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Ind. Eng. Chem. Res. 2004, 43, 7826-7842
Infinite-Dimensional State-Space (IDEAS) Approach to Globally Optimal Design of Distillation Networks Featuring Heat and Power Integration Konstantinos Holiastos and Vasilios Manousiouthakis* Chemical Engineering Department, University of California at Los Angeles, 5531 Boelter Hall, 405 Hilgard Avenue, Box 951361, Los Angeles, California 90095-1361
This work addresses the problem of synthesizing globally optimal distillation networks featuring heat and power integration. The problem is formulated within the infinite-dimensional statespace (IDEAS) conceptual framework, which accounts for all possible network alternatives and yields convex (linear) programming formulations. The method is illustrated on a case study involving the cryogenic separation of a nitrogen/oxygen mixture. 1. Introduction Material separation/purification and chemical reaction are the most common and important operations in the chemical process industry. Although a chemical reaction is not always employed (e.g., in oxygen/nitrogen production from air), a separation step is almost always present. The most important and widely used separation method for miscible liquids is distillation, which exploits vapor pressure differences of the substances to be separated. Many approaches exist for the design of both single distillation columns and networks thereof, featuring sidestreams, side columns, pump-arounds, etc. In one work, Sargent1 presented a general and flexible framework for distillation system design that applies to azeotropic systems. Another common approach2 involves the use of shortcut models for the construction of complex networks. Biegler et al. described many such models.3 Once a design is selected, refinement methods exist to optimize process conditions, e.g., through a parallel genetic algorithm4 where process variables depend at least partly on random processes. Collocation methods have also been successfully used for the simultaneous design and optimization of batch distillation columns.5 Finally, extremely detailed models exist6 for single distillation column analyses once important process parameters have been determined. The bulk of the operating cost of a distillation column is associated with a high (low) temperature heat source (sink) at the reboiler (condenser). Integration of these heat loads within the column itself is impossible because of entropy considerations. Various schemes have been devised to circumvent this problem. Distillation columns integrated with heat exchange networks seek to utilize the discarded heat in another part of the network. For example, Yeomans and Grossmann7 allowed heat integration in distillation processes through a superstructure approach, solved by a mixed integer nonlinear programming model. Bagajewicz and Manousiouthakis8 employed a mass and heat exchange network representation of distillation networks to derive designs featuring improved energy efficiency. Hao-Chieh Cheng and * To whom correspondence should be addressed. E-mail:
[email protected]. Tel.: (310) 206-0300. Fax: (310) 206-4107.
Luyben9 examined alternate distillation configurations with heat integration in mind and demonstrated a 3545% reduction in energy consumption for a benzene/ toluene/m-xylene system. Other often-studied integration approaches for distillation are multieffect designs, which take advantage of the boiling point’s dependence on pressure. The heat released in the condenser can be used elsewhere in the process (as in another reboiler), serving to bypass thermodynamic limitations inherent in single-pressure systems. Wankat10 employed multieffect systems to compare and contrast alternate designs and the conditions under which each is optimal. Agrawal11 presented a method to design multieffect distillation sequences with heat integration in mind. Phimister and Seider12 presented a novel semicontinuous pressure swing distillation system that uses time-dependent pressure changes in a single column to minimize capital cost. Another useful method for heat-integrated distillation design is the use of heat pumps. Heat pumps upgrade heat quality using work. The reduction in operating cost resulting from this heat upgrade can be larger than the expense of mechanical or electrical work input, although exact numbers depend on how the heat pump(s) is (are) integrated into the distillation network. The obvious attractiveness of such systems led a number of investigators to tackle this problem. Holiastos and Manousiouthakis13 gave an analytical expression for the global minimum hot/cold/electric utility cost of a two-stream heat pumping system that finds application on any distillation column. Optimal properties of heat-pumpassisted distillation systems were evaluated by Fonyo and Benko¨14 through analysis of a number of related case studies. Rivera-Ortega et al.15 used a combinatorial method to determine heat pump operating temperature limits for a distillation column, taking into account the fuel-to-cost ratio. Tufano16 stated that, by balancing heat pump and heat transformer action, a match can be achieved with distillation heat load requirements. In a dynamic simulation of a distillation/heat pump system where the heat pump operates between the condenser and reboiler,17 a steady state is observed for many different systems, with a typical heat pump coefficient of performance (COP) of 5-7. Although methods such as the ones above are useful, they do not address the distillation network synthesis
10.1021/ie010434i CCC: $27.50 © 2004 American Chemical Society Published on Web 10/26/2004
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Figure 1. General representation of the IDEAS approach to distillation network design.
