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Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 3, 1979
pressure over pure water at the same temperature fi
= fraction of total cations in solution represented by cation
v i = stoichiometric coefficient: see eq 1; also vij = v i
AG = Gibbs free energy change
+ vi
i
f, = fraction of total anions in solution represented by anion
I
j = ionic strength = ~ O . 5 m , Z I 2
K,, = solubility product for one equivalent of salt with examples shown in Table I m, = molality, g-mol (of the ion indicated by subscript)/1000 g of water n = number of moles of waters of hydration (crystallization)/g-equiv of solid double salt w = g of water per total equivalents of cations or anions in a saturated solution 2, = ionic charge on ion represented by subscript a = number of moles of simple salt A,B,/g-equiv of double salt (eq 2 must be satisfied) /3 = number of moles of simple salt C,,D,,/g-equiv of double salt (eq 2 must be satisfied) y = mean ionic activity coefficient of the ion pair, indicated by subscript, in the multicomponent solution under consideration
Literature Cited D'Ans, J., "Die Losungsgleichgewichthe der Systeme der Salzeozeanischer Salzablagerungen", Verlagsgesellschafl fur Ackerbau, Berlin, 1933. Denbeii, K. G., "Principles of Chemical Equilibrium", p 308, Cambridge Unhrersity Press, 1955. Kusik, C. L., Meissner, H. P., AIChE Symp. Ser. 173, 7 4 , 14-20 (1978). Meissner, H. P., "Processes and Systems in Industrial Chemistry", pp 46-61, Prentice-Hail. Englewood Cliffs, N.J., 1971. Meissner, H. P., Kusik, C. L., AIChf J., 18, 294 (1972). Meissner, H. P., Kusik, C. L., Ind. Eng. Chem. Process Des. Dev., 12, 205 (1973). Meissner, H. P., Tester, J. W., Ind. Eng. Chem. Process Des. Dev., 1 1 , 128 (1972). Meissner, H. P., et ai., AIChE J., 18, 661 (1972). Meissner, H. P., Kusik, C. L., Field, E. L., "Estimation of Phase Diagrams and Solubilities for Aqueous Multi-ion Systems", presented at 70th Annual AIChE Meeting, New York, N.Y., 1977. Seidell. S., Linke, W. F., "Solubilities", Voi. 11, 4th ed, pp 296-316, American Chemical Society, Washington, D.C., 1965.
Received for reuiew December 23, 1977 Accepted January 2, 1979
Use of the Vaporization Efficiency in Closed Form Solutions for Separation Columns Phillip C. Wankat" and Joseph Hubert School of Chemical Engineering, Purdue University, West Lafayette, Indiana 4 7907
Modified forms of the Winn, Fenske, and Kremser equations utilizing a constant vaporization efficiency are derived. The modified forms of the Smoker and Underwood equations are presented but not derived. These equations allow the rapid calculation of required number of stages without requiring the assumption of equilibrium stages. They also allow estimation of the average vaporization efficiencies from data on column performance, and then simulation of column performance under different operating conditions.
