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The selection of test mixtures for distillation columns
Chemical Engineering Science, 1967, Vol. 22, pp. 957-961. Pergamon Press Ltd., Oxford. Printed in Great Britain.
Tlie selection of test mixtures for distillation columns D. G. PEACOCK The School of Pharmacy, University of London, Brunswick Square, London, W.C.l (Received 8 February 1967) Abstract--The selection of mixtures for testing fractionating columns involves a number of factors, often mutually conflicting. The choice may be quite difficult. The tests involve the analysis of samples and analytical errors may seriously affect the results obtained. The importance of this factor depends not only on the accuracy of the analytical methods available but also on the range of compositions handled. The latter is itself dependent on the choice of mixture. This aspect of the selection is discussed. For a given separating power there is an optimum relative volatility of the chosen test mixture, at which analytical errors are least important. A chart is presented showing this optimum and the effect of departures from it, to provide a quantitative basis for this aspect of the choice.
The value for N obtained from Eq. (1) is susceptible to errors introduced either in the equilibrium data (i.e. in a) or in the sample analyses (xD and xB). COULSON and HERINGTON [3] have discussed the effect of errors in a; they and a number of subsequent workers have devised thermodynamic consistency tests for checking experimental vapourliquid equilibrium data. Errors in the sample analyses form the topic of this present work.
1. INTR~O~~TI~N Trrn TECHNIQUE of testing a distillation column to assess its separating power is well established [l, 2, 41. The column is operated with a suitable binary mixture, under total reflux, for long enough to allow a steady state to be reached. Samples are then taken of the liquid refluxing from the condenser to the column and (preferably) of the liquid returning from the column to the reboilett. The samples are analysed and the result calculated from Fenske’s equation :
2. ANALYTICAL ERRORS
(1) Although this process is essentially very simple, considerable care and attention to detail are necessary if meaningful results are to be obtained. FJXNSKE[l] has listed specific precautions to be taken to secure reliability.
t The alternative course of sampling from the reboiler itself is often taken. This is, however, somewhat less satisfactory, since the sample may be contaminated with tars or other non-volatile impurities and because an extra ideal stage is included in the system tested. This extra stage means some loss of accuracy in the result obtained for the column, and is particularly undesirable if columns equivalent to only 2 or 3 theoretical stages are under test. For longer columns, this latter point is of less importance and the criteria developed here for selecting test mixtures represent a useful working guide.
Errors in analysis can present serious difficulties in comparative tests of distillation equipment. This is especially true in industrial, large-scale testing, since questions of availability and disposability may place restrictions on the choice of mixtures to be used-restrictions largely absent where only laboratory-scale tests are involved. Furthermore, where comparative tests are being made, with two or more (possibly competing) designs, errors of analysis will generally operate independently on the separate tests made. Errors in a wilI more usually operate similarly on all tests, affecting the comparison less in consequence. The impact of analytical errors is particularly severe if extremes of concentration occur (i.e. if x, or (1 -xJ is very close to zero), or if a is so low that xi, and xD are not very different. In practice one tries to pick a mixture which avoids these difficulties and to work with the compositions
957
D. G. PEACCPZK
1 1 +_ -- @J2 + (In cr)’ xD2(1- xD)’ xB2(l - xa)’
and xD symmetrically placed about 50 per centas FENSKE [I] put it: “Whenever possible, the middle portion of the vapour-liquid diagram should be used”.
xB
For the present purpose, of examining the effect of analytical errors and the dependence of this effect on the value of a, the following assumptions are made:
xg=(l-xJ=x
$=Y.
constant and free from random error;
the relative amounts of the two components are so adjusted that xD is equal to (1 - xB), this common value being denoted by x. In fact this will only hold approximately true, but this approximation will not invalidate the results;
(4 the values of xD and xB are subject, inde-
(5)
from which it may be shown that
(a>the relative volatility TVis assumed to be
(b) to simplify the arithmetic, it is assumed that
(4)
ux
where:
‘lJ2. x(1-x&x/(1-x))
(7)
The variation of $ with x is shown in Fig. 1. A minimum of $ = 3.16 is observed at x = 0.824, and (as one would expect) x=0.5 and x= 1 are asymptotes.
pendently, to random errors which are assumed to be small, normally distributed, and of standard deviation + a,. Further, this variation cX is assumed to be approximately constant over the whole practicable range of compositions. This will be more or less true according to the method of analysis used ; it is certainly a good working approximation where analysis is based on the measurement of a physical property such as density or refractive index. Returning to Eq. (1), this may be expressed in the alternative form :
Top composition.