problem in its entirety. The infinite-dimensional statespace (IDEAS) approach to process synthesis avoids these problems. IDEAS is a general chemical process network design methodology that gives rise to infinitedimensional convex (linear) optimization problem formulations. It provides the globally minimum cost design over all possible network structures and applies to any type of unit operation, including heat pumps. So far, splitters, mixers, heat exchangers, mass exchangers, turbines, reactors, membrane processes, and distillation columns have been employed within the IDEAS framework; a complete list of references to the aforementioned works is given by Justanieah and Manousiouthakis.18 Two or more unit operations can also be easily combined, as is the case in the present work. An exact description of how or why IDEAS encompasses all design frameworks is deferred to later sections. For now, it suffices to state that it is possible not only to place heat pumps in an optimal manner within the distillation network, but also to design for such nontraditional design characteristics as side draw-offs, recycle streams, intermediate heat exchangers, and multiple feed locations at the same time. Typically, such problems require special modeling efforts; with IDEAS, all such design alternatives are addressed simultaneously. Because mass pinch analysis is no longer applicable to problems involving multicomponent mass exchange (such as distillation) or coupled mass and heat exchange, IDEAS employs a mass exchange network (MEN) representation that explicitly considers mass exchangers to be the building blocks of the MEN operator.19 In this work, distillation networks are represented, within the IDEAS framework, as interacting MENs and HENs (heat exchange networks), where the HEN is modeled using an optimal hot/cold/electric utility cost formulation developed by Holiastos and Manousiouthakis.20 2. Background The general structure of the IDEAS representation of a distillation network is shown in Figure 1, where the distribution network (DN) operator and the operator block (OP) are shown with their interconnections. An infinite number of streams correspond to each stream shown to connect the DN with the OP in Figure 1. External inlet and outlet streams enter and leave the DN operator at the left and top, respectively, and they are assumed to be finite in number. Mixing and splitting takes place in the DN. The mass exchange and heat exchange tasks are captured by distinct MEN and HEN + HEP (heat engine/pump network) operators. In this case, the OP block consists of both operators’ combined action.
Figure 2. IDEAS representation for the distillation with heat pumping design on which the formulation is based.
An alternative IDEAS representation is depicted in Figure 2. In this diagram, the streams entering and leaving the DN are all saturated and of a particular phase (liquid or vapor). A single stream is drawn for clarity. Similarly, a single stream represents all external inlets and outlets, regardless of phase. Symbols and parameters appearing in Figure 2 represent flow rate and property values and are defined in a subsequent subsection. In this representation, the MEN IDEAS operator is modeled by individual mass exchangers, whereas the HEN + HEP operator is modeled by an aggregate heat exchange/heat engine/heat pumping representation. Appendices A and B present a detailed mathematical description of the employed mass exchanger model. Section 3 outlines the mathematical formulation of IDEAS. Finally, section 4 presents an IDEAS design case study, and section 5 draws conclusions from this work. 3. IDEAS Representation 3.1. Definitions. Here, the IDEAS conceptual framework for the synthesis of distillation networks featuring heat and power integration is described. A detailed diagram of the IDEAS representation employed in this work is shown in Figure 2. The external inlet streams are assumed to be finite in