3
EngEtring ROCW development Interrelations Distillation Curves between I JOHN R. BOWMAN M E L L O N INSTITUTE, PITTSBURGH, PA, Formulas for true boiling point and volatility curves are dcrived from the composition distribution function treated in a previous paper. These are alternative methods for characterizing mixtures of an indefinite num- ber of components. The first moment of the composition distribution function is shown to be closely related to tho bubble point, Examples are presented for the case of simple batch distillation. By inversion of the formula for the true boiling point curve, that function can be com- puted from the simple distillation curve. N A previous paper (1) the concept of the composition dis- I tribution 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 dis- tillation 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 phys- ically as the density of material in the a direction where the con- centrations are ranged along a linear a scale. Actually, however, it has more the properties of a conventional fraction concentra- tion, because in nearly all equations the factor da divides out or is integrated. Where one or more components are present in finite concentra- tion, the function x has infinite peaks at 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 vola- tility 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 mix- tures 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 ' 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 experimen- tally. 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. VOLATlIJTY CURVES The volatility curve provides a means for specifying the com- position of a complex mixture and is a compromise between the mathematically fundamental composition distribution function and the practically convenient true boiling point curve. Con- sider a conventional true boiling point batch distillation-i~., one in which each component distills over pure, and the com- ponents 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 direc- tion. The volatility curve ia defined to be the plot of the relative volatility of the component coming overhead at any instant verRus the fraction of the original distilland charge remaining in the pot at the same instant, The mathematical expression for a vola- tility curve will be termed a volatility curve function. The vola- tility 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 vola- tility curve is usually more convenient to use than the function itself. Although this is unconventional in graphical representa- tion, it introduces no complication. Accordingly the functions will be written a = e(W) (1) w - W(a) (2) or 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 W(IY) = (3) 2622

Interrelations between Distillaton Curves

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Page 1: Interrelations between Distillaton Curves

EngEtr ing

R O C W development

Interrelations Distillation Curves

between

I

JOHN R. BOWMAN M E L L O N INSTITUTE, PITTSBURGH, PA,

Formulas for t rue boiling point and 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 moment of the composition distribution function is shown to be closely related t o tho bubble point, Examples a re presented for the case of simple batch distillation. By inversion of the formula for the t rue boiling point curve, that function can be com- puted from the simple distillation curve.

N A previous paper ( 1 ) the concept of the composition dis- I tribution 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 dis- tillation 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 phys- ically as the density of material in the a direction where the con- centrations are ranged along a linear a scale. Actually, however, it has more the properties of a conventional fraction concentra- tion, because in nearly all equations the factor da divides out or is integrated.

Where one or more components are present in finite concentra- tion, the function x has infinite peaks at 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 vola- tility 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 mix- tures 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 i t can be discontiriuous and is not directly observable. The most practical general characterization of the composition of complex

'

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 experimen- tally. 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.

VOLATlIJTY CURVES

The volatility curve provides a means for specifying the com- position of a complex mixture and is a compromise between the mathematically fundamental composition distribution function and the practically convenient true boiling point curve. Con- sider a conventional true boiling point batch distillation-i~., one in which each component distills over pure, and the com- ponents 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 direc- tion. 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 vola- tility curve will be termed a volatility curve function. The vola- tility 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 vola- tility curve is usually more convenient to use than the function itself. Although this is unconventional in graphical representa- tion, it introduces no complication. Accordingly the functions will be written

a = e (W) (1)

w - W ( a ) (2)

or

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

W(IY) = (3)

2622

Page 2: Interrelations between Distillaton Curves

November 1951 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y 2623 TRUE BOlLlNG POINT CURVES

Calculation of true boiling point curve functions for composi- tions 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 equili- brium data, but analytic expressions can be obtained if an addi- tional idealizing assumption is made. Let the vapor pressure- temperature function for the key component be aasumed to be

(4)

The previously made assumption of constant redative volatility gives

Therefore P(U, T) = UPO(T) ( 5 )

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.

