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Comparison of vaporization efficiencies with Murphree-type efficiencies in distillation — I
Chemical Engineering Science, 1970, Vol. 25. pp. 43 l-436.
Comparison
Pergamon Press.
Printed in Great Britain.
of vaporization efficiencies with Murphree-type efficiencies in distillation-I
C. D. HOLLAND Texas A & M University, College Station, Texas 77843, U.S.A. and K. S. McMAHON Monsanto Company, Texas City, Texas, U.S.A. (First received 16 January 1969; in revisedform 2 1August 1969) AMact-
Certain serious limitations of Murphree-type efficiencies for the description of the degree of completion of the transfer processes occurring on the plate of a distillation column in the process of separating a multicomponent mixture are presented. By comparison, vaporization efficiencies are shown to be non-zero, finite. and positive numbers, provided the components to which they apply are detectable and have equilibrium constants which are non-zero. finite, and positive numbers.
“Murphree-type efficiency” is used to denote that class of efficiencies defined in terms of a ratio of differences, such as those defined by Murphree[8], Hausen[S], Standart191, and Holland [6]. When a Murphree-type efficiency is applied to a multicomponent mixture, the possibility of obtaining a zero for either the numerator or the denominator exists. On the other hand, it is shown herein that vaporization efficiencies [2,3] are always non-zero, finite and positive numbers, provided the components to which they apply are detectable and have equilibrium constants which are non-zero, positive, and finite. Certain inadequacies of the Murphree-type efficiencies for the description of the degree of completion of the transfer processes occurring on the plate of a distillation column in the process of separating a multicomponent mixture have been demonstrated[6, 101. In particular, it was shown that for a set of apparently reasonable values for a Murphree-type efficiency, negative values were obtained for one or more of the component flow rates. In spite of these demonstrations, the serious limitations of the Murphreetype efficiencies in the analysis of the separation of multicomponent mixtures do not appear to be generally recognized. THE TERM
This paper calls attention to certain limitations of the Murphree-type efficiencies by the presentation of several examples and shows that for these same examples, the vaporization efficiencies are non-zero, finite and positive number. For purposes of demonstration, the Murphree efficiency for the vapor is selected, and it is shown that this efficiency may take on numbers ranging from - to +oo. From the examples produced, it becomes evident that in some instances, the Murphree efficiency tends to reflect the deviation of the Murphree model[8] from the actual behavior of a plate, rather than the degree of completion of the transfer processes. In other instances, Murphree efficiencies tend to reflect certain characteristics which are common for multicomponent systems but not for binary systems. Let the plates be numbered down from the top of a column. The Murphree plate efficiency for the vapor, EMji, for component i on plate j is defined as follows: E
,, = .bfn
Yli-Yj+l,i YX -Yj+l,i
(1)
where yfi is the actual mole fraction of component i in the well mixed vapor leaving plate j, yj+l,i is
431
C. D. HOLLAND
and K. S. McMAHON
the actual mole fraction of component i in the well mixed vapor entering plate j, and yi is the hypothetical mole fraction that component i would have if the vapor leaving plate j were in equilibrium with the liquid leaving, or more precisely, Yz = Kmixji .