number. The ith stream is represented as (u(i), R(i)), where u(i) (the flow rate) and R(i) are the ith elements of infinite sequences u and R, respectively. Because a finite number of external inlet streams are assumed, a position P exists beyond which all elements of u are identically zero (the absence of streams is modeled as a stream of zero flow rate.) Sequence u represents the flow rates of external inlet streams. Because all elements with indices beyond P must have zero value and because each stream is constrained in practice to have a finite flow rate, by definition (see the Nomenclature section), belongs to l1 and more specifically to ΩP ⊂ l1. As the discussion implies, ΩP is the set of sequences in l1 with only their first P entries being possibly nonzero. Sequence R holds in position i information that defines the thermal and compositional state of stream i. Because we consider an isobaric network and an unsaturated stream of NC components, the desired
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Table 1. External Inlet Flow Rates Sequence Pair (u, r): Mathematical Description
Table 2. Internal Flow Rates Sequence Pairs: Mathematical Description
description
symbol
belongs to set
flow sequence
state
flow rate, stream i flow rate sequence state vector, stream i sequence of state vectors
u(i) u ) {u(i)} R(i) R ) {R(i)}
R Ω P ⊂ l1 Q n ⊂ Rn ln∞
w1 ∈ l1 τ1 ∈ ln∞ v2 ∈ l1 φ2 ∈ ln∞ x1 ∈ l1 γ1 ∈ ln∞ x2 ∈ l1 γ2 ∈ ln∞
w1(i) ∈ R τ1(i) ∈ Qn ⊂ Rn v2(i) ∈ R φ2(i) ∈ Qn ⊂ Rn x1(i) ∈ R γ1(i) ∈ Qn ⊂ Rn x2(i) ∈ R γ2(i) ∈ Qn ⊂ Rn
information consists of NC - 1 mole fraction specifications and a specific enthalpy. The enthalpy, the NCth specification, exactly fixes the state of the stream. Each element in R is therefore, itself, a vector of size n ) NC. It is desirable that vector R(i) belong to a countably infinite set for IDEAS to have a standard mathematical representation. It is important that vector R(i) belong to Qn, the set of real and rational n∞ in dimensional vectors; to use the series notation ∑i)1 the IDEAS model, R(i) must belong to a countably infinite set, which Qn is. This is not a limitation, as an element in Qn can always be found to approximate any element in Rn (the set of real n-dimensional vectors) arbitrarily closely. We therefore have R(i) ∈ Qn ⊂ Rn. The sequence R of vectors R(i), on the other hand, must belong to l n∞, as all elements are nonzero and infinite in number. Table 1 summarizes the aforementioned set and element interrelationships. In an entirely similar manner, for the set (y, β), we have that y ∈ ΩK ⊂ l1, where K is the index beyond which all elements of y are zero (K is greater than or equal to the number of external outlets), and β ∈ l n∞ with β(i) ∈ Qn ⊂ Rn. All other system streams, except the external inlets and outlets, are assumed to be saturated. Distillation generally features saturated liquid and vapor streams inside the columns, so this simplification is justified. In any case, this restriction can be relaxed, albeit at increased computational effort. To maintain a saturated state during mixing in the DN, some heat inflow or outflow is permitted at these junctions. It should be noted here that this heat is taken into account thermodynamically in the pinch model. As a result of the saturated state of these streams, sequence pairs (u, R′), (y, β′), (w1, τ1), (v2, φ2), (x1, γ1), and (x2, γ2) differ from (u, R) and (y, β) in the following two respects: (i) Consider sequence pair (u, R′) as an example. As the streams exit the HEN + HEP operator, their flow rates are unchanged; however, the new state sequence R′ now contains only saturated stream information. Sequence R′ (along with β′, τ1, φ2, γ1, and γ2) now belongs to a strict subset of l n∞, with R′(i) ∈ Qn ⊂ Rn. (ii) Unlike u and y, sequences w1, v2, x1, and x2 are not restricted to have finite nonzero elements. IDEAS allows all possible internal configurations to ensure the global minimum. It does