A T - 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 dis- tillate compositions and, for ideal system, the bubble pbint of the composition ~ ( u ) is the root in T of the equation

P = J.’a, T)x(u)da ( 8 )

p - PO(T)Z (9)

Uging Equation 5 to eliminate p(u, T) gives the formula

where

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 ascloci- ated 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 com- ponent. 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 P I

Tb -- log - - B

Because the bubble point depends only on the first moment of the composition distribution function, i t does not fully charac- terize the composition. This is obvious from physical considera- tions; 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 deter- mined, and can be calculated by inversion of an integral trans- form. This is illustrated by the example of the next section.

EXAMPLE: SIMPLE DISTILLATION CURVES. In the previous paper (1 ), parametric formulas were derived for several limiting types of distillation processes. The one for simple .distillation- Le., batch distillation without a rectifying column-reduces to

and

In accordance with Equation 3, the first equation is readily transformed to express the volatility eurve function for the instantaneous distillate composition,

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 quanti- ties and graphical differentiation can be avoided by using the Stieltjes integrals

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., dis- tillate bubble point versus fraction remaining in the pot-the first moment, I , is required. From Equations 10 and 12

(17)

(11)

Page 3: Interrelations between Distillaton Curves

2624 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y Vd. 43, No. 11

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 com- position from the simple distillation curve is required, is some- what 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

The boundary conditions, from Equation 13 are

and W = 1 at J = 0 (20)

The solution is

(21)

If the simple diatillation curve function T D ~ ( W ) is known experimentally, the function In(W) can be computed from Equation 11. Two graphical or numerical integrations then permit the computation of the function J (W) by Equation 21. Putting the inverse, W ( J ) , of the last function in the left-hand side of Equation 13 or 16 gives the composition distribution or the volatility curve function for the original charge by inversion of the integral transform.

The calculation of J ( W ) and its inverse from the distilland temperature curve is simpler than from the distillate tempera- ture one. The function Zw(W) is calculated from Equation 11 as before, but using the liquid temperature data, and then the relation

J ( W ) = A’””- WIw(W)

derived in the previous paper ( I ) , is used. In this way, only one graphical integration is required.

Tn mathematical terminology, Equations 13 and 16 are the expressions for the Laplace and the Laplace-Stieltjes transforms, respectively, and the inversion problem-Le., the determination of the unknown function in the integrand, when the left-hand side is a known function-has received much attention from mathe- maticians in recent years (8, S, 6-8). several inversion formulas, expressing the unknown function explicitly, have been derived, Most of these, however, are wholly unsuitable for numerical computation, depending as they do on infinite or complex values of the known function, or on its derivative of infinite order, For the present problem, the inversion formula of Erd6lyi (4) ia the most convenient. It approximates the unknown function by a finite series of orthogonal functions, with coefficients derived from equally spaced ordinates of the known function. Another numerical method (2) involving successive approximations may be useful, especially if several components are present in finite concentrations.

If the number of components is finite, the inversion problem reduces to the determination of the concentrations, xi, and the relative volatilities, ai, from the Dirichlet series

n

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. In 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 Oif j3<an

m if > ma J --e- mePJW(J) = zn if j3 = an (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 Z I D ZW J n P = total pressure po( T ) = vapor pressure of the key component a t temperature

p ( 0 , T ) = vapor pressure of the component having relative vola-

T = temperature Tb = bubblepoint Tob = bubble point of distillate Trpa = bubble point of distilland W = fraction of charge remaining in pot Wo = fraction of charge remaining in pot in true boiling

point distillation of the original charge WD = fraction of charge remaining in pot in true boiling

point distillation of instantaneous distillate com-

= a constant, Equation 4 = a constant, Equation 4 = f is t moment of composition, Equation 10 = first moment of distillate composition = f is t moment of distilland composition = parameter without physical significance = number of components in a finite system

T

tility a: at temperature T

position 3 = compmition distribution function x~ za

= composition distribution function of distillate = composition distribution function of charge

a = relaiive volatility

LITERATURE CITED

Bowman, J. R., IND. ENQ. CHEM., 41,2004 (1949). Churchill, R. V., “Modern Operational Mathematics in Engineer-

ing,” New York, MoGraw-Hill Book Co., 1944. Doetsch, G., “Theorie und Anwendung der Laplace-Transforma-

tion,” 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,”

Widder, D. V., “The Laplace Transform,” Princeton University American Mathematical Society, 1943.

Press, 1941. (8) Wiener, N., “The Fourier Integral.” London, Cambridge Univer-

sity 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.