(2)
by Carey[2]. In 1949, Edmister[4] proposed a somewhat similar efficiency and called it an “absorption efficiency”. Later the vaporization efficiency was applied to the batch-steam distillation of multicomponent mixtures [7], and subsequently to the separation of multicomponent mixtures in conventional distillation columns [6, lo]. The vaporization efficiency Eji has been defined[6] as follows:
Since y$ was defined as a mole fraction, it follows that
Yji = EjiKji+
(5)
igI YZ = 1,
where Kji is evaluated at the actual temperature of the liquid leaving plate j and pressure of plate or j. For non-ideal solutions, the activity coefficient i &m~5i = 1. (3) yn is included in Equation (5) as a multiplier of i=l Kji. Thus, KMji is evaluated at that hypothetical From defining Eqs. (I), (2) and (5) the Murphree, modified Murphree, and vaporization bubble point temperature required to satisfy Eq. (3). The Murphree plate efficiency for efficiencies are interrelated in the following component i in the liquid was defined in a manner. manner similar to that shown for the vapor. J% _ Yff -Yj+l,i The liquid efficiency contained an xi instead of (6) yg, and it was computed on the basis of a hypoEnrJi - yji-Y.i+l,i thetical dew point temperature based on the actual yjl’s. The modified Murphree efficiency[6, IO] was defined by
(8)
@ =
Yri --Yj+1,i Yj*-Yj+l,i
(4)
where Yjc= Kjtxji, and the Kji’S are evaluated at the actual temperature of the liquid leaving plate j. In a program to determine efficiencies from plant data, this temperature convention certainly has practical significance. However, in a predictive method for the calculation of plate efficiencies, other temperature conventions (such as Murphree’s) may be chosen regardless of the actual temperature of the liquid leaving a plate. The vaporization efficiency was perhaps first employed by either McAdams in the analysis of the batch-steam distillation of a volatile component from a dissolved non-volatile component present in large amounts as described in the Chemical Engineers Handbook[3] or
provided, of course, that each of the respective efficiencies exist. Regardless of the model required to describe the precise mass and heat transfer processes occurring between the vapor and liquid phases on the plate of a distillation column, the vaporization efficiency as defined by Eq. (5) takes on bounded positive values (0 < E,, < a), provided the following conditions
432
0 < Xji 5 1 0 < Yji 5 1
(9)
0 < Kji < m are satisfied. These conditions
do not, however,
guarantee
Comparison of vaporization efficiencies-
the existence of the Murphree-type efficiencies. There follow several examples which demonstrate the limitations of the Murphree-type efficiencies in the description of the transfer processes that occur on the plate of a distillation column in the process of separating a multicomponent mixture. The first example is based on the fact that at total reflux, the denominator of the Murphree efficiency is close to zero for any component whose KMji is close to unity. At the operating condition of total reflux, the component-material balances take the form xji. Now consider the following example. Yj+l,i -
I
and (3) as follows: YB =
KMjixji= KMjiYj+l,i=
Yj+l,i*
(13)
Thus, *
Yji
Y5+l,i
uii- Yji
’
(14)
Since it is given by condition (2) that yz # yji, the inequality given by Eq. (12) follows immediately from Eq. (14). Thus, the numerator of Eq. (11) is seen to be finite while its denominator is zero. Since the division of a finite number by zero is undefined, EMji is undefined. Example 1 It should be observed that the condition Knrii = 1 is not at all severe for some component i of a Given: multicomponent mixture composed of distin(1) The column is at tota reflux (yj+,,i = xji) in the process of separating a multicomponent guishable components (the value of K at a given mixture. temperature and pressure differs for each compo(2) The actual vapor leaving plate j is not in nent). For it follows from Eq. (3) that for any set of xji’s, the K-values at the bubble point equilibrium with the liquid leaving yji # yz. (3) For some component i, KMji= 1. temperature must take on a set of numbers ranging from less than unity to greater than To prove: unity. Thus the occurrence of condition (3) is certainly a likely possibility. E,,ji is undefined. In spite of these difficulties, there exist several Proof: ingenious and rational ways to define bounds on Murphree-type efficiencies. For systems where Equations (1) and (2) may be combined and rearranged to give a pure component composition is approached, its Murphree efficiency may be difficult to determine directly, but the Murphree efficien(Yji)/(Yj+l,i) - 1 (10) cies of the impurities remain finite. For systems (KMj*Xji)l(Yj+l,i) 1* where several components are present in appreciable amounts and K for one or more components Since it is given that the column is at total passes through unity (at total reflux), sometimes reflux, Eq. (10) reduces to arbitrary limits may be used for the Murphree efficiencies to give a “reasonable” match with (Yji)l(Yj+l,i) 1 (11) MN the data. KMji- 1 * Note, however, that the value of Eji (given by To complete the proof, it is first necessary to Eq. (5)) for Example 1 is non-zero, finite and show that positive; that is,
EMji =
E
,,
-
=
-
yji+ Yj+l,i
1.
Eji
(12)
C&S.