not a priori disallow the existence of any state, as any state might be integral to the optimal structure. (It is only the problem statement that places strict limitations on the external inlet and outlet stream states.) The state sequences τ1, φ2, γ1, and γ2 then simply belong to l n1 . Table 2 summarizes the aforementioned discussion. Sequence pair (w2, τ2) is different from the pairs just described; it contains information on (i) the design specifications for the mass exchangers comprising the MEN operator and (ii) the streams entering these mass exchangers. In a sense, the elements of sequence pair (w2, τ2) specify the state of a mass exchanger rather than the state of a single stream. To accomplish this, the
following information must be contained in each element (w2(i), τ2(i)): (iii) For a mass exchanger that accepts s inputs, there must be s stream state specifications of the type (u(i), R(i)). (iv) Consider a mass exchanger design with no degrees of freedom. For a given set of inputs, various outputs might exist depending on the exchanger design (degree of freedom). For a given set of input streams, IDEAS incorporates all possible outputs by specifying all possible (infinite in number) degrees of freedom and generating an infinite number of exchangers. An nodimensional vector D denotes the aforementioned degrees of freedom and is part of sequence pair (w2, τ2), as is shown below (v) Specification of the no degrees of freedom and s input stream states does not fix the exchanger operation; the actual inlet and outlet flow rates can vary, but only in constant proportion to one another. At this point, the IDEAS model can be formulated in two ways: (i) all flow rates can be expressly given as a multiple of one flow rate (an optimization variable) or (ii) all flow rates can be denoted with separate symbols (indices) but must be constant multiples of one another; this condition is enforced via appropriate constraints. For notational purposes, the second option is chosen. The matrix R contains relative flow rate information. Its use in constraints is discussed in a future subsection. Formally, the special sequence pair (w2(i), τ2(i)) is defined as follows
w2(i) ) [w12(i) w22(i) w32(i) ... ws2(i)]T
(1)
As a reminder, s is the number of input streams for the ith mass exchanger. Element w2(i) ∈ Rs, and w2 ∈ l s1. Corresponding to w2 is τ2, the sequence
{τ2(i)} ) {δ1(i),δ2(i), ..., δs(i), R(i), D(i)}
(2)
where δj(i) is a vector of size n and is the state vector for the jth input stream of the ith mass exchanger, R(i) ∈ Q(s-1)×s defines a subspace of l s1 to which every w2(i) is restricted (to ensure that all inlet flows are constant multiples of one another), and D(i) ∈ Qno contains all design parameters for the mass exchange units (0 for a simple tray, 1 for a multiplate mass exchanger, etc.) A pair (w2(i), τ2(i)) then uniquely defines the exchanger design, inlet (and outlet) compositions, and relative flow rates. More information on how the various sequence pairs are interrelated is provided in the IDEAS constraints description; such a discussion is better placed there, rather than in the present definition subsection. It will be useful in future discussions of the HEN + HEP operator to distinguish hot and cold streams. It is obvious that a stream in this model is either hot or cold. Consider first sequence pair (w1, τ1); without loss of
Ind. Eng. Chem. Res., Vol. 43, No. 24, 2004 7829 Table 3. Definition of Hot and Cold Subsequences for Stream Sets Entering and Leaving the HEN + HEP Operator sequence
hot and cold subsequences
u y w1 x1 R R′ β β′ τ1 τ′1 γ1 γ′1
{uH, uC} {yH,yC} C {wH 1 , w1 } C {xH ,x } 1 1 {RH, RC} {R′H, R′C} {βH, βC} {β′H, β′C} C {τH 1 , τ1 } {τ′1H, τ′1C} C {γH 1 , γ1 } {γ′1H,γ′1C}
generality, the hot streams can be placed at the start of the sequence. The hot streams are then represented H by the subsequence wH 1 ) {w1 (i)} ⊂ w1, and similarly, the cold streams are represented by the subsequence wC1 ) {wC1 (i)} ⊂ w1. It is then clear that all of the following hold
