Ind. Eng. Chem. Res. 2002, 41, 2213-2225
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Visualization of High-Dimensional Systems via Geometric Modeling with Homogeneous Coordinates Christianto Wibowo Department of Chemical Engineering, University of Massachusetts, Amherst, Massachusetts 01003
Ka M. Ng* Department of Chemical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong
A framework is presented for the visualization of high-dimensional systems, particularly multicomponent phase diagrams. It is based on geometric modeling with homogeneous coordinates of the transformations among geometric varieties in high-dimensional space. By reduction of the dimensionality of the system through such transforms, the resulting linear cuts and projections, in their aggregate, provide a mental picture of the system in its entirety. A general way to perform cuts and projections is presented, including how to calculate the transforms and how to define the projective space. Specifically, a procedure is presented for the development of a set of canonical coordinates for the projective space with reduced dimensions. Also, it is shown that the seemingly different transformed coordinates for reactive systems in the literature can be unified under our framework. Examples are provided to highlight the procedure and to demonstrate the visualization of phase diagrams for nonreactive and reactive systems. Introduction High-dimensional systems derived from multivariate problems and multicomponent mixtures are ubiquitous in chemical engineering. Visualization is an effective way of understanding the functional relationships among the variables of such systems. A case in point is the use of phase diagrams to represent thermodynamic relationships such as phase and reaction equilibria. Such diagrams have been used extensively in equipment design and process synthesis.1 Vapor-liquid equilibrium (VLE), solid-liquid equilibrium (SLE), and liquidliquid equilibrium (LLE) data for two-, three-, and fourcomponent systems are routinely represented in twoand three-dimensional (2D and 3D) plots.2,3 Also, a great deal of effort has been invested to automate and ease the visualization of such diagrams.4,5 However, surprisingly little has been done on how to visualize systems with higher dimensions in a general way. We would argue that the only practical approach to visualize entities of dimensions four or higher is to reduce the dimensionality. This can be achieved by taking projections and cuts. Consider the following familiar low-dimensional example. The T-x diagram for a three-component system is conveniently represented using a triangular prism in three dimensions. The vertical axis is temperature, while the base is a ternary diagram. We can visualize this 3D diagram in two dimensions by looking at a polythermal projection or an isothermal cut.6 Obviously, a single projection or cut only captures part of the system under consideration. However, as demonstrated in Samant et al.,7 the user can form a mental picture of the phase diagram in its entirety by making a sequence of such projections and * Corresponding author. Tel: +852 2358-7238. Fax: +852 2358-0054. E-mail:
[email protected].
cuts. Clearly, it would be advantageous to obtain a deeper understanding of how to perform projections and cuts. Reduction of the dimensionality is not limited to nonreactive systems. An important application can be found in the development of reactive phase diagrams, which have been widely used for the synthesis of reactive separation processes.8-10 The number of independent reactions removes the same degrees of freedom, thus confining the possible compositions to a lower dimensional subspace. In principle, these compositions can be represented by a lower dimensional coordinate systemsa transformed coordinate systemsthat contains all of the original information in a simpler way.11 Similar transformed coordinates have been proposed based on the use of elements12 and ions.13 As will be shown below, from a geometrical viewpoint, despite the seemingly different approaches, such transformations are simply special kinds of projection from high to low dimensions. In this paper, the concepts of geometric modeling serve as the foundation for the visualization of nonreactive and reactive high-dimensional systems, particularly multicomponent phase diagrams. The different transformed coordinates for reactive systems will be viewed in a unified manner. Geometric modeling deals with mathematical statements and logical relationships that define geometrical objects and has been used in computer graphics to visualize 3D objects on 2D computer screens.14-16 The approach is equally applicable for higher dimensions using proper mathematical operations in performing the transformations. This paper is organized as follows. First, we summarize the basic concepts of geometric modeling, including the properties of homogeneous coordinates. More details can be found in linear algebra texts.17-19 Second, a systematic way of visualizing high-dimensional phase
10.1021/ie010507y CCC: $22.00 © 2002 American Chemical Society Published on Web 04/06/2002
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Table 1. Mathematical Representation of Linear Varieties form
expanded notation
functional
q11y1 + q12y2 + ... + q1nyn + q10y0 ) 0 q21y1 + q22y2 + ... + q2nyn + q20y0 ) 0 l qn-m,1y1 + qn-m,2y2 + ... + qn-m,nyn + qn-m,0y0 ) 0
parametric
y1 ) a11t1 + a21t2 + ... + am1tm + a01 y2 ) a12t1 + a22t2 + ... + am2tm + a02 l yn ) a1nt1 + a2nt2 + ... + amntm + a0n y0 ) a10t1 + a20t2 + ... + am0tm + a00
Figure 1. Geometric relationship between homogeneous and Cartesian coordinates.
diagrams through generalized cuts and projections is presented. A key component is the identification of the canonical coordinates for a projective subspace. Third, we show how the various transformed coordinates for reactive systems can be derived using geometric modeling, as well as the similarities and interconnections among them. Examples will be provided throughout to illustrate the concepts. Basic Concepts of Geometric Modeling with Homogeneous Coordinates Let P(x1,x2,...,xn) be a point in an n-dimensional inhomogeneous space. Using homogeneous coordinate representation, the coordinates of point P are (y1, y2, ..., yn, y0), where yi ) y0xi. Figure 1 illustrates the geometrical interpretation for a 2D system. The 2D inhomogeneous space is viewed as a subspace of a 3D homogeneous space. To each point P(x1,x2) in the subspace, there is a corresponding line OP′ in the homogeneous space that contains all points P′(Rx1,Rx2,R), where R is a constant. Point P is obtained as a result of the central projection of line OP′ onto the y0 ) 1 plane, with the origin as the center of projection. The homogeneous coordinate system has three important properties:
vector-matrix notation qi‚y ) 0; (1)
i ) 1, 2, ..., n - m
or QTy ) 0
(3) (4)
y ) t1a1 + t2a2 + ... + tmam + a0
(5)
(2)
