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Ind. Eng. Chem. Res. 2005, 44, 10016-10020
Phase Rule and the Degree of Freedom Analysis of Processes R. Ravi* Department of Chemical Engineering, Indian Institute of TechnologysMadras, Chennai-600036, India
D. P. Rao Department of Chemical Engineering, Indian Institute of TechnologysKanpur, Kanpur-208016, India
The importance of specifying the state of the inlet streams in determining the degrees of freedom (DOF) of a process is highlighted in the context of a single-stage separation unit. It is shown that, for the separation of a C-component mixture, the commonly accepted value of 2C + 6 as the DOF of an equilibrium single stage holds only if neither of the inlet streams lie on the phase envelope. The implications of this feature for the DOF of multistage processes is explored. The importance of a precise interpretation of the phase rule in arriving at the above results is emphasized. Introduction The degree of freedom (DOF) analysis is an important, indispensable component of chemical process design. The analysis, which essentially involves counting the number of process variables and the equations relating them, becomes particularly complicated for multistage separations processes. Hence, many approaches (Gilliland and Reed,1 Kwauk,2 Gandhi et al.,3 and Luyben4) have been developed by which the DOF for a complex system may be obtained from those of its constituent units. In this context, the DOF of a single equilibrium stage assumes special importance, because it is the primary element in staged separations processes. In this article, we examine a hitherto unnoticed implication of the phase rule for the DOF of a single stage and discuss its consequences for multistage columns. Specifically we show that, for the separation of a C-component mixture, the conventionally accepted value of DOF of 2C + 6 for a single equilibrium stage with two inlet streams is valid only if those streams are “strictly” single phase (i.e., their states do not lie on a phase envelope). If the state of one of the inlet streams lies on a phase envelope or is part of a two-phase system in equilibrium, then the DOF for such a stage is shown to be 2C + 5. If the states of both inlet streams of a stage lie on a phase envelope, as typically happens for a stage which is inside a column and which is flanked on each of its two sides by an equilibrium stage, then DOF turns out to be (2C + 4). These results are developed for single-stage processes employing a massseparating agent (MSA) as well as an energy-separating agent (ESA). We give a brief discussion of the relevance and implications of our results for the DOF analysis of a multistage column. The Degrees of Freedom of a Process The DOF of a process is usually defined as the number of independent variables (extensive as well as intensive) that must be specified so that the rest of the process variables are uniquely determined. We may * To whom correspondence is to be addressed. Tel.: 91-4422574167. Fax: 91-44-22570509. E-mail:
[email protected].
then regard the process itself as being completely characterized. Consider an isolated single equilibrium stage (Figure 1), where a feed stream (F) and a solvent stream (S), acting as an MSA, come into contact, which results in an extract (E) phase and raffinate (R) phase in equilibrium. The compositions of the E and R streams are represented by the (C - 1)-dimensional mole-fraction (or weight-fraction) vectors y and x, respectively. Normally, a value of 2C + 6 is assigned as the DOF for this system. What are the features of the process that must be specified before we assign a DOF to it? One, it is a single stage with two inlet streams, and two exit streams that are assumed to be in equilibrium. Furthermore, an implicit assumption is that the state of neither of the two inlet streams lies on a phase envelope. Thus a total of (C + 2) variables is assigned for each of the inlet streams with no restrictions on their intensive variables. What the above arguments imply is that, in addition to the features specified above, the state of the inlet streams must also be specified if an unambiguous value is to result for the DOF of a single stage. Before we consider the implications of this in detail, we discuss the phase rule because a precise interpretation of this rule will be crucial to our analysis. The Phase Rule We adopt, in spirit, the statement of the phase rule by Callen:5 for a system of C components with π phases coexisting in equilibrium, it is possible, arbitrarily, to pre-assign f ) C - π + 2 variables from the set
{T,P,xI,xII, ..., xπ}
(S1)
where xj is the (C - 1)-dimensional mole-fraction vector of the jth phase. The implication is that the remaining variables are then uniquely determined. The set defined by S1 takes into account the equality of temperature and pressure among the coexistent phases and obviates the need for consideration of mole fraction constraints. Note that Gibbs’ original statement of the phase rule (from Gibbs6) is in terms of the temperature, pressure, and chemical potentials of the C components. The derivations of both versions of the phase rule have been
10.1021/ie051025f CCC: $30.25 © 2005 American Chemical Society Published on Web 11/30/2005
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xC-1 ) f1(T,P,x1,..,xC-2) yi ) fi+1(T,P,x1,...,xC-2)
Figure 1. Single stage employing a mass-separating agent (MSA).