Vd. 25 No. 3-F
dk Kj+i
This inequality is shown to exist by commencing with Eq. (2), and then applying conditions (1) ,
=
433
=~
Yji
KjiYj+l,i’
(15)
Throughout the development, it is, of course, supposed that all of the mole fractions lie in the
C. D. HOLLAND
and K. S. McMAHON
open interval between 0 and 1 and that ZJ&is non-zero, finite and positive. If one accepts the possibility of the existence of a “pinch” [yji = yj+l,i and Xji = x~+~,L,for several j in an actual column in which-the vapor leaving each plate is not in equilibrium with the liquid leaving, then it can be shown that the Murphree efficiency takes on the value zero for each component which stays at constant composition in the pinch. More precisely, consider the following example. Example 2
leaving plate j is less than the hypothetical bubble point temperature (computed by use of Eq. (3)). For this case it is possible for the “Murphree driving force” (yi - KMjixji) to have a zero at any z in the closed interval 0 d z 5 zT. In other words, even though the transfer is represented by a diffusion model with an actual driving force greater than zero, the deviation of the actual temperature can make the Murphree driving force zero or negative. In Example 3, it is supposed that the zero in the Murphree driving force uccurs at z = zT. Example
Given: Cl)
Y5i =
t2)
Yz
#
3
Given:
Yl+l,i* Yji-
To prove:
tl)
Y5+l,i >
(2)
~5+1,t= KMjiXji*
Y5i.
To prove:
EM5,= 0.
EM5*is undefined.
Proof:
Proof:
From condition (2), it follows that Y5; -Y5i
+
0.
Then by condition (l), Y$-Y5+l,i
+
0
which is the denominator of EM,*. By use of this result and condition (1) it follows from Eq. (1) that Elcllr= 0. Note that E,, is non-zero, finite and positive for Example 2 as well as the remaining examples which follow. Numerous other examples for which the Murphree efficiency has a zero denominator may be produced by supposing that the behavior of an actual plate differs slightly from the one assumed in the development of the Murphree efficiency. In the following examples, the depth of liquid on a plate is denoted by z (At the surface of the liquid, z = 0, and at the floor of the plate z = zr). Also, at z = 0, yi =yji and at z= zr, Yi = Yj+l,i, and &xi = K5tx5i. Example 3 is formulated by first supposing that the transfer of component i is from the vapor to the liquid phase for all z (0 Z z 5 zT). Next, suppose that the actual temperature of the liquid 434
That EM5*is undefined follows immediately upon imposing conditions (1) and (2) on the definition Of EMjias given by Eq. (1). Another example of this type is formulated as follows. Suppose that the mass transfer of component i is from the liquid to the vapor phase (y5* > yj+l,i), and that the liquid leaves platej at an actual temperature which is greater than the bubble point temperature. If for component i, it is further supposed that at the bubble point temperature the equality K,+,jixji= yj+l,i occurs, then the Murphree efficiency for component i is undefined. The following examples are based on the experimental fact that composition profiles for certain components may pass through either a maximum or a minimum in columns in the process of separating multicomponent mixtures. In Example 4, it is supposed that maxima in the yt and Klxl profiles occur at some zk lying between 0 and zT, y5r # yj+l,i, component i is being transferred from the liquid to the vapor phase at all z, and that the actual temperature of the liquid leaving plate j is greater than its bubble point temperature. (This results in a Murphree
Comparisonof vaporization efficiencies- I similar to those demonstrated for Murphree efficiencies when applied to multicomponent systems.
driving force which is too small.) For component i, it is further supposed that the relationship KMjiXji = yj+l,i occurs.