Considering the variation of xB and xD, the standard deviation UNof N may be calculated from
(3)
xg
Fro. 1. The relationship between composition and sensitivity
to errors.
Thus a given level of uncertainty in analysis will have the least effect on the result, if the test 958
The selection of test mixtures for distillation columns
mixture and the column under test are so related as to give compositions X, and xB at or about this minimum point. Since there is a correspondence [Eq. (2) and (5)l between x and the product N . log a, this same requirement may be stated in terms of the latter, as in Fig. 2; ideally one would choose to have N. log,,cr = 1.34 and the effect of having values above or below this can be seen.
used. Figure 3 does, however, provide a quantitative basis for such a decision. It will be noted that if appropriately-calculated? confidence limits are known for x, and x,, then Eq. (6) enables confidence limits for N to be estimated. For example, if the 0.95level limits _+Sx are 10.01, the corresponding limits +dN/N are 0.01 II/. In the ideal case, then, such as the numerical example above, with c1= 1.29, the result N would be known to f3.16 per cent. Were CI= 1.03, however-the “five times minimum error” case-the error in N could well exceed 2 15 per cent. AcknowMgrneat-The author is indebted to Prof. Saunders for advice in the presentation of this paper.
L.
NOTATION
N
o-2
0.1
03
0.4
06
0.8
N lq, FIG.
xB composition of liquid returning from column to reboiler, in mole fraction of more volatile component
1 a
XD
2. This same relationship plotted directly in terms of N.logloa.
.Y= xD = (I4. THE PRACTICAL CHOICE For the purpose of comparing the suitability of alternative test mixtures, the foregoing relationships are more conveniently expressed in Fig. 3. Suppose, for example, comparative tests were to be made of columns for which results of about 12 theoretical plates were expected. For minimum sensitivity to analytical errors one would like to have a= 1.29 and anything from 1.16 to 1a48 would be within 20 per cent of this minimum and thus highly suitable. To use a mixture having o!= l-03, however, would bring a severe penalty; error sensitivity would be five times the ideal minimum. In practice, of course, there may be compelling reasons why such a mixture should nevertheless be
number of ideal theoretical plates to which the column under test is equivalent
composition of distillate, mole fraction of more volatile component
xB)
These are assumed approximateIy equal for the purposes of this paper u relative volatility of the binary test mixture used
C!; >
limits of error (confidence limits) Ofxg, xD and Of NIqxCtiVdy
J/ error
factor, identity (7)
defined
by
the
gX standard error of xB, xD values used in Eq. (1) bN standard error of resulting estimate of N
t The estimation of these limits demands some caution, particularly if they must be based on a small number of analyses. A full discussion of this is outside the present scope. 959
D. G.
PEACOCK
25
lo 8 6 5 4
3
2
I.5 I.4
I.3
I.2
I.03
I
2
3
4
5
678
IO
Number of
2x3
30
40
60
80
1025 loo
tkoreticol plates
FIQ. 3. Chart for selection of test mixtures, by the criterion of sensitivity of the result to analytical errors.