subjected to critical analysis.7,8 Furthermore, it is wellknown that the above version of the phase rule does not hold for a binary mixture exhibiting an azeotrope (see Tester and Model9). However, we ignore such complications and examine generic systems for which the rule as stated above holds. Some Implications of the Phase Rule for Two-Phase Systems In a system of two coexisting phases, the set described by S1 may be represented by
{T,P,x,y}
(S2)
where x and y denote the (C - 1)-dimensional molefraction vectors of the two phases. Since π ) 2, we obtain f ) C. What this means is that there are 2C - CdC constraints among the 2C variables. These are the equality of chemical potentials of the C species in the two phases and are conventionally expressed as
yi ) Kixi
(for i ) 1, ..., C)
(1)
However, because the mole fractions of all the C components in both the phases appear in these equations, one would have to include the mole fraction constraints C
xi ) 1 ∑ i)1
(2a)
(for i ) 1, ..., C - 1)
C
yi ) 1 ∑ i)1
(2b)
Although the form of eq 1 is useful, from a practical viewpoint, a more transparent set of equations results if we choose, for instance, the set
{T,P,x1,x2,..xC-2}
(S3)
as the C variables to be pre-assigned. The phase rule, as stated previously, then implies the existence of functions fi (where i ) 1, ..., C), such that the remaining C variables
{xC-1,y1,y2,...yC-1} are uniquely determined by
(3b)
The phase rule in the above form implies the existence of unique relations such as those given in eqs 3, regardless of the choice of the C-independent variables. The fact that this is not universally valid is borne out by the well-known example of a binary mixture exhibiting an azeotrope,9 where the set {T,P} is not a suitable choice, because multiple solutions exist for the vapor and liquid mole fractions. As stated previously, we do not consider such cases here. Degrees of Freedom of a Single Stage Consider stage 1 of a single-section counter-current column employing an MSA (Figure 2). On the face of it, it seems to be similar to the isolated single stage of Figure 1, with the E2 stream replacing the S stream. However, there is a crucial difference. The E2 stream is part of the two-phase system R2 and E2 and the appropriate constraints imposed by the phase rule must be taken into account. Here, the set of relevant variables are
{T2,P2,y2,x2} = { T2,P2,y2,1,...y2,C-1,x2,1,...,x2,C-1} (S4) only C of which may be independently chosen. (Note that two subscripts will be used on the compositions: the first to denote stage numbers and the second to denote the component.) Thus, although complete specification of the stream S in Figure 1 requires (C + 2) variables, including (C + 1) independent intensive variables and the flow rate S, only C of the intensive variables of E2snamely, {T2,P2,y2}smay be independently specified. Thus specifying the value of any subset of C variables in this set {T2,P2,y2}, which in turn is a subset of (S4), is sufficient to determine the value of the remaining variable in the set. Moreover, all other variables in the set S4 are also determined. For instance, we may choose T2,P2,y2,1,...y2,C-2 as the Cindependent variables. The phase rule then implies the following relations:
y2,C-1 ) g1(T2,P2,y2,1,...,y2,C-2) x2,i ) gi+1(T2,P2,y2,1,...,y2,C-2)
and
(3a)
(4a)
(for i ) 1, ..., C - 1) (4b)
If we consider the full set of eqs 4 to be associated with the second stage, then the DOF for the first stage equals that for an isolated single stage, namely 2C + 6, and on suitable specification of those variables, the first stage is completely characterized. That is, the E2 stream is completely determined or the variable {T2,P2,y2} numbering (C + 1) is determined without the use of any of eqs 4. Thus, the set of equations in eqs 4 for the second stage would constitute a set of C equations but in only (C - 1) variables, denoted by the vector x2, constituting an over-determined set of equations. Thus, eq 4a would have to be included in the set of equations for stage 1 resulting in a DOF value of 2C + 5. We now reinforce the above argument through an illustration. Illustration: Consider the liquid extraction, at constant temperature and pressure, of a ternary mixture A-B-C (where C is the solute and B is the solvent). We implement the first step of a stage by stage calcula-
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Figure 4. Stage inside a multistage column.
Figure 2. Terminal stage of a multistage column.
to actually locate E2. This extra piece of information is the equation
y2,B ) g1(T2,P2,y2,C)
(11)
We now need to specify
{T2,P2}
Figure 3. Sketch of liquid extraction of a ternary mixture.