Introduction
yAj
Accurate solutions for staged separation systems can be obtained on computers by using various stage-by-stage or successive approximation methods. Although the modern computer techniques are quite rapid, they still require too much computer time for problems such as preliminary economic estimates or reactor recycle calculations where a large number of iterations through the separator are required. For these problems, for preliminary hand calculations, and for simulating existing installations closed form solutions such as the Kremser, Smoker, Fenske or Underwood equations are very useful. These equations are usually derived with the following assumptions: constant flow rates, constant relative volatility or linear equilibrium isotherms, and equilibrium stages. The last assumption can be relaxed if a stage efficiency can be incorporated. However, the usual Murphree efficiency is usually difficult to utilize (the Kremser equation is an exception (King, 1971)). An alternate efficiency which has frequently been used in multicomponent separation calculations is the vaporization efficiency. The vaporization efficiency, EAl, can be defined as (Holland, 1975) where the vapor is assumed to form an ideal solution. We will rewrite eq 1 as 0019-7882/79/1118-0394$01.00/0
=
EA]KAjXAj
(2)
where KAj = KAj*yAj includes the ideal solution K value and the activity coefficient evaluated using the mole fractions of the liquid leaving plate j . The vaporization efficiencies are bounded and can be estimated from pilot or plant tests, or from models of separation plates (Holland, 1975). The vaporization efficiencies also have the advantage that they are easy to incorporate in calculations. For binary systems the vaporization efficiencies are not equal unless the stage is an equilibrium stage. In this paper the vaporization efficiency will be used to derive modified forms of the Winn and Fenske equations for total reflux in distillation columns and the Kremser equation for linear equilibria. The modified forms of the Smoker and Underwood equations will be presented without derivation. The use of these equations to determine the vaporization efficiencies from operating data is also developed. A sample calculation using these equations will be presented. T o t a l R e f l u x Equations
Equations for total reflux in distillation have been presented by Winn (1958) and Fenske (1932) (or see King, 1971; or Smith, 1963). Both of these equations are easily modified to include the vaporization efficiency. The derivation will be done for the Winn equation since the Fenske equation is a special case of the Winn equation. 0 1979 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 3, 1979 395
For the Winn equation the equilibrium expression is assumed t o obey an equation of the form CYAB=
PKBb-'
(3)
and
If b = 1this simplifies to a constant relative volatility and the Fenske equation will result. The equilibrium equation can be written as v *
NMin
-Aj- - KA = PKBb
(4)
xAj
The vapor leaving a stage is not in equilibrium but satisfies eq 2 where E A and E B are assumed to be constant. For total reflux the operating equation simplifies to Y A j = xAj+l (5) Starting a t the bottom of the column in the reboiler eq 2 is valid for both components A and B. Taking the ratio of Y A / Y B we ~ obtain
Substituting in eq 5 which becomes
YAR =
XAl (7)
For stage 2 the ratio of vapor compositions can be obtained from eq 2.
-YA -1 - P EAIXA1 Ynib
EBibX~ib
Substituting eq 7 into eq 8 we obtain
Continuing up the column to the distillate product by a similar procedure we obtain
XBdb
lYBtb]
(&)( %)-(2)(%)($) (10)
where t refers to the top stage. If E A and Eg are constant or a geometric average is used, eq 10 simplifies to
=
In ( ~ A B E A I E B )
(14)
Equations 13 and 14 are the modified form of the Fenske equation for constant relative volatility and constant vaporization efficiencies. They reduce to the original forms of the Fenske equation if EA = EB = 1.0.An equation similar to eq 13 is given by Holland (1975, p 304). Equation 10 is a general form of the Winn equation (or Fenske if b = 1) and is valid for variable equilibrium properties and vaporization efficiencies. Equations 11 and 14 use constant or average values of the relative volatilities and the vaporization efficiencies. If the efficiencies or relative volatilities are not constant it is clear from eq 10 that a geometric average of P(EA/Egb) should be used in eq 11 to 14. In the preceding derivation the number of components was not specified. Thus these equations are valid for an arbitrary number of components. These equations can also be written in terms of recovery fractions which may be more convenient in solving problems. Modified Kremser Equation The Kremser equation (Kremser, 1930) is very useful for separations where the equilibrium equation can be approximated as linear. The modified derivation to incorporate the vaporization efficiency follows the derivation of King (1971). The flow rates are assumed to be constant, the equilibrium relationship is of linear form Y * A , = mXAj + b (15) and the vaporization efficiencies are constant and are represented as y ~=jEAY*A, = EA(mXA, + b) (16) The column is assumed to be isothermal and heats of absorption, mixing, and solution are all assumed to be negligible. The derivation is done for a column where the stream with flowrate V and mole fractions Y enters stage 1and leaves stage N. The stream with flowate L and mole fractions X flows countercurrently. The derivation can be considered to be for absorption, but the results hold for any separation where the assumptions are valid. Writing a mass balance around stage N we obtain
Equations 16 and 17 can be combined and rearranged to where NMinis the minimum number of equilibrium stages including partial reboiler and partial condenser. Solving eq 11 for N M i n Writing a mass balance for stage N - 1, combining with eq 16 and rearranging, we obtain
If EA = E B = 1, eq 11 and 12 reduce to the original Winn equations. If b = 1, eq 3 reduces to a constant relative volatility. If we call (YAB = p, eq 11 and 1 2 reduce to
Multiplying eq 18 and 19 gives
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Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 3, 1979
The general expression can be derived by repeating this procedure.