CONCLUSIONS
Example 4 Given: (1)
Yji f
(2)
K.afji%i
Y1+1,i* =
Yj+l,i-
To prove: EM*iis undefined. Proof: The proof follows immediately upon imposing conditions (1) and (2) on the definition of E,, given by Eq. (1). Next suppose that component i is being transferred from the vapor to the liquid phase along with several other components, each at a different rate. Certainly, if the component is transferred at all, the efficiency of transfer is not zero. However, for component i, the mole fraction in the vapor yi has the value yl+l,i at z = zT, passes through a maximum at some zk (0 < zk < zT), as certain of the other components concentrate the vapor, and returns to its initial value yj+l,i at z = 0; that is, for component r, Yji= Yj+l,i* Further, suppose that the plate is not perfect, in terms of Murphree concepts; that is, Yi + Yji- Formally, this example reduces to Example 2, and it follows that for component i, EMji = 0. Although the Murphree efficiency for the vapor was singled out for examination in the above analyses, other Murphree-type efficiencies are subject to similar difficulties in the analysis of the behavior of plates in the process of separating multicomponent mixture. Recently, Hausen [5] and Standart [9] proposed the use of efficiencies which were of the same general form as the Murphree efficiency, a ratio of differences, but differing from the Murphree efficiencies by the fact that the respective efficiencies for the vapor and liquid phases were equal. However, because of the form of these efficiencies (the ratio of two differences), they have limitations
The examples presented demonstrate that in some instances the Murphree efficiency cannot reflect the efficiency of the transfer processes occurring on the plate of a multicomponent distillation column; whereas, for all cases where the conditions of Eq. (9) are satisfied, the vaporization efficiencies are non-zero, finite and positive and have the potential for reflecting the efficiency of the transfer processes occurring. The demonstrated characteristics of the Murphree efficiencies suggests that all correlations purporting to given even approximate values of these efficiencies for multicomponent systems, such as Ref. [ 11, should be re-examined. In view of the general applicability of the vaporization efficiency, it appears plausible to suggest the use of his efficiency as the base upon which to build future correlations. Acknowledgments-Support of this research by the National Science Foundation and the Monsanto Company is gratefully acknowledged. The helpful suggestions of Professor E. C. Klipple are also appreciated.
435
NOTATION
vaporization efficiency; defined by Eq. (5) Murphree efficiency for vapor; defined by Eq. (1) modified Murphree efficiency; defined by Eq. (4) the K-value for component i; evaluated at the pressure of plate j and at the temperature of the liquid leaving platej the K-value for component i; evaluated at the pressure of platej and at the corresponding bubble point temperature of the actual liquid leaving platej mole fraction of component i in the actual liquid leaving platej mole fraction of component i in the actual vapor leaving platej
C. D. HOLLAND
and K. S. McMAHON
mole fraction that component i would have if the vapor leaving platej were in equilibrium with the liquid leaving. YJI the value of the product Kjlxjl
yz
Subscripts i component number, (1 5 i S c) j plate number; the plates are numbered down from the top of a column
REFERENCES
[II
Bubble-Tray
Design Manual. American Institute of Chemical Engineers 1958.
121 CAREY J. S., Sc.D. Thesis. Mass. Inst. Tech. 1930. [3] Chemical Engineers'Handbook(Edited by PERRY J. H.), 3rd Edn., p. 582. McGraw-Hill [4] EDMISTER W. C., Petrol. Engr 1949 21 1 C-45. [5] HAUSEN H., Chemie-lngr-Tech. 1953 25 747. [6] HOLLAND C. D., Multicomponent Distillation. Prentice-Hall 1963. [7] HOLLAND C. D. and WELCH N. E., Petrol. Refiner 1957 36 5,251. [8] MURPHREE E. V., Ind. Engng Chem. 1925 17 747. [9] STANDART Cl., Chem. Engng Sci. 1964 20 611. [lo] TAYLOR D. L., M.S. Thesis, Texas A &M University 1962.
1950.
Resume-Cet article presente certaine limitations serieuses des rendements du type Murphree pour la description du degre d’achevement des processus de transfert se produisant sur la plaque dune colonne a distillation, au tours du pro&de de separation d’un melange a composants multiples. Par comparaison, on montre que les rendements de vaporisation sont des nombres non-zero, dtfinis et positifs, a condition que les composants auxquels ils s’appliquent puissent etre detect& et ont des constantes d’equilibre qui soient des nombres non-zero, definis et positifs. Zusammenfassung- Es wird auf verschiedene ernsthafte Beschrankungen der Wirkungsgrade nach Murphree bei der Beschreibung des Staffaustauschgrades auf dem Boden einer Destillationskolonne bei der Trennung einer aus mehreren Bestandteilen zusammengesetzten Mischung hingewiesen. Fiir Vergleichszwecke wird gezeigt, dass die Verdampfungswirkungsgrade ungleich Null, endliche und positive Zahlen sind, vorausgesetzt die Bestandteile auf die sie sich beziehen, sind nachweisbar und haben Gleichgewichtskonstanten, die ungleich Null, endliche und positive Zahlen sind.