(H1) The homogeneous coordinates (y1, y2, ..., yn, y0) and (λy1, λy2, ..., λyn, λy0), where λ is a constant, represent the same point in the inhomogeneous space. (H2) The coordinates (y1, y2, ..., yn, 0) describe a point at infinity in the direction of vector [y1, y2, ..., yn]T in the inhomogeneous space. (H3) If the homogeneous axis is chosen to be y0 ) n yi, then (x1, x2, ..., xn) become the normalized values ∑i)1 of (y1, y2, ..., yn). As will be shown, these properties, particularly the third one, turn out to be very useful in the representation of chemical systems. Table 1 (eqs 1-5) summarizes the mathematical representation of linear varieties using homogeneous coordinates. A linear variety of dimension m (a line if m ) 1, a plane if m ) 2, or a hyperplane if m g 3) in an n-dimensional inhomogeneous space can be described using a set of n - m linear equations (eq 1). Alternatively, it can also be viewed as a linear combination of m + 1 vectors, all of which are linearly independent (eq 2). These vectors form a set of basis vectors defining the subspace. Therefore, we have two important properties of linear varieties of dimension m in an n-dimensional inhomogeneous space: (L1) It is an intersection of n - m hyperplanes, each of dimension n - 1. (L2) It can be defined by m + 1 basis vectors of dimension n + 1 or by m + 1 points whose position vectors are linearly independent. The intersection between two linear varieties of dimension m1 and m2, respectively, in an n-dimensional space is described by a set of 2n - m1 - m2 equations with n + 1 variables. It can be written as
DTy ) 0
(6)
where D ∈ R(n+1)×(2n-m1-m2) contains all of the coefficients. On the basis of the existence of solutions, we have the following rule to determine the properties of the intersection: (L3) (a) If rank(DT) > n + 1, then the two linear varieties do not intersect. (b) If rank(DT) ) n + 1, then the two linear varieties intersect at exactly one point. (c) If rank(DT) + a ) n + 1, where a > 0, then the two linear varieties intersect at a linear variety of dimension a. Note that if matrix D is nonsingular, then rank(DT) ) 2n - m1 - m2. Cuts and Projections to a Lower Dimensional Subspace It is impossible to view an n-dimensional system in its entirety on a piece of paper when n g 4. To get
Ind. Eng. Chem. Res., Vol. 41, No. 9, 2002 2215
around this problem, the strategy is to examine a set of lower dimensional subspaces to form a mental picture of the complete space. This can be achieved in two ways. First, only a part of the system is viewed and the rest is ignored, to get a cut. Geometrically, an m-dimensional cut can be considered as the intersection between the original object with a variety of dimension m. This paper only considers linear cuts, for which this variety is a hyperplane. Second, all points in the n-dimensional system are projected to an m-dimensional projective space, where m < n. The minimum number of such projections required to completely represent the phase behavior of an n-component system on a piece of paper (m ) 2) is
NP )
( ) n-1 2
2
+ k; k )
{
0 3 /4
if n is odd if n is even
(7)
The proof was given in work by Samant et al.7 and will not be repeated here. Cuts. A point in the original n-dimensional system appears in a cut if and only if it lies on the cutting m-dimensional hyperplane. A variety of dimension m′ in the original system appears in the cut as its intersection with the cutting hyperplane. If the variety in question is also a hyperplane, the intersection can be expressed in the form of eq 6. The existence and dimensionality of the intersection can then be determined using rule L3. If n - m > 1, the process of obtaining an mdimensional cut can be viewed as n - m successive cuts, each of which reduces the dimensionality of the system by 1. The equation for the m-dimensional cutting hyperplane can be obtained by expressing it as an intersection of n - m hyperplanes of dimension n - 1 (property L1). Projections. Representation of projections using geometric modeling hinges on two issues: how to do a projection and how to define the subspace where we want to project to. A projection can be mathematically represented using matrices14,20
y′ ) Py
(8)
where y ∈ R(n+1)×1 and y′ ∈ R(m+1)×1 are the position vectors of a point and its image, respectively, and P ∈ R(m+1)×(n+1) is the projection matrix. Table 2 (eqs 9-13) Table 2. Projection Matrices central projection from dimension n to a linear projective space of dimension n - 1 P ) (cTk)I - ckT parallel projection from dimension n to a linear projective space of dimension n - 1 P ) (wTk)I - wkT orthogonal projection from dimension n to a linear projective space of dimension n - 1 P ) I - kkT multiple orthogonal projection from dimension n to a linear projective space of dimension m P ) I - K(KTK)-1KT K ) [k1 k2 ... kn-m] Legend c ) position vector of the center of projection k ) unit normal vector to the projective space ki ) unit normal vector to hyperplane i w ) unit vector parallel to the direction of projection
(9)
(10)
(11)
(12) (13)
Figure 2. Multiple orthogonal projection from 3D to 1D.
summarizes the projection matrices to be used in eq 8 for three basic types of projection. The derivations are given in the appendix. In performing a projection, we need to select the projective subspace and the center of projection (in central projections) or direction of projection rays (in parallel projections). In the case of orthogonal projection, the projective subspace is always orthogonal to the projection rays. Therefore, once the projective subspace is specified, the direction of projection rays cannot be chosen arbitrarily and vice versa. In this paper, we only consider linear projective subspaces. Identification of Canonical Coordinates for a Projective Subspace Let us consider an n-dimensional system represented by a set of homogeneous coordinates {y1, y2, ..., yn, y0}. The simplest way to reduce the dimensionality of this system is by performing a multiple orthogonal projection to an m-dimensional projective subspace. Figure 2 illustrates a 3D example. Planes 1 and 2 are the planes orthogonal to q1 and q2, respectively. To maintain clarity, only a portion of these planes lying inside the given cube is drawn. The intersection of these two planes, line OA, is the projective space in question. Under the multiple orthogonal projection, point X is first projected in the direction of q1 to X′ and then in the direction of q2 to X′′ on line OA. Note that point X′ is not necessarily on plane 1. The m-dimensional projective subspace is defined by the intersection of n - m hyperplanes of dimension n - 1 (property L1). The directions of the projection rays are given by the normal vectors to these hyperplanes, q1, q2, ..., qn-m. Note that these projection rays are linearly independent. The projective subspace can be defined by a set of basis vectors (property L2). In Table 3, we propose a procedure which guarantees the identification of a convenient set of basis vectors for a given m-dimensional linear subspace in an n-dimensional inhomogeneous space. Indeed, there are infinitely many
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Table 3. Procedure for Determining Basis Vectors step 1 step 2 step 3 step 4
Select n - m variables as reference variables. Without loss of generality, let these be ym+1, ym+2, ..., yn. Set y1 ) 1 and y2, ..., ym , y0 ) 0. Solve eq 1 to obtain the values of ym+1, ym+2, ..., yn. Together with the values set in step 2, they are the coordinates of a basis vector. Repeat steps 2 and 3 to obtain the other m basis vectors, each time by setting an element of {y2, y3, ..., ym, y0} to unity and all others to zero.
