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Division by dni and restriction to constant T , n/p(= nV ) , and n; ( j
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# i ) leads to:
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For simplicity of notation, the partial derivatives in the following development are written without subscripts, and are understood to be at constant T, n/p(= n V ) , and n; Thus, with P = (nZ)RTl(nlp),
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Combination of Eqs (1448)and (1449)yields:
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142 VLEfrom Cubic Equations of State
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Equation (663a), written for the mixture and multiplied by n, is differentiated to give the first term on the right:
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a(nGR/~~) - a(nz) ani ani
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1 - ln(1 - p b ) - n [a ln(l - pb) ~ ani
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aI Iq in +-a aniz] - n q - -anii
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where use has been made of Eq (1447) The equation for ln $i now becomes:
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= - 1 - ln(1 - pb)Z - n a(nz) ani
a ln(1 - pb)
This reduces to:
=- - nq- - ln(1 - pb)Z - ijil 1 - pb ani ani
a(pb)
All that remains is evaluation of the two partial derivatives The first is:
The second follows from differentiation of Eq (662a) After considerable algebraic reduction this yields:
a 1 - -> a(~b ani ani (1
+ opb)(l + cpb) - nb (1 + apb)(l + cpb)
Substitution of these derivatives in the equation for In $i reduces it to: 16,
b'[b-y
b I - pb
+ cpb)(l + apb)
- ln(1 - pb)Z - qi I
Reference to Eq (661) shows that the term in square brackets is Z - 1 Therefore, bi lnq5i = -(Z - 1) - ln(1 - pb)Z - q i I b However, whence Thus,
CHAPTER 14 Topics in Phase Equilibria
Because experience has shown that Eq (1442) is an acceptable mixing rule for parameter b, it is here adopted as appropriate for present purposes Whence,
a(nb)
The equation for In $iis therefore written:
where I is evaluated by Eq (662b) Equation (1136) is a special case for pure species i Application of Eq (1450) requires prior evaluation of Z at the conditions of interest by an equation of state This may be accomplished for a vapor phase by solution of Eq (1438) and for a liquid phase by solution of Eq (1439) Parameter q is defined in relation to parameters a and b by Eq (1441) The relation of partial parameter q ito lii and bi is found by differentiation of this equation, written:
Whence,
a(nq)
[F],,nj
= q (I+:
(1451)
Any two of the three partial parameters form an independent pair, and any one of them can be found from the other two'
8 ~ e c a u sq , a , and b are not linearly related, ifi e
# a,/gi RT
142 VLE from Cubic Equations of State
CHAPTER 14 Topics in Phase Equilibria
Equation (1450) provides the means to evaluate di, and is the basis for the solution of VLE problems A useful procedure makes use of Eq (1429), rewritten as yi = KixiBecause yi = 1,
Kixi = 1
(1452)
where Ki, the K-value, is given by:
Thus for bubblepoint calculations, where the liquid-phase composition is known, the problem is to find the set of K-values that satisfies Eq (1453) A block diagram of a computer program for BUBL P calculations is shown by Fig 148
9 ~ H Sage, B L Hicks, and W
N Lacey, Industrial and Engineering Chemistry, vol 32, pp 1085-1092,1940
142 VLE from Cubic Equations of State
CHAPTER 14 Topics in Phase Equilibria
Although the linear mixing rule for b [Eq (1442)] has proved generally acceptable, the quadratic mixing rule for a [Eq (1443)] is often unsatisfactory An alternative is a mixing rule for q that incorporates activity-coefficient data The connection between activity coefficients and equation-of-state parameters is provided by activity-coefficient and fugacity-coefficient definitions; thus,
Whence,
In yi = 1 n 6 ~
(1454)
142 VLE from Cubic Equations of State
ines
where yi, $i, q5i are all liquid-phase properties evaluated at the same T and P Subtracting and Eq (1136) from Eq (14,50) gives:
where symbols without subscripts are mixture properties Solution for q i yields:
CHAPTER 14 Topics in Phase Equilibria
Because qi is a partial property, the summability equation applies: 4=
C xiqi
(1456)
Equations (1455) and (1456) together constitute a thermodynamically sound mixing rule for q Application of Eq (1455) requires prior evaluation of Z and Zi from the equation of state These quantities are also required for evaluation of 1 and Ii by Eq (662b) However, the equation of state contains q, evaluated from the qi values through Eq (1456) Equations (1455) and (1456), together with Eq (1438) or Eq (1439) and the necessary auxiliary equations, must therefore be solved simultaneously for {Zi), Z, {Ii), I , {qi), and q, either by iteration or by the equation-solving feature of a software package The results make possible the calculation of $i values by Eq (1450) A choice must be made of an equation of state Only the Soave/Redlich/Kwong and PengJRobinson equations are treated here, and they usually give comparable results A choice must also be made of a correlating equation for the liquid-phase composition dependence of In yi The Wilson, NRTL, and UNIQUAC equations (Sec 122) are of general applicability; for binary systems the Margules and van Laar equations may also be used The equation selected depends on evidence of its suitability to the particular system treated The required input information includes not only the known values of T and {xi}, but also estimates of P and {yi},the quantities to be evaluated These require some preliminary calculations:
1 For the chosen equation of state (with appropriate values of a, r , and a),for each species Q, find values of bi and preliminary values of qi from Eqs (1434) and (1435) 2 If the vapor pressure PiSat for species i at temperature T is known, determine a new value for qi by iterative solution of Eq (1437) at P = Pisat with pi from Eq (1432), Zi and Ii for both liquid and vapor phases from Eqs (1431), (1433), and (662b) 3 A reasonable estimate of P is given by the sum of known or estimated PiSat values, each weighted by its known liquid-phase mole fraction 4 For each species i at the given T and estimated P , find liquid-phase values for Zi and Ii from Eqs (1433) and (662b) 5 For each species i at the given T and estimated P , find vapor-phase values for Zi and Ii from Eqs (1431) and (662b) 6 For each pure species i evaluate $ and $, by Eq (1136) : 7 An initial estimate of the vapor-phase composition is based on the assumption that both the liquid and vapor phases are ideal solutions Each fugacity coefficient is then given by $i = $i, and Eq (1453) can be written (Ki yi/xi):
Since these values are not constrained to sum to unity, they should be normalized to yield the initial estimate of vapor-phase composition The essential step in the iterative process of Fig 148 is evaluation of {$f} and
{$y}
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