w1 )
{wH 1,
wC1 }
outlet stream temperatures, respectively. Similar expressions exist for the sequence pairs (u, R) and (y, β′) but are not explicitly written. Each of these expressions contributes to the temperature interval diagram, formed by the union of {Ta(R(j))}, {Ta(β′(j))}, {Ta(τ1(j))}, {Ta(R′(j))}, {Ta(β(j))}, and {Tb(γ1(j))} for all j ∈ {1, ..., ∞}, which is next rank ordered by size to form an l∞ sequence{T′(i)} that defines the temperature interval edges. In other words, {T′(i)} contains all temperatures in the DN/HEN + HEP/MEN IDEAS model, ordered numerically. If the model contains multiple streams with the same temperature, this temperature appears only once in{T′(i)}. The above definitions pave the way for specification of a set of problem parameters that aid in the correct accounting of energy in the HEN + HEP operator. Each such parameter is assigned to a temperature interval for each stream and has the value of 1 if the stream exists in the interval and 0 if the stream does not exist in the interval λH(j,i) )
{
H 1 if Ta(τH 1 (j)) g T′(i) and Ta(γ1 (j)) e T′(i+1) ∀j ∈ {1, ..., ∞} ∀i ∈ {1, ..., ∞} 0 otherwise
(3)
C {w1(i)} ) {{wH 1 (i)}, {w1 (i)}}
(8) (4)
λC(j,i) )
{
The elements of τ1, which form pairs with those of w1, then can be grouped and expressed as follows
τ1 )
{τH 1,
τC1 }
1 if Ta(γC1 (j)) g T′(i) and Ta(τC1 (j)) e T′(i+1) ∀j ∈ {1, ..., ∞} ∀i ∈ {1, ..., ∞} 0 otherwise
(5)
C {τ1(i)} ) {{τH 1 (i)}, {τ1 (i)}}
(6)
In analogy to (w1, τ1), similar sets are defined for other sequence pairs entering or leaving the HEN + HEP operator. Table 3 summarizes this notation; it uses only the simpler relations of eqs 3 and 5. The HEN + HEP operator utilizes the heat and power integration model presented in ref 20, which requires a temperature interval diagram; alternate power integration models exist,21-23 but the one chosen has the advantage of growing in size linearly with the number of temperature intervals, without committing a priori to a predetermined network structure. Process temperatures can be identified using information (e.g., composition and enthalpy) contained in any of the vector subsequences. Let these temperatures be uniquely defined by a map Ta(‚) such as the following (for subsequence τ1)
(9) H
κ (j,i) )
{
1 if Ta(RH(j)) g T′(i) and Ta(R′H(j)) e T′(i+1) ∀j ∈ {1, ..., ∞} ∀i ∈ {1, ..., ∞} 0 otherwise
(10) κC(j,i) )
{
1 if Ta(R′C(j)) g T′(i) and Ta(RC(j)) e T′(i+1) ∀j ∈ {1, ..., ∞} ∀i ∈ {1, ..., ∞} 0 otherwise
(11) µH(j,i) )
{
1 if Ta(β′H(j)) g T′(i) and Ta(βH(j)) e T′(i+1) ∀j ∈ {1, ..., ∞} ∀i ∈ {1, ..., ∞} 0 otherwise
(12) C
Ta(‚)
τ1(j) 98 Ta(τ1(j))
(7)
The function Ta(‚) determines the temperature of each stream as it enters the HEN + HEP block. One can then define inlet and outlet temperatures for each stream j by using the appropriate subsequence element as an argument for Ta(‚). For example, in, say, {w1, τ1} H H H (j) ) Ta(τH Tw 1 (j)) Tw1,out(j) ) Ta(γ1 (j)) 1,in
∀j ∈ {1, ..., ∞} C C Tw (j) ) Ta(τC1 (j)) Tw (j) ) Ta(γC1 (j)) 1,in 1,out
∀j ∈ {1, ..., ∞} Here, the subscripts in and out denote the inlet and
µ (j,i) )
{
1 if Ta(βC(j)) g T′(i) and Ta(β′C(j)) e T′(i+1) ∀j ∈ {1, ..., ∞} ∀i ∈ {1, ..., ∞} 0 otherwise
(13) These parameters are extensively used in the formulation that follows; two additional types of variables must be defined first, however. All internal DN flows with the external outlet, the HEN + HEP operator, or the MEN operator as their destination, labeled henceforth zf,g(i,j), represent the flow from the junction associated with g(j) to the junction associated with f(i). For example, a flow originating from the external inlet j that feeds MEN stream i would be denoted zv2,u(i,j). These are flows internal to the DN that realize the
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Figure 3. Temperature-enthalpy plot for a saturated vapor nitrogen/oxygen mixture.