other possible sets of basis vectors because any linear combinations of these vectors can form a basis set as well. To describe objects lying entirely in the m-dimensional projective subspace, it is both redundant and inconvenient to use the original set of n + 1 coordinates. Instead, we can use a set of m + 1 coordinates {Y1, Y2, ..., Ym, Y0}, which will be referred to as canonical coordinates. In general, the relationship between the canonical and original coordinates (y1, y2, ..., yn, y0) can be written in matrix form as
Notice that ∆ is idempotent (that is, ∆2 ) ∆) and P ) I - ∆ (eq 12). According to a theorem of linear algebra,18 this implies that ∆ is symmetric and the range of P, R(P), is equal to the null space of ∆, N(∆). Because the projective space contains all vectors y′ for which y′ ) Py (eq 8), it is equal to R(P) by definition. Therefore, the projective space is equal to N(∆). Let us now define matrix Λ ∈ R(n+1)×(m+1), whose columns are the basis vectors of the projective space (which is also the null space of ∆). By definition,
∆Λ ) 0
(20)
However, because ∆ is symmetric, it is also true that
ΛT ∆ ) 0
(21)
Substituting P ) I - ∆ into eq 18, we obtain
[u1′ u2′ ‚‚‚ um′ u0′ ] ) [u1 u2 ‚‚‚ um u0 ] ∆[u1 u2 ‚‚‚ um u0 ] (22) Substitution into eq 17 yields
ψ′ ) Zψ
(14)
ψ ) [y1 y2 ‚‚‚ yn y0 ]T
(15a)
Premultiplying eq 23 with ΛT and substituting eq 21 gives
ψ′ ) [Y1 Y2 ‚‚‚ Ym Y0 ]T
(15b)
ΛT[u1 u2 ‚‚‚ um u0 ]ψ′ ) ΛTψ
where
and Z ∈ R(m+1)×(n+1) is a linear transformation matrix. The use of homogeneous coordinates affords this convenient representation. Note that the symbol ψ has been used instead of y to represent the vectors in order to distinguish eq 14 from eq 8. Equation 8 gives the coordinates of the image of a point undergoing a geometric transformation, whereas eq 14 returns the new coordinates of a point in the canonical coordinate system. The transformation matrix Z can be determined as follows. First, we recognize that the position vector of any point on the projective subspace can be written in two ways:
y ) Y1u1′ + Y2u2′ + ... + Ymum′ + Y0u0′ ) y1u1 + y2u2 + ... + ynun + y0u0 (16) where ui′ and ui are the unit vectors of the m-dimensional subspace and the original n-dimensional space, respectively. Equation 16 can be written in matrix form as
[u1′ u2′ ‚‚‚ um′ u0′ ]ψ′ ) [u1 u2 ‚‚‚ un u0 ]ψ (17) The most straightforward way (but not the only way) to obtain the set of basis vectors {u1′, u2′, ..., um′, u0′} is to take m + 1 unit vectors of the original space and perform multiple orthogonal projection to the m-dimensional subspace. Without loss of generality, {u1, u2, ..., um, u0} can be taken as the set of choice. Thus,
[u1′ u2′ ‚‚‚ um′ u0′ ] ) P[u1 u2 ‚‚‚ um u0 ]
(18)
where P ∈ R(n+1)×(n + 1) is the projection matrix from eq 12. Let us now define ∆ ∈ R(n+1)×(n+1) as
∆ ) K(KTK)-1KT
(19)
[u1 u2 ‚‚‚ um u0 ]ψ′ - ∆[u1 u2 ‚‚‚ um u0 ]ψ′ ) ψ (23)
(24)
The basis vectors of the projective subspace can be found using the steps in Table 3. Following these steps, we obtain m + 1 sets of linear equations in ym+1, ym+2, ..., yn. The solution to each of these sets of linear equations is
biT ) -θiTQref-1; i ) 0, 1, 2, ..., m
(25)
where
θi ) [q1,i q2,i ‚‚‚ qn-m,i ]T; i ) 0, 1, 2, ..., m
[
q1,m+1 q Qref ) 1,m+2 l q1,n
[
q2,m+1 q2,m+2 l q2,n
‚‚‚ ‚‚‚ l ‚‚‚
qn-m,m+1 qn-m,m+2 l qn-m,n
]
Therefore, we can write
1 0 ΛT ) l 0 0
0 1 l 0 0
‚‚‚ ‚‚‚ l ‚‚‚ ‚‚‚
0 0 l 1 0
b11 b21 l bm1 b01
‚‚‚ b1,n-m ‚‚‚ b2,n-m l ‚‚‚ bm,n-m ‚‚‚ b0,n-m
0 0 l 0 1
]
(26)
(27)
(28)
Notice that because [u1, u2, ..., um, u0] is just the minor of an identity matrix obtained by eliminating the (m + 1)th to the nth columns, it can be easily verified that for ΛT given by eq 28 the left-hand side of eq 24 reduces to ψ′. Therefore, comparison with eq 14 concludes that Z ) ΛT. In summary, if the unit vectors of the projective subspace are obtained via a multiple orthogonal projection of m + 1 unit vectors of the original space (as expressed by eq 18), then the canonical coordinates obtained by substituting Z into eq 14 are
Ind. Eng. Chem. Res., Vol. 41, No. 9, 2002 2217
Yi ) yi - θiTQref-1yref; i ) 0, 1, 2, ..., m
(29)
where θi and Qref are defined by eqs 26 and 27, respectively, and
yref ) [ym+1 ym+2 ‚‚‚ yn ]T
(30)
xi )
[
(
)]
0 ∆Hm,i 1 1 1 exp ; i ) 1, 2, ..., S γi R Tm,i T
(32)
The remaining C - S components are unsaturated, implying that they can be present in an amount smaller than its saturation value. We can simply write
From eq 29, it follows that the corresponding set of inhomogeneous coordinates is
xi ) ti-S; i ) S + 1, S + 2, ..., C - 1
Yi xi - θiTQref-1xref ) Xi ) Y0 1 - θ TQ -1x
which is constrained within the boundaries to the saturation variety:
0
ref
(31)
(33)
ref
This is the general form of canonical coordinates obtained from a multiple orthogonal projection. It is interesting to note that a linear coordinate transformation in homogeneous space does not necessarily correspond to a linear coordinate transformation in the inhomogeneous space, as is clearly shown by eq 31. This is, of course, another advantage afforded by using homogeneous coordinates in geometric modeling. Also note that by selection of a different set of unit vectors of the projective subspace (u1′, u2′, ..., um′, u0′), a different set of canonical coordinates can be obtained. Applications to Chemical Systems It is a common practice to represent the phase behavior of chemical systems in mole fraction coordinates. For a C-component system, there are C - 1 independent mole fractions, constituting an inhomogeneous coordinate system. In the corresponding homogeneous coordinate representation, we use the number of moles of C - 1 components {n1, n2, ..., nC-1} along with the total number of moles, nTOT, as the axes, so as to take advantage of the properties of homogeneous coordinates. Therefore, we have a total of C independent coordinates. The total number of moles is taken as the homogeneous axis, such that the normalized mole fractions appear naturally in the corresponding inhomogeneous coordinates. To illustrate the use of cuts and projections for visualizing chemical systems, we now discuss several applications along with relevant examples. Phase Diagrams for Nonreactive Systems. As a representative phase diagram for nonreactive systems, let