REFERENCES
FENSKEM. R., LAWROSKIS. and TONGBER~C. O., Ind. Engng Chem. 1938 30 297. [2] FAY J. W. J., Rep. Prog. Chem. 1944 40 219. [3] ~OULSONE. A. and HERINGXBNE. F. G., Trans. Faraday Sot. 1948 44 629. E. F. G., Proceedings of the International Symposium on Distillation, Brighton, p. 110, The Institution [4] ~IUNQTON Chemical Engineers 1960. [l]
960
of
The selection of test mixtures for distillation columns R&m&--La selection des melanges pour l’essai des colonnes de fractionnement se base sur un certain nombre de facteurs, qui sont souvent en opposition mutuelle. Le choix peut etre assez diflicile. Les essais nkcessitent l’analyse des 6chantillons et les erreurs d’analyse peuvent affecter skieusement les rksultats obtenus. L’importance de ce facteur depend non seulement de la precision de-s methodes analytiques disponibles, mais aussi de la gamme des compositions trait&es. Cette demiere depend elle-meme du choix du melange. On discute cet aspect de la selection. Pour un pouvoir de separation don& il existe une volatilite relative optimum du melange choisi pour l’essai, a laquelle les erreurs d’analyse ont une signification minimale. Un tableau est p&se& indiquant cet optimum ainsi que l’effet des deviations, afin de fournir une base quantitative pour cet aspect de la selection. Auswahl von Mischungen zur Prtifung von Fraktionierkolonnen hHngt von Zllsamlnenf assang-Die einer Anzahl oftmals einander widersprechender Faktoren ab. Die Wahl kann sich als recht schwierig erweisen. Zu den Tests gehiirt die Analyse von einzelnen Proben, und analytische Fehler kiinnen die Endergebnisse sehr weitgehend beeinflussen. Die Redeutung dieses Faktors hgngt nicht nur von der Genauigkeit der zur Verftigung stehenden analytischen Verfahren ab, sondem such vom Rereich der untersuchten Gem&he- der seinerseits wieder von der gewahlten Mischung abh%ngt. Dieser Aspekt der zu treffenden Auswahl wird besprochen. Rei einem gegebenen Trennaufwaud gibt es einen Restwert der relativen Fhichtigkeit f* die gewahlte Testmischung, bei dem sich analytische Fehler am wenigsten auswirken. Ein Diagramm zeigt diesen optimafen Wert und den Effekt eines Abgehens davon, und liefert eine quantitative Grundlage fur diesen Aspekt der Auswahl.
961
Tlie selection of test mixtures for distillation columns D. G. PEACOCK The School of Pharmacy, University of London, Brunswick Square, London, W.C.l (Received 8 February 1967) Abstract--The selection of mixtures for testing fractionating columns involves a number of factors, often mutually conflicting. The choice may be quite difficult. The tests involve the analysis of samples and analytical errors may seriously affect the results obtained. The importance of this factor depends not only on the accuracy of the analytical methods available but also on the range of compositions handled. The latter is itself dependent on the choice of mixture. This aspect of the selection is discussed. For a given separating power there is an optimum relative volatility of the chosen test mixture, at which analytical errors are least important. A chart is presented showing this optimum and the effect of departures from it, to provide a quantitative basis for this aspect of the choice.
The value for N obtained from Eq. (1) is susceptible to errors introduced either in the equilibrium data (i.e. in a) or in the sample analyses (xD and xB). COULSON and HERINGTON [3] have discussed the effect of errors in a; they and a number of subsequent workers have devised thermodynamic consistency tests for checking experimental vapourliquid equilibrium data. Errors in the sample analyses form the topic of this present work.
1. INTR~O~~TI~N Trrn TECHNIQUE of testing a distillation column to assess its separating power is well established [l, 2, 41. The column is operated with a suitable binary mixture, under total reflux, for long enough to allow a steady state to be reached. Samples are then taken of the liquid refluxing from the condenser to the column and (preferably) of the liquid returning from the column to the reboilett. The samples are analysed and the result calculated from Fenske’s equation :
2. ANALYTICAL ERRORS
(1) Although this process is essentially very simple, considerable care and attention to detail are necessary if meaningful results are to be obtained. FJXNSKE[l] has listed specific precautions to be taken to secure reliability.
t The alternative course of sampling from the reboiler itself is often taken. This is, however, somewhat less satisfactory, since the sample may be contaminated with tars or other non-volatile impurities and because an extra ideal stage is included in the system tested. This extra stage means some loss of accuracy in the result obtained for the column, and is particularly undesirable if columns equivalent to only 2 or 3 theoretical stages are under test. For longer columns, this latter point is of less importance and the criteria developed here for selecting test mixtures represent a useful working guide.