T o obtain an expression for the terminal concentrations we add eq 2 1 written for each stage with j ’ s varying from N - 1 to 0.
E( 4)” mEAV (22)
,,=I
This can be rearranged by dividing eq 22 by (1+ eq 22). ‘Ai”
- tu&‘
YA..- ~ E A X -A < EA~
-
(23)
The sum of the power series is given as k
Car’ =
i=O
a ( 1 - rk+l) 1-r
for different stages are multiplied together and then added for varying values of n. The appropriate average vaporization efficiency is not obvious, but a geometric average appears to be appropriate for terms in the group L / (mEAV). Use of a geometric mean also agrees with Holland’s (1975, Chapter 10) modeling of an absorber. Small variations in the equilibrium parameter m can also be averaged using a geometric average. The term (-mEAXA,n - bEA)was derived using a vaporization efficiency for stage N (top of the column). Since the number of components was not specified, these equations are valid for multicomponent systems. Graphical plots of these equations have been developed for the usual Kremser equation (see King, 1971) and can be used inserting the vaporization efficiency in the appropriate places. Other Closed Form Equations The two derivations presented serve to show the modifications necessary to incorporate the vaporization efficiency into the standard solutions for separation processes. For the Underwood and Smoker equations only the results will be given (the derivations are available on request from the authors). Underwood (1944, 1945, 1946a,b, 1948) developed equations for binary and multicomponent distillation a t finite reflux ratios (see King, 1971). For the modified equation the system is assumed to have constant relative volatility, constant vaporization efficiencies, and constant molal overflow. The defining equation for Underwood’s variable 4 in the rectifying section is
for Irl < 1. Hence for L / m E A V < 1, eq 23 becomes
yA;.
-
L mEAV
f
L
\N+l
[ mEAV}
The two roots to this equation are bounded
EAQA > 4 1 > EBQB> 1 - 1
-I
If L/mEAV > 1 then the numerator and denominator of eq 23 can be divided by (L/mEAWNand eq 24 can be derived by following the same procedure. If (L/mEAV) = 1 the right-hand side of eq 24 reduces to N/(N+ 1). Equation 24 is a modified form of the Kremser equation which is useful for determining the outlet concentration if N is fixed. An alternate form can be obtained by subtracting 1.0 from both sides of eq 24 and rearranging.
$2
>0
(28)
The roots are used to find the number of stages in the rectifying section, NR, from the equation
( 42/@
1lNR =
{ [ a A E A X A f / ( o A E A - 4 111 + - 4 I)I)/I[aAEAXAf/(aAEA - 4 211 + [ a B E B X B f / ( a B E B - 4211) (29)
[aBEBXBf/(aBEB
where XArand x B i are the liquid compositions on the feed stage, not the feed compositions. In the stripping section the appropriate equations are -V’ =
aAEA’BXA,b
(a&*’
-
o(BEB’BXB,b + 4’) 4’) -
(30)
and ( 4 2 ’ / 4 1’INS =
42’11 + 4 2/11 I/([EA”YA’XA,f/ (aAEA’ 1 + [ ~ B E B ’ X (a&’ B , ~ / - 4 1’) I I (31)
( [ E A ’ ~ A X A , f / ( ~ A E A’
[aBEB’XB,f/( a B E B ’
Equation 25 can be solved for N.
When EA= 1,eq 24 to 26 reduce to the appropriate forms of the Kremser equation. For the forms of the Kremser equation (eq 24 to 26) a constant vaporization efficiency was assumed. In the derivation of these equations the vaporization efficiencies
4 1’)
-
Comparison of eq 27 to 31 with the corresponding equations for the usual Underwood method shows that a A and cyB are everywhere replaced by a A E A and a B E B or aAEA’ and CYBEB‘. These equations are easy to use for the design problem where the distillate and bottoms compositions, the reflux ratio, feed composition, feed rate, feed temperature, and the optimum feed locations are specified. The flow rates in the column are determined by overall mass balances and mass and energy balances at the feed plate. The optimum feed plate location specification is used to find the composition on the feed plate by finding the intersection of the two operating lines. Then
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 3, 1979 (DXA,d/
xAf
=
v) + (BXA,b/V?