us consider the SLE phase diagram of a multicomponent molecular system. Using the framework of Samant et al.,7 such a phase diagram can be represented as a digraph. The vertexes represent different points on the phase diagram, including pure components, eutectic points, and saturation points. They are labeled after the saturated components. Unsaturated components, if present, are written as subscripts. For example, vertex AB indicates a binary mixture where both A and B are saturated. The vertex at which three components (A, B, and C) are present but only A and B are saturated is labeled as ABC. Two vertexes are said to be adjacent to each other if they are connected by an edge. Furthermore, these vertexes and edges define the boundaries of regions of higher dimensions, inside which one or more components are saturated. Such regions are referred to as saturation varieties. The procedure for calculating these varieties is discussed elsewhere.7 To calculate cuts, we need to express the saturation varieties in a functional form. If components 1 to S are saturated, then
0 e xi e xsat i ; i ) S + 1, S + 2, ..., C - 1 (34) Together these equations form a system of equations with C - S parameters (that is, t1 to tC-S-1 plus the temperature T). Therefore, for an isothermal or polythermal phase diagram, eqs 32-34 represent a saturation variety of dimension C - S - 1. Using this functional form, it is possible to calculate cuts of any dimension of the saturation varieties. Because the primary objective of performing cuts is to visualize the phase diagram, we are mostly interested in 2D cuts. In such a cut, we expect to see vertexes and edges that are part of the various saturation varieties in the original phase diagram. Because for ideal systems the saturation varieties in a polythermal phase diagram are nearly linear, we can use rule L3 to determine the dimensionality of the intersection. Unless the cutting plane (m1 ) 2) coincides with a saturation variety (m2 ) C - S - 1), the rank of matrix DT in eq 1 is always equal to C + S - 3 (note that n ) C - 1). Thus, according to rule L3c, S ) 2 if we set a ) 1. In other words, we have just shown that the edges in a 2D cut must be a part of the saturation varieties where two components are cosaturated. Therefore, the 2D cut of a polythermal SLE phase diagram can be generated by identifying all saturation varieties where two components are cosaturated, and then those cosaturation lines can be determined on the cutting plane using eqs 32-34. Let us now consider the projection of the (C 1)-dimensional phase diagram to an m-dimensional projective space. This can be done in several ways. One way is to use repeated central or parallel projections, or combinations of the two, each time reducing the dimensionality of the system by 1. A central projection with the reference component apex as the center of projection is known as the Ja¨ necke projection, which is widely used in the literature.21-24 Another way is to use an orthogonal projection, in which case the dimensionality can be reduced directly to the desired value. To obtain such a projection, we need to define C - m - 1 projection rays. A convenient choice of projection rays {q1, q2, ..., qC-m-1} turns out to be
qj,i )
{
1, if i ) j + m or i ) 0 0, otherwise
(35)
because this choice leads to Qref ) I (eq 27), θ0 ) [1, 1, ..., 1]T, and θi ) 0 for i ) 1, 2, ..., m (eq 26). When the number of moles is substituted in place of y in eq 29, it is found that the projective space is defined by the following set of m + 1 canonical coordinates
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ψ′ ) [n1 n2 ‚‚‚ nm nTOT -
∑
n j ]T
(36)
j)m+1
The corresponding inhomogeneous coordinates are
xi
Xi )
; i ) 1, 2, ..., m
C-1
1-
∑
(37)
xj
j)m+1
In other words, the new set of coordinates is obtained by neglecting components m + 1 to C - 1 and normalizing the remaining mole fractions such that they always sum to unity. We again refer to these neglected components as reference components. Note that if the mole fraction of a reference component is unity, the denominator in eq 37 becomes zero. Therefore, pure reference components cannot appear in this projection. It is interesting to note that for the projection rays represented by eq 35, if the orthogonal projective space dimensionality is reduced by 1, such a projection corresponds to a Ja¨necke projection in the corresponding inhomogeneous (mole fraction) space, with the reference component vertex as the center of projection. As indicated by eq 7, multiple projections are required to get a complete view of the phase diagram. The simplest way to get the other projections is by choosing a different set of reference components. Obviously, other choices of projection rays, which lead to different sets of canonical coordinates, are also possible. For example, one can choose the projection rays to be parallel to the axes of the original coordinate system. Let us now illustrate these concepts using an example. Example 1: SLE Phase Diagram of a Quinary System. Consider a system of five components: A, B, C, D, and E. We would like to visualize the polythermal SLE phase diagram for this system by taking different cuts and projections. A complete representation of the phase diagram requires a 4D inhomogeneous coordinate system (C ) 5). Although other combinations are equally good, {xB, xC, xD, xE} is arbitrarily chosen as the set of inhomogeneous coordinates to be used in the phase diagram. Simple eutectic and ideal behavior is assumed. The data for use in eq 32 are given in Table 4. Figure 3 shows the binary cosaturation lines on various cuts of the phase diagram taken at different values of xD and xE ) 0.2. The vertexes in the cuts are labeled according to the convention in work by Samant et al.7 At xD ) 0.1, the cut shows that both D and E are unsaturated regardless of the mole fractions of A, B, and C, even at the triple saturation point ABCDE (Figure 3a). However, at xD ) 0.55, the situation is different. Point ABCDE disappears because, with the presence of such a large amount of D, cosaturation of A, B, and C is not possible (Figure 3b). Instead, three vertexes where D is saturated appear in the cut. At xD ) 0.57, it is not even possible for just A and C to be cosaturated. Point ACDBE disappears and is replaced by ADCE and CDAE (Figure 3c). The cut at xD ) 0.65 reveals that, at this concentration of D, A can no longer be saturated. Therefore, point DAE emerges in Figure 3d. We will now illustrate how to calculate the projection of any point in the phase diagram by means of a projection matrix. Using the procedure of Samant et al.,7 the eutectic points and adjacency information for this five-component system can be obtained (Table 5). For