tion, starting from the first stage, with the graphical procedure described by Treybal10 (Figure 3) adding further insight. Assume that the following variables are specified:
F,TF,PF,zF,C,zF,B,T1() TF),P1() PF),E1,y1,C (S5) First, the E1 stream may be located graphically on the extract part of the phase envelope. This translates analytically as
y1,B ) g1(T1,P1,y1,C)
(5)
The R1 stream may then be located on the raffinate portion of the phase envelope following the tie-line through E1. This essentially amounts to the use of the equations
x1,B ) g2(T1,P1,y1,C)
(6)
x1,C ) g3(T1,P1,y1,C)
(7)
Equations 5-7 are special cases of eqs 4 for a ternary system. The stream E2 is then located graphically by finding the difference point, ∆, through the following equations:
F - E1 ≡ ∆ ) R1 - E2
(8)
FzF,C - E1y1,C ≡ ∆y∆ ) R1x1,C - E2y2,C
(9)
FzF,B - E1y1,B ≡ ∆x∆ ) R1x1,B - E2y2,B
(10)
The coordinates of the difference point (x∆,y∆) may be determined from the left-hand side of the above set of equations. The right-hand side of the equations imply that three pointss∆, R1, and E2slie on a straight line. However, another piece of information is required, i.e., that E2 lies on the extract portion of the phase envelope,
(S6)
Assuming T2 ) T1 ) TF and P2 ) P1 ) PF, we may locate E2 at the intersection of ∆R1 and the extract portion of the phase envelope, which is represented by eq 11. The amounts of R1 and E2 may be found graphically. Analytically, the above problem involves the determination of the four unknowns (R1, E2, y2,B, and y2,C), using the four equations described as eqs 8-11. Equation 11 is entirely analogous to eq 4a and is absolutely essential to characterize stage 1 completely. Furthermore, among all the equations, it is this equation that would not figure in the set of equations for an isolated single stage. We may now easily obtain the DOF for this stage. The number of variables specified in sets S5 and S6 is 11 ) 2(3) + 5, and not 2(3) + 6. The one variable not included abovesthe heat rate associated with stage 1smay be determined from the energy balance, which has not been written above. (We have adopted the convention of neglecting work interactions in the DOF analysis of separations processes.) It is now straightforward to extrapolate to a stage that is entirely inside a column (stage ‘j’ in Figure 4). Here, both of the entering streams are part of a two-phase system. Thus, there would be two additional equations, compared to that of the isolated single stage of Figure 1. Thus, the DOF of such a stage would be 2C + 4. The aforementioned results also have a bearing on separation processes such as distillation accomplished by an energy-separating agent (ESA). There the isolated single stage usually has a single feed and the reported DOF (see Seader and Henley11) is C + 4. This, however, is valid only if the feed is subcooled or superheated. The value will be one less if the feed is a saturated liquid or saturated vapor. This is most easily seen in the case of a binary mixture. Suppose the feed is a saturated liquid at temperature TF, pressure PF, and composition zF (this latter item is the mole fraction of the more-volatile component). In addition to the set of equations for the case where the feed is, for example, a subcooled liquid, we then would have to include the equation, namely,
TF ) TL(xF,PF)
(12)
where TL(x,PF) is the bubble-point curve at pressure PF. For a stage inside a column employing an ESA, which is flanked on both sides by other equilibrium stages, the DOF would still be 2C + 4. In the same vein, in a singlesection countercurrent column, the terminal stages of
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the column would have a DOF of 2C + 4 or 2C + 5, depending on whether the external input stream is in a saturated state or not. Specification of Variables (Single Stage) The previously derived results have a bearing on the specification of variables. For instance, consider the scheme shown in Figure 1. Because the DOF for this process is 2C + 6, complete specification of the feed and solvent streams, each involving (C + 2) independent variables, would mean that two additional variables need to be specified for complete characterization of the single stage. The same cannot be said for either the extract or the raffinate stream (see Table 1 of Gilliland and Reed1). The crucial difference is that the extract and raffinate streams are each to be regarded as a part of a two-phase system R and E, for which the arguments made for the R2 and E2 streams of Figure 2 apply. Thus, complete specification of, for example, the raffinate stream, contributes not (C + 2) but only (C + 1) variables to the total DOF of 2C + 6. These are any C intensive variables and the rate R of the raffinate stream. Furthermore, of the (C + 5) variables still to be specified, we cannot include any of the intensive variables of the extract stream or the remaining intensive variable of the raffinate stream, because they are determined by phase-equilibrium relations such as those presented in eq 3. The aforementioned example highlights the importance of the restrictions provided by the phase rule. A more common example is provided by mole fraction constraints. For instance, for the flash vaporization of a C-component feed mixture in the subcooled state, complete specification of the feed implies a specification of F,TF,PF,zFi (for i ) 1, ..., C). However, the total number of variables specified is to be counted as C + 2 and not C + 3, because only C - 1 of the mole fractions are independent. If the latter choice is made, we would get the value of DOF as C + 5 rather than the correct value, C + 4 (see pages 165 and 258 in the work by Seader and Henley11). The aforementioned discussion highlights the importance of the qualification “independent” in the definition of the DOF. When a set of, for example, f′ variables associated with a process are specified, it is necessary to examine whether there are any restrictions among these variables. If there are r restrictions among these variables, then we may regard only f′ - r variables as being specified and contributing to the DOF of the process. Degrees of Freedom of a Multistage Column We present a modification of a commonly employed procedure (see Kwauk2) to arrive at the DOF of a multistage column from the results for a single stage. The modified procedure has the advantage of not having to involve the overcounting that results from the interconnecting streams that are common to adjacent stages. In addition, it would enable us to bring out our point of view with greater clarity. For concreteness, we consider the single-section counter-current column (Figure 5) employing an MSA. The feed stream and the solvent stream are strictly single-phase, with neither of them lying on the phase envelope. We consider the first stage. As shown previously, the DOF for this stage is 2C + 5. Thus, with the
Figure 5. Single-section column employing an MSA.