436
Comparison
Pergamon Press.
Printed in Great Britain.
of vaporization efficiencies with Murphree-type efficiencies in distillation-I
C. D. HOLLAND Texas A & M University, College Station, Texas 77843, U.S.A. and K. S. McMAHON Monsanto Company, Texas City, Texas, U.S.A. (First received 16 January 1969; in revisedform 2 1August 1969) AMact-
Certain serious limitations of Murphree-type efficiencies for the description of the degree of completion of the transfer processes occurring on the plate of a distillation column in the process of separating a multicomponent mixture are presented. By comparison, vaporization efficiencies are shown to be non-zero, finite. and positive numbers, provided the components to which they apply are detectable and have equilibrium constants which are non-zero. finite, and positive numbers.
“Murphree-type efficiency” is used to denote that class of efficiencies defined in terms of a ratio of differences, such as those defined by Murphree[8], Hausen[S], Standart191, and Holland [6]. When a Murphree-type efficiency is applied to a multicomponent mixture, the possibility of obtaining a zero for either the numerator or the denominator exists. On the other hand, it is shown herein that vaporization efficiencies [2,3] are always non-zero, finite and positive numbers, provided the components to which they apply are detectable and have equilibrium constants which are non-zero, positive, and finite. Certain inadequacies of the Murphree-type efficiencies for the description of the degree of completion of the transfer processes occurring on the plate of a distillation column in the process of separating a multicomponent mixture have been demonstrated[6, 101. In particular, it was shown that for a set of apparently reasonable values for a Murphree-type efficiency, negative values were obtained for one or more of the component flow rates. In spite of these demonstrations, the serious limitations of the Murphreetype efficiencies in the analysis of the separation of multicomponent mixtures do not appear to be generally recognized. THE TERM
This paper calls attention to certain limitations of the Murphree-type efficiencies by the presentation of several examples and shows that for these same examples, the vaporization efficiencies are non-zero, finite and positive number. For purposes of demonstration, the Murphree efficiency for the vapor is selected, and it is shown that this efficiency may take on numbers ranging from - to +oo. From the examples produced, it becomes evident that in some instances, the Murphree efficiency tends to reflect the deviation of the Murphree model[8] from the actual behavior of a plate, rather than the degree of completion of the transfer processes. In other instances, Murphree efficiencies tend to reflect certain characteristics which are common for multicomponent systems but not for binary systems. Let the plates be numbered down from the top of a column. The Murphree plate efficiency for the vapor, EMji, for component i on plate j is defined as follows: E
,, = .bfn
Yli-Yj+l,i YX -Yj+l,i
(1)
where yfi is the actual mole fraction of component i in the well mixed vapor leaving plate j, yj+l,i is
431
C. D. HOLLAND
and K. S. McMAHON
the actual mole fraction of component i in the well mixed vapor entering plate j, and yi is the hypothetical mole fraction that component i would have if the vapor leaving plate j were in equilibrium with the liquid leaving, or more precisely, Yz = Kmixji .