Errors in analysis can present serious difficulties in comparative tests of distillation equipment. This is especially true in industrial, large-scale testing, since questions of availability and disposability may place restrictions on the choice of mixtures to be used-restrictions largely absent where only laboratory-scale tests are involved. Furthermore, where comparative tests are being made, with two or more (possibly competing) designs, errors of analysis will generally operate independently on the separate tests made. Errors in a wilI more usually operate similarly on all tests, affecting the comparison less in consequence. The impact of analytical errors is particularly severe if extremes of concentration occur (i.e. if x, or (1 -xJ is very close to zero), or if a is so low that xi, and xD are not very different. In practice one tries to pick a mixture which avoids these difficulties and to work with the compositions
957
D. G. PEACCPZK
1 1 +_ -- @J2 + (In cr)’ xD2(1- xD)’ xB2(l - xa)’
and xD symmetrically placed about 50 per centas FENSKE [I] put it: “Whenever possible, the middle portion of the vapour-liquid diagram should be used”.
xB
For the present purpose, of examining the effect of analytical errors and the dependence of this effect on the value of a, the following assumptions are made:
xg=(l-xJ=x
$=Y.
constant and free from random error;
the relative amounts of the two components are so adjusted that xD is equal to (1 - xB), this common value being denoted by x. In fact this will only hold approximately true, but this approximation will not invalidate the results;
(4 the values of xD and xB are subject, inde-
(5)
from which it may be shown that
(a>the relative volatility TVis assumed to be
(b) to simplify the arithmetic, it is assumed that
(4)
ux
where:
‘lJ2. x(1-x&x/(1-x))
(7)
The variation of $ with x is shown in Fig. 1. A minimum of $ = 3.16 is observed at x = 0.824, and (as one would expect) x=0.5 and x= 1 are asymptotes.
pendently, to random errors which are assumed to be small, normally distributed, and of standard deviation + a,. Further, this variation cX is assumed to be approximately constant over the whole practicable range of compositions. This will be more or less true according to the method of analysis used ; it is certainly a good working approximation where analysis is based on the measurement of a physical property such as density or refractive index. Returning to Eq. (1), this may be expressed in the alternative form :
Top composition.
Considering the variation of xB and xD, the standard deviation UNof N may be calculated from
(3)
xg
Fro. 1. The relationship between composition and sensitivity
to errors.
Thus a given level of uncertainty in analysis will have the least effect on the result, if the test 958
The selection of test mixtures for distillation columns
mixture and the column under test are so related as to give compositions X, and xB at or about this minimum point. Since there is a correspondence [Eq. (2) and (5)l between x and the product N . log a, this same requirement may be stated in terms of the latter, as in Fig. 2; ideally one would choose to have N. log,,cr = 1.34 and the effect of having values above or below this can be seen.
used. Figure 3 does, however, provide a quantitative basis for such a decision. It will be noted that if appropriately-calculated? confidence limits are known for x, and x,, then Eq. (6) enables confidence limits for N to be estimated. For example, if the 0.95level limits _+Sx are 10.01, the corresponding limits +dN/N are 0.01 II/. In the ideal case, then, such as the numerical example above, with c1= 1.29, the result N would be known to f3.16 per cent. Were CI= 1.03, however-the “five times minimum error” case-the error in N could well exceed 2 15 per cent. AcknowMgrneat-The author is indebted to Prof. Saunders for advice in the presentation of this paper.
L.
