can be analyzed, single terms being peeled off in order of decreacing relative volatility. The analysis is simplified if the relative volatilities df the components are known in advance, but thL information is mt necessary. NOMENCLATURE
A B
= a constant, Equation 4 = a constant, Equation 4
ID
= = = = =
Z
=
ZW J n
P
po( T ) p(0,T )
T Tb Tob Trpa W
Wo
= = = = = =
WD
=
3
= = = =
x~ za a
f i s t moment of composition, Equation 10 first moment of distillate composition f i s t moment of distilland composition parameter without physical significance number of components in a finite system total pressure = vapor pressure of the key component a t temperature T = vapor pressure of the component having relative volatility a: at temperature T temperature bubblepoint bubble point of distillate bubble point of distilland fraction of charge remaining in pot fraction of charge remaining in pot in true boiling point distillation of the original charge fraction of charge remaining in pot in true boiling point distillation of instantaneous distillate composition compmition distribution function composition distribution function of distillate composition distribution function of charge relaiive volatility LITERATURE CITED
Bowman, J. R., IND. ENQ.CHEM.,41,2004 (1949). Churchill, R.V.,“Modern Operational Mathematics in Engineering,” New York, MoGraw-Hill Book Co., 1944. Doetsch, G., “Theorie und Anwendung der Laplace-Transformation,” Berlin, Springer, 1937. Erdblyi, A., Phil. Mag., 34,533 (1943).
Pease, J., Math. Phya., 16,202 (1938). Shohat, J. A., and Tamarkin, J. D., “The Problem of Moments,” American Mathematical Society, 1943. Widder, D. V., “The Laplace Transform,” Princeton University Press, 1941. (8)Wiener, N.,“The Fourier Integral.” London, Cambridge University Press, 1933.
RECEIVXD June 28, 1950. Presented heforg the Division of Industrial and Engineering Chemistry at the 116th Meeting of the AMEBICAN CEEMICAL SOCIETY, Atlantio City, N. J.