(L‘/V? - (L/v)
(32)
Equations 27,29,30, and 31 can then be solved for the 4’s and NR and Ns. The Underwood equations can easily be extended to multicomponent systems by adding terms for the added components (Underwood, 1948). This extension is straightforward, but their use for multicomponent systems is much more complicated than the binary solutions. The approximate solution method shown by King (1971) simplifies the calculations and is easily adapted to use with vaporization efficiencies. Smoker (1938) developed an alternative equation for binary distillation with constant molar overflow and constant relative volatilities (see Smith, 1963). The Smoker equation is easily modified to include constant vaporization efficiencies. In the rectifying section the modified Smoker equation for the optimum feed plate location is
Smoker equation shows that a(EA/&) or @(EA’/&’) replaces CY everywhere in the equations corresponding to eq 33-36. The modified Smoker equation reduces to the modified Fenske equation, eq 14, a t total reflux (this is most clearly seen from the generalized solution given by Smith (1963)). The vaporization efficiencies and relative volatilities were assumed constant in the derivations of the Underwood and Smoker equations. The derivations do not indicate the appropriate averages to use if the volatility or efficiency are not constant. Since the total reflux Winn and Fenske equations used a geometric mean, it is probably reasonable to use a geometric mean of amEA/ E B for the Underwood and Smoker equations also. The use of a geometric mean is consistent with Holland’s (1975, Chapter 10) use of a geometric mean for modeling a distillation column. The geometric averages should be estimated separately for the stripping and enriching sections. Efficiency Determination and Column Simulation An additional use of these equations is as a rapid method of determining an average vaporization efficiency from column operating data. For a distillation column at total reflux where the Fenske form is valid eq 13 can be rearranged to XAd/XBd
_ E E AB
+
where c = 1 (&A/&) - l ) h , xfis the composition of the more volatile component on the feed plate given by eq 32, and h is the X value of the point of intersection of the pseudo-equilibrium line and the operating line. h is the root between 0 and 1 of the equation
-
In (PEA/EB~) In (PI
(38)
When b = 1 and P = CYAB, eq 38 simplifies to the corresponding form for the Fenske equation.
Eo =
Comparison of these results with the usual form of the
(37)
The ratio E A / E B can be rapidly estimated from distillate and bottoms compositions measured at total reflux. From the development of eq 13 it is clear that this ratio is a geometric mean ratio. The relationship between the geometric mean ratio of vaporization efficiencies and the overall efficiency for a total reflux distillation is also easily determined. From the Winn eq 12, Nm can be determined for equilibrium stages ( E A = E B = 1.0)and for real stages. Then by dividing these two equations Nequi1
where C ’ = 1 + (cY(EA’/EB’) - l)k’, and h’is the root between 0 and l of the equation
l/NM1n
( x A R / x BAB R)
Eo--= NAd For the stripping section the modified Smoker equation is
397
In ( ~ A B E A I E B ) 1n AB)
(39)
In these equations EA/EBb and EA/EBare the geometric mean values. For absorbers an average value of EA can be determined from operating data by solving eq 24, 25, or 26 for E A . A trial-and-error solution method is probably required since an explicit solution for EA does not appear to be feasible. A relationship between the overall efficiency and the vaporization efficiency can be obtained by dividing eq 26 written for equilibrium stages by eq 26 written for real stages. Unfortunately, this result does not appear to simplify. Operating data from distillation columns at finite reflux ratios can also be used to estimate the ratio of E A / E B from either the modified Smoker or modified Underwood equations. Again a trial-and-error solution of the appropriate equation for E A / & will be required. These closed form solutions are easy to use to calculate the geometric mean vaporization efficiencies from operating data. Since the mean vaporization efficiencies should
398
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 3, 1979