Figure 6. Single-section column employing an energy-separating agent (ESA).
specification of a suitable set of (2C + 5) variables, the first stage is completely characterized and the R1 and E2 streams, the input streams to stage 2, are completely determined. Complete specification of R1 and E2 implies the specification of 2(C + 1) ) (2C + 2) independent variables. The DOF for stage 2 is 2C + 4; therefore, two additional variables are required to characterize it completely. This will be the case for each of the stages from 2 to N - 1. For the Nth stage, complete specification of the RN-1 and EN streams again means the specification of (2C + 2) independent variables. Because the DOF for stage N is 2C + 5, three additional variables characterize that stage completely. Thus, the DOF for the column is
(2C + 5) + 2(N - 2) + 3 + 1 ) 2N + 2C + 5 with the last term of “1” on the left-hand side of the above equation referring to the specification of the number of stages N in the column. The above result matches exactly with that reported in the literature by Kwauk,2 although that result has been obtained with a DOF of 2C + 6 for all single stages. To resolve this matter, we note that if one were to repeat the above analysis using the value of 2C + 6 for a single stage and the associated assumptions therein, we would get, for the column, a value for DOF of
(2C + 6) + 2(N - 1) + 1 ) 2N + 2C + 5 This is because the additional DOF counted in the first stage is compensated by the undercounting for the last stage. Furthermore, for stages 2 through N, (C + 2)
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variables are assigned for each stream. Thus, a combination of all these features leads to the correct final value.
and the implications of these results for the DOF analysis of a multistage column are discussed.
Such fortuitous circumstances may not always occur. For instance, in a single-section distillation column (Figure 6) that has a saturated liquid and a saturated vapor stream as its inlet streams, the DOF for each stage is 2C + 4, leading to a column DOF of
Literature Cited
(2C + 4) + 2(N - 1) + 1 ) 2N + 2C + 3 However, if we adopt a value of 2C + 6 for each stage, then the DOF would continue to be 2C + 2N + 5, which is an overestimation of two. This is due to the neglect of the analogue of eq 12 for the two streams entering the column, namely, L0 and VN+1. Conclusions It has been shown that the commonly adopted degrees of freedom (DOF) value of 2C + 6 for a single equilibrium stage applies only when the inlet streams are strictly single phase and cannot be considered as part of a two-phase system. When one or both of the inlet streams is part of a two-phase system, the DOF value reduces to 2C + 5 and 2C + 4, respectively. The key role that the phase rule has in determining these results
(1) Gilliland, E. R.; Reed, C. E. Degrees of Freedom in MultiComponent Absorption and Rectification Columns. Ind. Eng. Chem. 1942, 34, 551. (2) Kwauk, M. A System for Counting Variables in Separation Processes. AIChE J. 1956, 2, 240. (3) Gandhi, K. S.; Chandra, S.; Dryden, C. E. Calculation Methods for Complex Flow Diagrams. Chem. Age India 1964, 15, 11. (4) Luyben, W. L. Design and Control Degrees of Freedom. Ind. Eng. Chem. Res. 1996, 35, 2204. (5) Callen, H. B. Thermodynamics and an Introduction to Thermostatics; Wiley: New York, 1985. (6) Gibbs, J. W. The Scientific Papers of J. Willard Gibbs; Dover: New York, 1961; Vol. 1. (7) Feinberg, M. On Gibbs’ Phase Rule. Arch. Rat. Mech. Anal. 1979, 70, 219. (8) Dunn, J. E.; Fosdick, R. L. The Morphology and Stability of Material Phases. Arch. Rat. Mech. Anal. 1980, 74, 1. (9) Tester, J. W.; Modell, M. Thermodynamics and its Applications; 3rd Edition; Prentice Hall: Englewood Cliffs, NJ, 1996. (10) Treybal, R. E. Mass Transfer Operations; McGraw-Hill: Singapore, 1981. (11) Seader, J. D.; Henley, E. J. Separation Process Principles; Wiley: New York, 1998.
Received for review September 13, 2005 Revised manuscript received October 24, 2005 Accepted November 3, 2005 IE051025F