(2)
by Carey[2]. In 1949, Edmister[4] proposed a somewhat similar efficiency and called it an “absorption efficiency”. Later the vaporization efficiency was applied to the batch-steam distillation of multicomponent mixtures [7], and subsequently to the separation of multicomponent mixtures in conventional distillation columns [6, lo]. The vaporization efficiency Eji has been defined[6] as follows:
Since y$ was defined as a mole fraction, it follows that
Yji = EjiKji+
(5)
igI YZ = 1,
where Kji is evaluated at the actual temperature of the liquid leaving plate j and pressure of plate or j. For non-ideal solutions, the activity coefficient i &m~5i = 1. (3) yn is included in Equation (5) as a multiplier of i=l Kji. Thus, KMji is evaluated at that hypothetical From defining Eqs. (I), (2) and (5) the Murphree, modified Murphree, and vaporization bubble point temperature required to satisfy Eq. (3). The Murphree plate efficiency for efficiencies are interrelated in the following component i in the liquid was defined in a manner. manner similar to that shown for the vapor. J% _ Yff -Yj+l,i The liquid efficiency contained an xi instead of (6) yg, and it was computed on the basis of a hypoEnrJi - yji-Y.i+l,i thetical dew point temperature based on the actual yjl’s. The modified Murphree efficiency[6, IO] was defined by
(8)
@ =
Yri --Yj+1,i Yj*-Yj+l,i
(4)
where Yjc= Kjtxji, and the Kji’S are evaluated at the actual temperature of the liquid leaving plate j. In a program to determine efficiencies from plant data, this temperature convention certainly has practical significance. However, in a predictive method for the calculation of plate efficiencies, other temperature conventions (such as Murphree’s) may be chosen regardless of the actual temperature of the liquid leaving a plate. The vaporization efficiency was perhaps first employed by either McAdams in the analysis of the batch-steam distillation of a volatile component from a dissolved non-volatile component present in large amounts as described in the Chemical Engineers Handbook[3] or
provided, of course, that each of the respective efficiencies exist. Regardless of the model required to describe the precise mass and heat transfer processes occurring between the vapor and liquid phases on the plate of a distillation column, the vaporization efficiency as defined by Eq. (5) takes on bounded positive values (0 < E,, < a), provided the following conditions
432
0 < Xji 5 1 0 < Yji 5 1
(9)
0 < Kji < m are satisfied. These conditions
do not, however,
guarantee
Comparison of vaporization efficiencies-
the existence of the Murphree-type efficiencies. There follow several examples which demonstrate the limitations of the Murphree-type efficiencies in the description of the transfer processes that occur on the plate of a distillation column in the process of separating a multicomponent mixture. The first example is based on the fact that at total reflux, the denominator of the Murphree efficiency is close to zero for any component whose KMji is close to unity. At the operating condition of total reflux, the component-material balances take the form xji. Now consider the following example. Yj+l,i -
I
and (3) as follows: YB =
KMjixji= KMjiYj+l,i=
Yj+l,i*
(13)
Thus, *
Yji
Y5+l,i
uii- Yji
’
(14)
Since it is given by condition (2) that yz # yji, the inequality given by Eq. (12) follows immediately from Eq. (14). Thus, the numerator of Eq. (11) is seen to be finite while its denominator is zero. Since the division of a finite number by zero is undefined, EMji is undefined. Example 1 It should be observed that the condition Knrii = 1 is not at all severe for some component i of a Given: multicomponent mixture composed of distin(1) The column is at tota reflux (yj+,,i = xji) in the process of separating a multicomponent guishable components (the value of K at a given mixture. temperature and pressure differs for each compo(2) The actual vapor leaving plate j is not in nent). For it follows from Eq. (3) that for any set of xji’s, the K-values at the bubble point equilibrium with the liquid leaving yji # yz. (3) For some component i, KMji= 1. temperature must take on a set of numbers ranging from less than unity to greater than To prove: unity. Thus the occurrence of condition (3) is certainly a likely possibility. E,,ji is undefined. In spite of these difficulties, there exist several Proof: ingenious and rational ways to define bounds on Murphree-type efficiencies. For systems where Equations (1) and (2) may be combined and rearranged to give a pure component composition is approached, its Murphree efficiency may be difficult to determine directly, but the Murphree efficien(Yji)/(Yj+l,i) - 1 (10) cies of the impurities remain finite. For systems (KMj*Xji)l(Yj+l,i) 1* where several components are present in appreciable amounts and K for one or more components Since it is given that the column is at total passes through unity (at total reflux), sometimes reflux, Eq. (10) reduces to arbitrary limits may be used for the Murphree efficiencies to give a “reasonable” match with (Yji)l(Yj+l,i) 1 (11) MN the data. KMji- 1 * Note, however, that the value of Eji (given by To complete the proof, it is first necessary to Eq. (5)) for Example 1 is non-zero, finite and show that positive; that is,
EMji =
E
,,
-
=
-
yji+ Yj+l,i
1.
Eji
(12)
C&S.