to severely limit the design, and heat input or removal should be made available. It should be emphasized that these point heat loads are integrated into the HEN + HEP operator at the temperature of the mixing junction to which they correspond. Point heat load quantities appear next in the DN and HEN + HEP models, along with an explanation of the notation involved. 3.3. DN Operator Model. The DN model consists of eqs 14-28 below. The reader is encouraged to refer to Figure 2 when analyzing these and other equations that follow. Equations 14-16 are the overall mass balances on the splitting operations of the DN inlet streams in sets u, x1, and x2, respectively. Each DN inlet stream k ∈ {1, ..., ∞} is split into potentially infinite streams, each of which feeds DN exit streams in sets y, w1, and v2. ∞
mixing and splitting that comprise operator inlets and outlets, respectively. Subsequently, the sequence {zf,g(i,j)} belongs to l1(2). A final set of variables arises from the nature of the heat and power integration model in ref 20. For each interval i, every flow j (whether hot or cold) is free to split in the HEN + HEP operator. Each fraction participates in the HEN or the HEP subnetwork that belongs to the HEN + HEP operator. Depending on the hot/cold nature of the stream and the type of network in which it participates, the variables representing these flows are as follows: uH,HEN(i,j), uC,HEN(i,j), yH,HEN(i,j), (i,j), and wC,HEN (i,j) for the HEN and yC,HEN(i,j), wH,HEN 1 1 (i,j), uH,HEP(i,j), uC,HEP(i,j), yH,HEP(i,j), yC,HEP(i,j), wH,HEP 1 and wC,HEP (i,j) for the HEP. The superscripts are self1 explanatory based on the prior discussion. Index j denotes the stream, and i denotes the stream interval. Whereas this discussion introduces the variables, a subsequent subsection gives the HEN + HEP model and the stream split equations. 3.2. Point Heat Loads. The assumptions made in this IDEAS design methodology necessitate the presence of point heat loads (hot and cold) at each mixing junction forming streams in (w1, τ1), (v2, φ2), and (y, β′). Only in this manner can the energy balance be satisfied while the streams are simultaneously kept in a saturated state. To understand why such a requirement exists, consider Figure 3, which is a plot of the saturated vapor enthalpy as a function of temperature for the distillation example to follow. This is a nitrogen/oxygen mixture, and temperature variations in the saturated state reflect changes in composition. Suppose that two saturated vapor streams (A and B in Figure 3) mix to form stream C. By the lever rule (which is a manifestation of the mass and energy balances), the thermal identity of the product stream C lies somewhere on the line AB, as shown in the figure. If the saturation curve is convex (or concave), point C cannot lie on the curve, that is, the state of the mixed stream cannot be saturated. Generalizing this argument to mixtures of more than two streams, the final stream will again not fall on the curve because multistream mixing can be modeled as a series of binary stream mixes. For the case of Figure 3, the mixed stream is superheated. Heat must be removed to bring it to a saturated state. If the saturation enthalpy curve is not convex or concave, it is possible that, for specific flow rates and compositions, the final stream is saturated. However, such conditions are likely
∞
∞
zyu(i,k) + ∑ zw u(i,k) + ∑ zv u(i,k) ∑ i)1 i)1 i)1
u(k) )
1
2
∀k ∈ {1, ..., ∞} (14)
∞
x1(k) )
∞
zyx (i,k) + ∑ zw x ∑ i)1 i)1 1
1 1
∞
x2(k) )
∑ i)1
∞
∑ i)1
2 1
∀k ∈ {1, ..., ∞} (15)
∞
zyx2(i,k) +
zv x (i,k) ∑ i)1
(i,k) +
∞
zw1x2(i,k) +
zv x (i,k) ∑ i)1 2 2
∀k ∈ {1, ..., ∞} (16)