Figure 3. Cuts of the five-component SLE phase diagram (example 1): (a) at xD ) 0.1, xE ) 0.2; (b) at xD ) 0.55, xE ) 0.2; (c) at xD ) 0.57, xE ) 0.2; (d) at xD ) 0.65, xE ) 0.2. Table 4. Heat of Fusion and Melting Point Data for Example 1 component
∆H0m (J/mol)
Tm (K)
A B C
32 000 40 000 38 000
450 500 480
component
∆H0m (J/mol)
Tm (K)
D E
27 000 23 000
400 350
visualization, it is desired to project the phase diagram to a 2D subspace using a multiple orthogonal projection. We will use a homogeneous coordinate system with coordinates {nB, nC, nD, nE, nTOT}, taking D and E as reference components. Let us choose the projection rays to be those defined in eq 35, that is,
q1 ) [0 0 1 0 1 ]T
(38a)
q2 ) [0 0 0 1 1 ]T
(38b)
Equation 12 is used to determine the projection matrix. The unit normal vectors k1 and k2 in this case are
k1 ) [0 0 (1/2)x2 0 (1/2)x2 ]T
(39a)
k2 ) [0 0 0 (1/2)x2 (1/2)x2 ]T
(39b)
which are obtained by simply normalizing q1 and q2. After substituting k1 and k2 into eq 13 to obtain K, we arrive at the final result
[
1 0 P) 0 0 0
0 1 0 0 0
0 0 1 /3 1 /3 -1/3
0 0 1 /3 1 /3 - 1 /3
0 0 -1/3 -1/3 1 /3
]
(40)
P can be used with eq 8 to calculate the projection of any point in the phase diagram. For example, the projection of point ABCDE (0.009, 0.017, 0.210, 0.711,
Ind. Eng. Chem. Res., Vol. 41, No. 9, 2002 2219 Table 5. Location of Vertices and Adjacency Information for Example 1 vertex
xA
xB
xC
xD
xE
adjacent to
A B C D E AB AC AD AE BC BD BE CD CE DE ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE ABCD ABCE ABDE ACDE BCDE ABCDE
1 0 0 0 0 0.757 0.670 0.246 0.078 0 0 0 0 0 0 0.564 0.231 0.076 0.221 0.075 0.056 0 0 0 0 0.209 0.074 0.055 0.055 0 0.054
0 1 0 0 0 0.242 0 0 0 0.395 0.080 0.016 0 0 0 0.168 0.055 0.014 0 0 0 0.067 0.015 0.010 0 0.049 0.013 0.009 0 0.010 0.009
0 0 1 0 0 0 0.330 0 0 0.605 0 0 0.124 0.028 0 0.268 0 0 0.088 0.024 0 0.113 0.027 0 0.018 0.083 0.024 0 0.017 0.018 0.017
0 0 0 1 0 0 0 0.754 0 0 0.920 0 0.876 0 0.231 0 0.715 0 0.690 0.000 0.216 0.820 0 0.228 0.226 0.659 0 0.214 0.212 0.223 0.210
0 0 0 0 1 0 0 0 0.922 0 0 0.984 0 0.972 0.770 0 0 0.910 0 0.900 0.728 0 0.958 0.762 0.756 0 0.889 0.722 0.717 0.749 0.711
AB, AC, AD, AE AB, BC, BD, BE AC, BC, CD, CE AD, BD, CD, DE AE, BE, CE, DE A, B, ABC, ABD, ABE A, C, ABC, ACD, ACE A, D, ABD, ACD, ADE A, E, ABE, ACE, ADE B, C, ABC, BCD, BCE B, D, ABD, BCD, BDE B, E, ABE, BCE, BDE C, D, ACD, BCD, CDE C, E, ACE, BCE, CDE D, E, ADE, BDE, CDE AB, AC, BC, ABCD, ABCE AB, AD, BD, ABCD, ABDE AB, AE, BE, ABCE, ABDE AC, AD, CD, ABCD, ACDE AC, AE, CE, ABCE, ACDE AD, AE, DE, ABDE, ACDE BC, BD, CD, ABCD, BCDE BC, BE, CE, ABCE, BCDE BD, BE, DE, ABDE, BCDE CD, CE, DE, ACDE, BCDE ABC, ABD, ACD, ABCDE ABC, ABD, ACD, ABCDE ABD, ABE, ADE, ABCDE ACD, ACE, ADE, ABCDE BCD, BCE, BDE, ABCDE ABCD, ABCE, ABDE, ACDE, BCDE
1.000) is a point with homogeneous coordinates (0.009, 0.017, -0.026, -0.026, 0.026). Clearly, after transformation, this point as well as all other points lies on a 3D subspace. Rather than using the 5D coordinate system to locate points in the projection, a set of 3D canonical coordinates is used. According to eq 36, a set of such coordinate axes is {N1, N2, N0}, corresponding to nB, nC, and nTOT - nD - nE, respectively. Note that both point ABCDE and its projection have the same canonical coordinates, that is, (0.009, 0.017, 0.079). The corresponding inhomogeneous canonical coordinates are
X1 )
xB 1 - xD - xE
(41a)
X2 )
xC 1 - xD - xE
(41b)
The inhomogeneous coordinates for point ABCDE in the projection are thus (0.112, 0.208). Projections of all other points can be obtained in the same way. Figure 4 shows the projection of the entire phase diagram, obtained using the projection rays given in eq 38. As mentioned, a different set of projection rays can be chosen. For example, we can alternatively select two that are parallel to the nD and nE axes, that is,
q1 ) [0 0 1 0 0 ]T
(42a)
q2 ) [0 0 0 1 0 ]T
(42b)
The use of eq 29 leads to N1 ) nB, N2 ) nC, and N0 ) nTOT. Therefore, X1 ) xB and X2 ) xC. In other words, the projection is obtained by simply neglecting the mole fractions of components D and E without renormalization. The resulting projection (Figure 5) provides a
Figure 4. Projection of the five-component SLE phase diagram to the normalized mole fraction space (example 1).
different view of the phase diagram compared to the one shown in Figure 4. Example 2: Cruickshank Projection. Cruickshank et al.25 proposed an orthogonal projection that they found to be useful for representing LLE data of fourcomponent systems. We will show that it falls within our framework. Figure 6 depicts this projection in the inhomogeneous mole fraction space, where {xA, xB, xC} is taken as the set of coordinate axes. Note that all possible compositions of this four-component system are contained in the triangular pyramid ABCD. The projective space has the form of an equilateral parallelepiped (Figure 6a). For convenience, it can be stretched to take the form of a square, and two transformed coordinates, X1 and X2, can be assigned (Figure 6b). We would like to obtain the relationship between these coordinates and the original ones. Because points E and F coincide in the projection, we can say that the direction of the projection rays is given
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Ind. Eng. Chem. Res., Vol. 41, No. 9, 2002
Figure 5. Orthogonal projection of the five-component SLE phase diagram in the direction of axes nD and nE (example 1). The inset shows the details at the lower left corner of the projection. Figure 7. Cruickshank projection of the four-component LLE phase diagram (example 2). Table 6. LLE Tie-Line Data for Example 226 aqueous phase
Figure 6. Cruickshank projection: (a) projection rays; (b) projective space.