Vd. 25 No. 3-F
dk Kj+i
This inequality is shown to exist by commencing with Eq. (2), and then applying conditions (1) ,
=
433
=~
Yji
KjiYj+l,i’
(15)
Throughout the development, it is, of course, supposed that all of the mole fractions lie in the
C. D. HOLLAND
and K. S. McMAHON
open interval between 0 and 1 and that ZJ&is non-zero, finite and positive. If one accepts the possibility of the existence of a “pinch” [yji = yj+l,i and Xji = x~+~,L,for several j in an actual column in which-the vapor leaving each plate is not in equilibrium with the liquid leaving, then it can be shown that the Murphree efficiency takes on the value zero for each component which stays at constant composition in the pinch. More precisely, consider the following example. Example 2
leaving plate j is less than the hypothetical bubble point temperature (computed by use of Eq. (3)). For this case it is possible for the “Murphree driving force” (yi - KMjixji) to have a zero at any z in the closed interval 0 d z 5 zT. In other words, even though the transfer is represented by a diffusion model with an actual driving force greater than zero, the deviation of the actual temperature can make the Murphree driving force zero or negative. In Example 3, it is supposed that the zero in the Murphree driving force uccurs at z = zT. Example
Given: Cl)
Y5i =
t2)
Yz
#
3
Given:
Yl+l,i* Yji-
To prove:
tl)
Y5+l,i >
(2)
~5+1,t= KMjiXji*
Y5i.
To prove:
EM5,= 0.
EM5*is undefined.
Proof:
Proof:
From condition (2), it follows that Y5; -Y5i
+
0.
Then by condition (l), Y$-Y5+l,i
+
0
which is the denominator of EM,*. By use of this result and condition (1) it follows from Eq. (1) that Elcllr= 0. Note that E,, is non-zero, finite and positive for Example 2 as well as the remaining examples which follow. Numerous other examples for which the Murphree efficiency has a zero denominator may be produced by supposing that the behavior of an actual plate differs slightly from the one assumed in the development of the Murphree efficiency. In the following examples, the depth of liquid on a plate is denoted by z (At the surface of the liquid, z = 0, and at the floor of the plate z = zr). Also, at z = 0, yi =yji and at z= zr, Yi = Yj+l,i, and &xi = K5tx5i. Example 3 is formulated by first supposing that the transfer of component i is from the vapor to the liquid phase for all z (0 Z z 5 zT). Next, suppose that the actual temperature of the liquid 434
That EM5*is undefined follows immediately upon imposing conditions (1) and (2) on the definition Of EMjias given by Eq. (1). Another example of this type is formulated as follows. Suppose that the mass transfer of component i is from the liquid to the vapor phase (y5* > yj+l,i), and that the liquid leaves platej at an actual temperature which is greater than the bubble point temperature. If for component i, it is further supposed that at the bubble point temperature the equality K,+,jixji= yj+l,i occurs, then the Murphree efficiency for component i is undefined. The following examples are based on the experimental fact that composition profiles for certain components may pass through either a maximum or a minimum in columns in the process of separating multicomponent mixtures. In Example 4, it is supposed that maxima in the yt and Klxl profiles occur at some zk lying between 0 and zT, y5r # yj+l,i, component i is being transferred from the liquid to the vapor phase at all z, and that the actual temperature of the liquid leaving plate j is greater than its bubble point temperature. (This results in a Murphree
Comparisonof vaporization efficiencies- I similar to those demonstrated for Murphree efficiencies when applied to multicomponent systems.
driving force which is too small.) For component i, it is further supposed that the relationship KMjiXji = yj+l,i occurs.