NOTATION
N
o-2
0.1
03
0.4
06
0.8
N lq, FIG.
xB composition of liquid returning from column to reboiler, in mole fraction of more volatile component
1 a
XD
2. This same relationship plotted directly in terms of N.logloa.
.Y= xD = (I4. THE PRACTICAL CHOICE For the purpose of comparing the suitability of alternative test mixtures, the foregoing relationships are more conveniently expressed in Fig. 3. Suppose, for example, comparative tests were to be made of columns for which results of about 12 theoretical plates were expected. For minimum sensitivity to analytical errors one would like to have a= 1.29 and anything from 1.16 to 1a48 would be within 20 per cent of this minimum and thus highly suitable. To use a mixture having o!= l-03, however, would bring a severe penalty; error sensitivity would be five times the ideal minimum. In practice, of course, there may be compelling reasons why such a mixture should nevertheless be
number of ideal theoretical plates to which the column under test is equivalent
composition of distillate, mole fraction of more volatile component
xB)
These are assumed approximateIy equal for the purposes of this paper u relative volatility of the binary test mixture used
C!; >
limits of error (confidence limits) Ofxg, xD and Of NIqxCtiVdy
J/ error
factor, identity (7)
defined
by
the
gX standard error of xB, xD values used in Eq. (1) bN standard error of resulting estimate of N
t The estimation of these limits demands some caution, particularly if they must be based on a small number of analyses. A full discussion of this is outside the present scope. 959
D. G.
PEACOCK
25
lo 8 6 5 4
3
2
I.5 I.4
I.3
I.2
I.03
I
2
3
4
5
678
IO
Number of
2x3
30
40
60
80
1025 loo
tkoreticol plates
FIQ. 3. Chart for selection of test mixtures, by the criterion of sensitivity of the result to analytical errors.
REFERENCES
FENSKEM. R., LAWROSKIS. and TONGBER~C. O., Ind. Engng Chem. 1938 30 297. [2] FAY J. W. J., Rep. Prog. Chem. 1944 40 219. [3] ~OULSONE. A. and HERINGXBNE. F. G., Trans. Faraday Sot. 1948 44 629. E. F. G., Proceedings of the International Symposium on Distillation, Brighton, p. 110, The Institution [4] ~IUNQTON Chemical Engineers 1960. [l]
960
of
The selection of test mixtures for distillation columns R&m&--La selection des melanges pour l’essai des colonnes de fractionnement se base sur un certain nombre de facteurs, qui sont souvent en opposition mutuelle. Le choix peut etre assez diflicile. Les essais nkcessitent l’analyse des 6chantillons et les erreurs d’analyse peuvent affecter skieusement les rksultats obtenus. L’importance de ce facteur depend non seulement de la precision de-s methodes analytiques disponibles, mais aussi de la gamme des compositions trait&es. Cette demiere depend elle-meme du choix du melange. On discute cet aspect de la selection. Pour un pouvoir de separation don& il existe une volatilite relative optimum du melange choisi pour l’essai, a laquelle les erreurs d’analyse ont une signification minimale. Un tableau est p&se& indiquant cet optimum ainsi que l’effet des deviations, afin de fournir une base quantitative pour cet aspect de la selection. Auswahl von Mischungen zur Prtifung von Fraktionierkolonnen hHngt von Zllsamlnenf assang-Die einer Anzahl oftmals einander widersprechender Faktoren ab. Die Wahl kann sich als recht schwierig erweisen. Zu den Tests gehiirt die Analyse von einzelnen Proben, und analytische Fehler kiinnen die Endergebnisse sehr weitgehend beeinflussen. Die Redeutung dieses Faktors hgngt nicht nur von der Genauigkeit der zur Verftigung stehenden analytischen Verfahren ab, sondem such vom Rereich der untersuchten Gem&he- der seinerseits wieder von der gewahlten Mischung abh%ngt. Dieser Aspekt der zu treffenden Auswahl wird besprochen. Rei einem gegebenen Trennaufwaud gibt es einen Restwert der relativen Fhichtigkeit f* die gewahlte Testmischung, bei dem sich analytische Fehler am wenigsten auswirken. Ein Diagramm zeigt diesen optimafen Wert und den Effekt eines Abgehens davon, und liefert eine quantitative Grundlage fur diesen Aspekt der Auswahl.
961