by vector EF. Using homogeneous coordinates and setting nTOT ) 2 for numerical convenience, we find that the coordinates of points E (xC ) xD ) 0.5) and F (xA ) xB ) 0.5) are (0, 0, 1, 2) and (1, 1, 0, 2), respectively. Thus, the direction of the projection ray is
q ) [1 1 -1 0 ]T
(43)
Following the general procedure described previously, we arrive at the following set of transformed coordinates: N1 ) nA + nC, N2 ) nB + nC, and N0 ) nTOT. The corresponding inhomogeneous coordinates are X1 ) xA + xC and X2 ) xB + xC. This set of transformed coordinates has been used to represent experimental LLE
organic phase
xA
xB
xC
xD
xA
xB
xC
xD
0.878 0.825 0.789 0.707
0 0.0528 0.0865 0.154
0.0343 0.0373 0.0401 0.0608
0.0872 0.0854 0.0844 0.0778
0.14 0.183 0.236 0.315
0 0.114 0.223 0.311
0.859 0.700 0.534 0.359
0.0013 0.0029 0.0062 0.0150
data by Mar Olaya et al.26 Some tie-line data are listed in Table 6, and the Cruickshank projection is shown in Figure 7. From examples 1 and 2, it is clear that different choices of projection rays would lead to different figures of the projection and different sets of transformed coordinates. Depending on the application, one projection may be preferable to others. Phase Diagrams for Reactive Systems. As mentioned, the presence of reaction confines the possible compositions to a lower dimensional subspace. It is often desirable to project the phase diagram to the lower dimensional subspace in such a way that the new composition variables take the same numerical values before and after reaction.8 In other words, they are reaction invariant. It will be shown that the above general procedure can be used to derive such a set of transformed coordinates. Consider a system involving C components and R independent reactions. The chemical reactions can be described as
ν1jA1 + ν2jA2 + ... + νCjAC ) 0; j ) 1, 2, ..., R
(44)
where νij is the stoichiometric coefficient of component i in reaction j. It has a negative value for reactants and a positive value for products. Changes in composition due to these reactions can be described as R
ni ) ni,0 +
νijξj; ∑ j)1
i ) 1, 2, ..., C
(45a)
R
nTOT ) nTOT,0 +
νTOT,jξj ∑ j)1
(45b)
where ξj is the molar extent of reaction j. In vector form,
Ind. Eng. Chem. Res., Vol. 41, No. 9, 2002 2221
n ) n0 + ξ1q1 + ξ2q2 + ... + ξRqR
(46)
qj ) [ν1j ν2j ‚‚‚ νC-1,j νTOT,j ]T
(47)
where
Equation 46 is basically the parametric representation of a linear variety of dimension R, with ξ1, ξ2, ..., ξR as the parameters. We will call this stoichiometric variety, which describes composition changes due to reactions. The most important advantage of using homogeneous (number of moles) coordinates instead of inhomogeneous (mole fraction) coordinates is that stoichiometric varieties are always linear and parallel. Figure 8 illustrates this point with an example of a three-component system with one reaction: A + B ) C. Because R ) 1, the stoichiometric variety is a line. Starting from an initial composition of 1 mol of A and 1 mol of B (point P1), the composition moves along line P1Q1 as the reaction progresses. Therefore, P1Q1 is a stoichiometric line. Similarly, P2Q2 and P3Q3 are two other stoichiometric lines parallel to P1Q1 for different initial compositions. Note that all possible compositions of this system lie above plane OAB (nTOT ) nA + nB). In other words, they satisfy
nTOT > nA + nB
(48)
When the entire space is centrally projected onto the plane nTOT ) 1, we obtain the triangle Q2A′B′ as the inhomogeneous composition space. Note that the projected stoichiometric lines P1′Q1′, P2′Q2′, and P3′Q3′ are not parallel in this inhomogeneous space. To obtain a (C - R)-dimensional projective subspace, we perform a multiple orthogonal projection such that the stoichiometric variety described by eq 46 disappears. In other words, the projection rays follow the directions of {q1, q2, ..., qR}. From eq 29, we obtain that the set of canonical coordinates describing the projective subspace is
Ni ) ni - νiTVref-1nref; i ) 1, 2, ..., C - R - 1 (49a) N0 ) nTOT - νTOTTVref-1nref
(49b)
nref ) [nC-R nC-R+1 ‚‚‚ nC-1 ]T
(50)
where
νiT ) [νi1 νi2 ‚‚‚ νiR ]; i ) 1, 2, ..., C - R - 1, TOT (51)
[
νC-R,1 ν Vref ) C-R +1,1 l νC-1,1
νC-R,2 νC-R +1,2 l νC-1,2
‚‚‚ ‚‚‚ l ‚‚‚
νC-R,R νC-R +1,R l νC-1,R
]
TOT
ref
Table 7. Data for Example 324 saturation composition 0 ∆Hm,i 1 1 1 xsat exp i ) γi R Tm,i T
[
(
)]
component
∆H0m (J/mol)
Tm (K)
C E standard Gibbs free energy of reaction 0 ∆Hm,i 1 1 1 xsat exp i ) γi R Tm,i T
20 756 17 917
650 640
[
)]
(
reaction
E1
E2
E3
A+B)C C+D)E
-12.125 13.264
7164.6 -3314.8
-8 × 105 8 × 104
If C - R g 4, it is still impossible to completely view the reaction-invariant phase diagram on a piece of paper. In such cases, the projections described in example 1 can be used to further reduce the dimensionality. The number of projections required is still given by eq 7 but with n replaced by C - R.7 Let us now consider an example to illustrate this. Example 3: SLE Phase Diagram of a SevenComponent System with Two Reactions. Consider a system of seven components: A, B, C, D, E, F, and G, with two reactions:
A+B)C C+D)E
(54)
(52)
The corresponding inhomogeneous coordinates are
Ni xi - νiTVref-1xref ) Xi ) N0 1 - ν TV -1x
Figure 8. Stoichiometric lines in homogeneous and Cartesian spaces.
(53)
ref
which turn out to be exactly the same as the mole fraction transformed coordinates obtained by algebraically eliminating the extents of reaction.11
where C and E have limited solubility in the reaction mixture. It is desirable to show the regions where the solution is saturated with any or both of these two components, at a temperature of 310 K. The relevant physical data are given in Table 7. The readers are referred to Berry and Ng9 for the calculation procedure. We choose to use {nB, nC, nD, nE, nF, nG, nTOT} to represent compositions in homogeneous space. Using eq 49 and choosing C and E to be the reference components, the transformed coordinates obtained after the projection of the stoichiometric variety are
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Ind. Eng. Chem. Res., Vol. 41, No. 9, 2002
NB ) nB + nC + nE ND ) nD + nE NF ) nF NG ) nG N0 ) nTOT + nC + 2nE
(55)
Then, taking a multiple orthogonal projection with F and G as reference components, we obtain the following set of inhomogeneous transformed coordinates:
X1 )
xB + xC + xE 1 + xC + 2xE - xF - xG
X2 )
x D + xE 1 + xC + 2xE - xF - xG
(56)
Figure 9. Projection of the seven-component reactive phase diagram (example 3).
The phase diagram obtained through this way of projection is depicted in Figure 9. As discussed previously, pure F and G cannot and do not appear in this projection.