CONCLUSIONS
Example 4 Given: (1)
Yji f
(2)
K.afji%i
Y1+1,i* =
Yj+l,i-
To prove: EM*iis undefined. Proof: The proof follows immediately upon imposing conditions (1) and (2) on the definition of E,, given by Eq. (1). Next suppose that component i is being transferred from the vapor to the liquid phase along with several other components, each at a different rate. Certainly, if the component is transferred at all, the efficiency of transfer is not zero. However, for component i, the mole fraction in the vapor yi has the value yl+l,i at z = zT, passes through a maximum at some zk (0 < zk < zT), as certain of the other components concentrate the vapor, and returns to its initial value yj+l,i at z = 0; that is, for component r, Yji= Yj+l,i* Further, suppose that the plate is not perfect, in terms of Murphree concepts; that is, Yi + Yji- Formally, this example reduces to Example 2, and it follows that for component i, EMji = 0. Although the Murphree efficiency for the vapor was singled out for examination in the above analyses, other Murphree-type efficiencies are subject to similar difficulties in the analysis of the behavior of plates in the process of separating multicomponent mixture. Recently, Hausen [5] and Standart [9] proposed the use of efficiencies which were of the same general form as the Murphree efficiency, a ratio of differences, but differing from the Murphree efficiencies by the fact that the respective efficiencies for the vapor and liquid phases were equal. However, because of the form of these efficiencies (the ratio of two differences), they have limitations
The examples presented demonstrate that in some instances the Murphree efficiency cannot reflect the efficiency of the transfer processes occurring on the plate of a multicomponent distillation column; whereas, for all cases where the conditions of Eq. (9) are satisfied, the vaporization efficiencies are non-zero, finite and positive and have the potential for reflecting the efficiency of the transfer processes occurring. The demonstrated characteristics of the Murphree efficiencies suggests that all correlations purporting to given even approximate values of these efficiencies for multicomponent systems, such as Ref. [ 11, should be re-examined. In view of the general applicability of the vaporization efficiency, it appears plausible to suggest the use of his efficiency as the base upon which to build future correlations. Acknowledgments-Support of this research by the National Science Foundation and the Monsanto Company is gratefully acknowledged. The helpful suggestions of Professor E. C. Klipple are also appreciated.
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NOTATION
vaporization efficiency; defined by Eq. (5) Murphree efficiency for vapor; defined by Eq. (1) modified Murphree efficiency; defined by Eq. (4) the K-value for component i; evaluated at the pressure of plate j and at the temperature of the liquid leaving platej the K-value for component i; evaluated at the pressure of platej and at the corresponding bubble point temperature of the actual liquid leaving platej mole fraction of component i in the actual liquid leaving platej mole fraction of component i in the actual vapor leaving platej
C. D. HOLLAND
and K. S. McMAHON
mole fraction that component i would have if the vapor leaving platej were in equilibrium with the liquid leaving. YJI the value of the product Kjlxjl
yz
Subscripts i component number, (1 5 i S c) j plate number; the plates are numbered down from the top of a column
REFERENCES
[II
Bubble-Tray
Design Manual. American Institute of Chemical Engineers 1958.
121 CAREY J. S., Sc.D. Thesis. Mass. Inst. Tech. 1930. [3] Chemical Engineers'Handbook(Edited by PERRY J. H.), 3rd Edn., p. 582. McGraw-Hill [4] EDMISTER W. C., Petrol. Engr 1949 21 1 C-45. [5] HAUSEN H., Chemie-lngr-Tech. 1953 25 747. [6] HOLLAND C. D., Multicomponent Distillation. Prentice-Hall 1963. [7] HOLLAND C. D. and WELCH N. E., Petrol. Refiner 1957 36 5,251. [8] MURPHREE E. V., Ind. Engng Chem. 1925 17 747. [9] STANDART Cl., Chem. Engng Sci. 1964 20 611. [lo] TAYLOR D. L., M.S. Thesis, Texas A &M University 1962.
1950.
Resume-Cet article presente certaine limitations serieuses des rendements du type Murphree pour la description du degre d’achevement des processus de transfert se produisant sur la plaque dune colonne a distillation, au tours du pro&de de separation d’un melange a composants multiples. Par comparaison, on montre que les rendements de vaporisation sont des nombres non-zero, dtfinis et positifs, a condition que les composants auxquels ils s’appliquent puissent etre detect& et ont des constantes d’equilibre qui soient des nombres non-zero, definis et positifs. Zusammenfassung- Es wird auf verschiedene ernsthafte Beschrankungen der Wirkungsgrade nach Murphree bei der Beschreibung des Staffaustauschgrades auf dem Boden einer Destillationskolonne bei der Trennung einer aus mehreren Bestandteilen zusammengesetzten Mischung hingewiesen. Fiir Vergleichszwecke wird gezeigt, dass die Verdampfungswirkungsgrade ungleich Null, endliche und positive Zahlen sind, vorausgesetzt die Bestandteile auf die sie sich beziehen, sind nachweisbar und haben Gleichgewichtskonstanten, die ungleich Null, endliche und positive Zahlen sind.
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