be approximately constant when modest changes in the operating variables are made the closed form equations can then be used for simulating column performance. This semiempirical approach should be both convenient and accurate. Example Calculation The use of these equations is straightforward and follows the use of the corresponding equations with equilibrium stages (see King (1971) or Smith (1963)). As a simple example consider a distillation column separating benzene and toluene at total reflux. A distillate composition of 99 mol % benzene and a bottoms composition of 99.5 mol % toluene are desired. The average relative volatility is approximately 2.5. Holland (1975, p 337) calculates the vaporization efficiency for benzene for this system. The vaporization efficiency for toluene was calculated from the data given by Holland and from Holland's convention for stage temperature (eq 11-1,p 328). Since the y's must sum to unity the relationship for temperature is
will stimulate the collection and use of vaporization efficiencies. Nomenclature b = equilibrium parameter defined in eq 3 for relative volatility or equilibrium parameter in linear equilibria, eq 15 B = bottoms flow rate, mol/time C = constant in Smoker equation = 1 + (a(EA/EB)- l ) k C' = constant in Smoker equation = 1 + (a(EA'/EB' - 1)k' D = distillate flow rate, mol/time E = vaporization efficiency defined by eq 1 and 2 Eo = overall efficiency defined in eq 38 k = x root of eq 34 k ' = x root of eq 36 K = Y / X , K value for equilibrium L = liquid molar flow rate in column, mol/time m = linear equilibrium parameter in eq 15 N = number of stages V = vapor molar flow rate in column, mol/time X = liquid mole fraction Y = vapor mole fraction 0 Greek Letters 1 = XEiyiKiXi (40) a A B = relative volatility = KA/KB i=l (3 = equilibrium parameter defined in eq 3 on each stage. As Holland notes this will give a different y = activity coefficient C#I = Underwood's variable for rectifying section defined by temperature than the Murphree definition which uses a eq 27 bubble point calculation. For gas and liquid film coef4' = Underwood's variable for stripping section defined by ficients both equal to 1.0, Holland's (1975, p 337) geometric eq 30 mean value of EB = 0.938. The corresponding geometric mean value of ET was calculated to be 1.250. At total Subscripts and superscripts reflux eq 14 is used to find the number of stages. A,B = components b = bottoms d = distillate In 0 ~ ~J ~ ~ ~ : ~ I =~ideal5 L in = inlet stream = 15.72 stages NMin = In [2.5(0.938)/(1.2500)] j,N,p,t,l,2 = stage location Min = minimum number of stages (total reflux) This compares to 10.79 stages if the stages are equilibrium out = outlet stream stages. Similar examples are easily developed for the other R = reboiler or rectifying section equations. S = stripping section ' = values in stripping section For this example the overall efficiency can be calculated * = equilibrium values from eq 39 as Literature Cited In [(2.5)(.938)/(1.250)] Eo = = 0.687 Fenske, M. R., Ind. Eng. Chern., 24, 482 (1932). In (2.5) Holbnd, C. D., "Fundamentals and Modeling of Separadon Rocesses: Absorption Distillation, Evaporation and Extraction", Prentice-Hall, Englewood Cliffs, N.J., This agrees with Eo calculated as Nequil/Nactual. Chapters 9, 10, and 11, 1975. King, C. J., "Separation Processes", Chapters 8 and 9, McGraw-Hill, New York, Conclusions N.Y., 1971. Kremser, A,, Natl. Petrol. News, 22(21), 42 (May 21, 1930). Modified forms of the Fenske, Kremser, Smoker, UnSmith, 6. D., "Design of Equilibrium Stage Processes", pp 155-160, 296-309, derwood, and Winn equations incorporating a vaporization McGraw-Hill, New York, N.Y., 1963. Smoker, E. H., Trans. AIChE, 34, 165 (1938). efficiency are easy to derive. These equations are then easy Underwood, A. J. V., J . Inst. Pet., 30, 225 (1944). to use to determine mean efficiencies from operating Underwood, A. J. V., J . Inst. Pet., 31, 11 1 (1945). systems. Once the vaporization efficiency is known these Underwood, A. J. V., J . Inst. Pet., 32, 598 (1946a). Underwood, A. J. V., J . Inst. Pet., 32, 614 (1946b). equations allow rapid prediction of column performance. Underwood, A. J. V., Chern. Eng. Prog., 44, 603 (1948). Rapid design of new systems can also be accomplished Winn, F. W., Pet. Refiner, 37(5), 216 (1958). utilizing these equations if the vaporization efficiency is known. Hopefully, the availability of rapid methods of Received for reuieu; December 27, 1977 Accepted March 31, 1979 calculating the vaporization efficiency from column data
1
1