the total number of moles of element i. We then have
On Transformed Coordinates for Reacting Systems
In accordance with the normalization factor in their transformed coordinates, we take
From the discussion in the previous sections, it is clear that the role of transformed coordinates in reactive phase diagrams is to represent compositions in the reaction-invariant projective space. They are basically derived from a set of arbitrarily chosen basis vectors defining the projective space. Therefore, it is not surprising that different sets of transformed (or canonical) coordinates have been proposed.8,12,13 In this section, we show that all can be derived from geometric modeling. There are two unifying features among the canonical coordinates, namely: (1) They must describe the same projective space because they are all reaction invariant. Different canonical coordinates originate from different choices of basis vectors. Recall that a set of basis vectors describing a linear subspace may be chosen arbitrarily. Furthermore, different choices of reference components would also lead to different sets of such coordinates. (2) They can be obtained from the original (number of moles) coordinates through a linear transformation in the homogeneous space. Therefore, a linear transformation matrix Z, as defined by eq 14, can be used to describe the relationship between the canonical and original coordinates. It was shown that the transformed coordinates proposed by Ung and Doherty can be obtained from a set of basis vectors obtained using the four-step procedure in Table 3. Equation 49 can be cast in the form of eq 14 to identify Z. Let us now look at the other transformed coordinates in more detail. Transformed Coordinates of Pe´ rez Cisneros, Gani, and Michelsen. The element mole fraction coordinates of Pe´rez Cisneros et al. do not depend on the reaction because the elements are preserved. The development begins with the selection of C - R “elements” which make up the C components. These are chosen such that they do not change during reaction. We can relate the number of moles of the elements to that of the components by defining eij as the number of moles of element i in 1 mol of component j and Ni as
C
Ni )
∑ j)1
C-1
eijnj )
(eij - ei,C)nj + ei,CnTOT ∑ j)1
(57)
C-R
N0 )
Ni ∑ i)1
(58)
Substituting eq 57 into eq 58, we get
N0 ) (eTOT,1 - eTOT,C)n1 + (eTOT,2 - eTOT,C)n2 + ... + (eTOT,C-1 - eTOT,C)nC-1 + eTOT,CnTOT (59) where C-R
eTOT,j )
eij ∑ i)1
(60)
It must be realized that there are only C - R independent coordinates. These can be taken as {N1, N2, ..., NC-R-1, N0}. Therefore, the set of transformed coordinates can be expressed in the generic form of eq 14, with n and N in place of y and Y, respectively, and Z replaced by
[
ZPC ) e11-e1C e21-e2C
e12-e1C e22-e2C
l l eC-R-1,1-eC-R-1,C eC-R-1,2-eC-R-1,C eTOT,1-eTOT,C eTOT,2-eTOT,C
‚‚‚ ‚‚‚ l ‚‚‚ ‚‚‚
e1,C-1-e1C e2,C-1-e2C
e1C e2C
l l eC-R-1,C-1-eC-R-1,C eC-R-1,C eTOT,C-1-eTOT,C eTOT,C
]
(61)
From this matrix, we can recover the C - R basis vectors by reading across each row. For certain systems, these transformed coordinates are the same as those of Ung and Doherty.12 An example is a four-component system involving A, B, AB, and an inert I, with one reaction, A + B ) AB. In general, the two transformation matrices are different but can be obtained from each other by linear combinations. For example, consider a four-component system with one reaction, A + BC ) AB + C. Table 8 shows the set of transformed coordinates obtained using the two pro-
Ind. Eng. Chem. Res., Vol. 41, No. 9, 2002 2223 Table 8. Transformed Coordinates for Reacting System A + BC ) AB + C transformed mole fractions (Ung and Doherty) (C as the reference component)
element mole fractions (Pe´rez Cisneros et al.)
N 1 ) nA + nC N2 ) nBC + nC N3 ) nAB - nC N0 ) nTOT
N1′ ) nA + nAB N2′ ) nBC + nAB N3′ ) nBC + nC N0′ ) nTOT + nBC + nAB
N1′ ) N1 + N3 N2′ ) N2 + N3 N3′ ) N2 N0′ ) N0 + N2 + N3
(62)
In other words, one set of coordinates is a linear combination of the other because they describe the same projective space. Transformed Coordinates of Samant and Ng. For conjugate salt systems, Samant and Ng13 proposed a set of transformed coordinates based on ionic concentrations. For a system containing m cations (M1, M2, ..., Mm), n anions (N1, N2, ..., Nn), and an inert I, we can write their transformed coordinates in homogeneous form as
R(Mi) ) zMi[Mi]; i ) 1, 2, ..., m
(63a)
R(Ni) ) zNi[Ni]; i ) 1, 2, ..., n
(63b)
R(I) ) 1
(63c)
R0 )
n
zM [Mi] ) ∑zN [Ni] ∑ i)1 i)1 i
i
(63d)
R(Mi) ) R(Ni) )
∑ βM ,jnj;
mI j)1
i
zNiC-1
∑ βN ,jnj;
mI j)1
i
i ) 1, 2, ..., m
(64a)
i ) 1, 2, ..., n
(64b)
C-1 MI MI R(I) ) nC ) (nTOT nj) mI mI j)1
∑
) ∑ (∑
)
zNi βNi,jnj (64d) i)1mI n
where βij is the number of moles of ion i in 1 mol of salt j. This set of transformed coordinates can be expressed in the generic form of eq 14, with the y’s replaced by the number of moles of the salts and ZSN ∈ R(m+n+2)×(mn+1) defined by
[
zM1βM1,2/mI
l zMmβMm,1/mI zN1βN1,1/mI
l zMmβM1m,2/mI zN1βN1,2/mI
l zNnβNn,1/mI -MI/mI
l zNnβNn,2/mI -MI/mI
m
∑z
‚‚‚ l ‚‚‚ ‚‚‚ l ‚‚‚ ‚‚‚
m
MiβMi,1/mI
i)1
∑z
MiβMi,2/mI
i)1
zM1βM1,C-1/mI l zMmβMm,C-1/mI zN1βN1,C-1/mI l zNnβNn,C-1/mI -MI/mI
0 l 0 0 l 0 MI/mI
m
‚‚‚
∑z i)1
MiβMi,C-1/mI
0
]
(65)
Note that because of the neutrality condition (eq 63d) the elements of the last row of matrix Z can be expressed either as a sum over all anions or a sum over all cations. As discussed previously, the basis vectors can be recovered by reading across the rows. The selection of the proper transformed coordinates to use really depends on the application. For example, in the analysis of reactive separations where molecular species are involved, it is more convenient to use transformed mole fraction coordinates. On the other hand, in the SLE calculations for a conjugate salt system, we need to use solubility products, which are expressed in terms on ionic concentrations. Therefore, the ionic transformed coordinates are the natural choice for this application. Conclusions
where the z’s are the magnitudes of the ionic charges. Because compositions are represented using ionic concentrations, which are not affected by reactions, this set of coordinates is obviously reaction invariant. It is similar to the element mole fraction coordinates of Pe´rez Cisneros et al., with the ions as well as the inert serving as elements. It is possible to express the transformed variables of eq 63 in terms of the number of moles of salt, although such an expression is undesirable because it is the ions not the salts that are physically present in the solution. Note that the anions and cations can form a total of mn simple salts, making the total number of components in the system C ) mn + 1. It is possible to express eq 63a-d in terms of the number of moles of the salts, n1, n2, ..., nC-1, and the total number of moles nTOT.
zMiC-1
(
zM i C-1 βMi,jnj ) i)1 mI j)1 m
∑ ∑ j)1
ZSN ) zM1βM1,1/mI
posed methods. By observation,
m
C-1
R0 )
Visualization of high-dimensional systems is a powerful means for conveying geometrical features that are hard to grasp when only numerical data are present. We have presented a general way of visualizing such high-dimensional entities through linear projections and cuts. On the basis of geometric modeling with homogeneous coordinates, such cuts and projections can be represented as linear transformations in a simple yet elegant way. Although for the sake of being more concrete multicomponent phase diagrams are emphasized, the procedure described here is obviously applicable to other multidimensional systems as well. This framework enables us to obtain a concise geometric interpretation of the transformed coordinates reported in the literature, including the Ja¨necke projection and the projection for reacting systems. It would be interesting to use the general procedure described in this paper to discover other transforms, which may be more useful and convenient than the ones presented here, depending on the application on hand. Also, it would be beneficial to extend this study to nonlinear cases that are common in chemical engineering. Efforts in these directions are underway. Acknowledgment
(64c)
Financial support from the National Science Foundation (Grant CTS-9908667) is gratefully acknowledged.
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Appendix: Derivation of Projection Matrices Consider a point X in an n-dimensional inhomogeneous space. Using homogeneous coordinates, its position vector is y ) [y1, y2, ..., yn, y0]T. It is desired to obtain its projection onto an (n - 1)-dimensional hyperplane described by
k‚y ) 0
(A1)
where k is a unit normal to the hyperplane. Let us call the projection X′, and its position vector y′. Case 1: Central Projection. Let the center of projection be point C, whose position vector is c. Because all projection rays must pass through C, points C, X, and X′ must be collinear. We can therefore write
y ) φy′ + (1 - φ)c
(A3)
Because X′ is on the hyperplane, its position vector must obey eq A1, that is, y′‚k ) 0. Therefore,
φ)1-
y‚k c‚k
(c‚k)y - (y‚k)c (c - y)‚k
T
y′ ) (c k)Iy - (ck )y
(A5)
(A6)
Notice that cTk is a scalar, and the identity matrix is inserted to represent the scalar multiplication in matrix form. Putting eq A6 in the form of eq 8, we obtain the projection matrix P as given by eq 9. From eq A5, we also see that if y ) c, the denominator is zero and the projection cannot be calculated. Consequently, point C does not appear in the projection. Case 2: Parallel Projection. Let w be the unit vector parallel to the direction of projection rays. Because points X and X′ are connected by a projection ray, the relationship between y and y′ can be written as
y′ ) y + λw
y′‚k ) y‚k + λw‚k
(A8)
Again, because X′ is on the hyperplane, y′‚k ) 0 (eq A1). Thus, we get
y‚k w‚k
(A11)
Again, putting this in the form of eq 8, we obtain the projection matrix P as given by eq 11. The projection matrix for a multiple orthogonal projection to an m-dimensional subspace can be derived in the same fashion. We first express the projective space as an intersection of n - m hyperplanes (property L1):
ki‚y ) 0; i ) 1, 2, ..., n - m
(A12)
y′ ) y + λ1k1 + λ2k2 + ... + λn-mkn-m (A13) or using matrix notation
y′ ) y + Kλ
(A14)
where K is defined by eq 13 and
λ ) [λ1 λ2 ‚‚‚ λn-m ]T
(A15)
Next, we premultiply eq A14 with KT to obtain
KTy′ ) KTy + KTKλ
(A16)
Because the projective space is the intersection of the n - m hyperplanes in eq A12, point X′ must also lie on each hyperplane. Therefore, y′‚ki ) 0 ∀ i, or KTy′ ) 0. We can thus write
KTKλ ) -KTy
(A17)
Solving for the λ’s and substituting into eq A14, we arrive at the expression for the projection matrix as given in eq 12.
(A7)
where λ is the distance between X and X′. Taking a dot product with k, we obtain
λ)-
y′ ) Iy - (kkT)y
Because the directions of projection rays follow the directions of the k’s, we write
Note that the denominator of eq A5 is a scalar. Because of property H1, we can multiply the result in eq A5 by this scalar (except when it is equal to zero). We also note that (a‚b)c can be written in matrix form as (cbT)a, where cbT is a matrix whose element in row i and column j is cibj. Thus, we obtain T
(A10)
Putting this in the form of eq 8, we obtain the projection matrix P as given by eq 10. Note that eq A10 has the same form as eq A6, with c replaced by w. The physical interpretation is that a parallel projection is equivalent to a central projection where the center of projection is moved to infinity. Case 3: Orthogonal Projection. In an orthogonal projection, the projection ray is orthogonal to the hyperplane of eq A1. Therefore, we can replace w in eq A7 with k. Through steps similar to those in case 2 and by taking into account that k‚k ) 1, we get
(A4)
Substitution of eq A4 into eq A2 yields
y′ )
y′ ) (wTk)Iy - (wkT)y
(A2)
where φ is a scalar. Taking a dot product with k, we obtain
y‚k ) φy′‚k + (1 - φ)c‚k
expression for y′. After algebraic manipulations similar to those of case 1, we have
(A9)
Equation A9 can be substituted into eq A7 to get an
Notation a ) constant in rule L3c bi ) (bij) ) vector defined by eq 25 C ) number of components D ) (dij) ) coefficient matrix, defined by eq 6 eij ) number of moles of element i in 1 mol of component j ∆Hm,i ) heat of fusion of component i, J‚mol-1 i, j, k ) indices k ) constant in eq 7 ki ) (ki,j) ) unit normal vector to hyperplane i K ) (kij) ) coefficient matrix, defined by eq 13 m, n ) dimension
Ind. Eng. Chem. Res., Vol. 41, No. 9, 2002 2225 m ) number of cations mI ) mass of inert I, kg MI ) molecular weight of inert I, kg‚mol-1 [Mi] ) molality of ion Mi, mol‚kg-1 n ) number of anions ni ) number of mole coordinates Ni ) transformed number of mole coordinates NP ) number of figures for complete representation of the phase diagrams P ) projection matrix qi ) (qi,j) ) normal vector to hyperplane i R ) number of independent reactions R ) universal gas constant, ) 8.314 J‚mol-1‚K-1 R(X) ) ionic transformed coordinate related to species X S ) number of saturated components ti ) parameter T ) temperature, K Tm,i ) melting point of component i, K u ) (ui) ) unit vector V ) (νij) ) matrix of stoichiometric coefficients, defined by eq 52 w ) (wi) ) unit vector parallel to the direction of the projection xi ) mole fraction of component i xi ) inhomogeneous coordinate Xi ) transformed inhomogeneous coordinate y ) (yi) ) position vector Yi ) canonical coordinate zi ) magnitude of the ionic charge Z ) coordinate transformation matrix γi ) activity coefficient of component i R, φ, λ ) constants βij ) number of moles of ion i in 1 mol of salt j ∆ ) matrix defined by eq 19 Λ ) matrix containing basis vectors of projective space νij ) stoichiometric coefficient of component i in reaction j ψ ) (yi) ) vector of homogeneous coordinates ψ′ ) (Yi) ) vector of transformed homogeneous coordinates ξ ) molar extent of reaction Superscripts T ) transpose sat ) saturated Subscripts ref ) reference PC ) Perez Cisneros et al. SN ) Samant and Ng TOT ) total
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Received for review June 14, 2001 Revised manuscript received February 1, 2002 Accepted February 6, 2002 IE010507Y