Equations 17-19 are total mass balances for the mixing of internal DN streams to compose the DN exit streams found in sets y, w1, and v2. These streams are formed by mixing potentially infinite steams from sets u, x1, and x2. ∞
y(k) )
∑ i)1
∞
zyu(k,i) +
∑ i)1
∞
w1(k) )
∑ i)1 ∑ i)1
zyx (k,i) ∑ i)1 2
∀k ∈ {1, ..., ∞} (17)
∞
zw1u(k,i) +
∞
v2(k) )
∞
zyx1(k,i) +
∑ i)1
∞
zw1x1(k,i) +
∑ i)1
1 2
∀k ∈ {1, ..., ∞} (18)
∞
zv2u(k,i) +
zw x (k,i) ∑ i)1
∞
zv2x1(k,i) +
zv x (k,i) ∑ i)1 2 2
∀k ∈ {1, ..., ∞}, ∀l ∈ {1, ..., s} (19)
The component and energy balances (applicable only for mixing junctions) are succinctly written below by use of the property vectors R(i), γ1(i), and γ2(i). As stated earlier, these vectors are assumed to contain the NC 1 mole fractions in the first NC - 1 positions and the specific enthalpy in the last position n ) NC. Any other information that might potentially be necessary for the particular IDEAS model can be incorporated here. For example, the entropy information is not needed for most implementations, but it is necessary in the current heat and power integration model. Mixing operations often require heat input or removal before the final stream’s enthalpy satisfies saturation conditions. Such heats are supplied by or given to the HEN + HEP operator and are integrated into the network. From a heat integration point of view, a heat
Ind. Eng. Chem. Res., Vol. 43, No. 24, 2004 7831
input to the HEN + HEP subnetwork (hot load) results from heat removal from a mixing junction in the DN. This is why, in the energy balances, hot loads are subtracted from the mixing junction and cold loads are added to it. Equations 26-28 below are vector equations, so the heat terms appear as appropriately placed elements in a vector q that otherwise consists of zero entries. Vector q then participates in the balance equations. It will be more important in future equations to label a heat load according not to the junction at which it occurs, but rather to the temperature interval. This approach is useful for developing the HEN + HEP operator model. Because mixing balances require heat loads based on junction identities, the following correlation relates heat loads in junction k to (hot and cold) heat loads in interval i. The definitions below set all elements of vector q individually, although only the position n is nonzero. Here, the mapping Ta(‚) again determines junction temperatures that are then correlated to the position in the temperature interval list. Variables q are then used in mixing and splitting balances, whereas variables Q are used in cascade equations and any equations based on the temperature interval model.
qCy,p(k)
{
}
{
}
(21)
{
}
(22)
{
}
(23)
{
}
0 if p * n ) QCy (i), i: Ta(β′C(k)) ) T′(i) if p ) n
0 if p * n C (k) ) QC (i), i: T (γC(k)) ) T′(i) qw 1,p if p ) n w1 a 1 0 if p * n qvC2,p(k) ) QC (i), i: T (φC(k)) ) T′(i) if p ) n v2 a 2
qH y,p(k) )
0 if p * n H QH (i), i: T (β′ (k)) ) T′(i) if p ) n y a
0 if p * n H (k) ) QH (i), i: T (γH(k)) ) T′(i) qw 1,p if p ) n w1 a 1
{
(20)
(24)
}
0 if p * n qvH2,p(k) ) QH (i), i: T (φH(k)) ) T′(i) if p ) n v2 a 2 (25) Equations 26-28 are vector equations that enforce the composition and energy balances. Let the pth element of a(i) be denoted by ap(i), and similarly for γ1(i) and γ2(i). Then ∞
β′p(k) y(k) )
∞
Rp(i) zyu(k,i) + ∑γ1,p(i) zyx (k,i) + ∑ i)1 i)1 1
∞
γ2,p(i) zyx (k,i) + qCy,p(k) - qH ∑ y,p(k) i)1 2
∀k ∈ {1, ..., ∞ }, ∀p ∈ {1, ..., n}
(26)
∞
∞
Rp(i) zw u(k,i) + ∑γ1,p(i) zw x (k,i) + ∑ i)1 i)1
τ1,p(k) w1(k) )
1
1 1
∞
C H γ2,p(i) zw x (k,i) + qw ∑ ,p(k) - qw ,p(k) i)1 1 2
1
1
∀k ∈ {1, ..., ∞ }, ∀p ∈ {1, ..., n} ∞
φ2,p(k) v2(k) )
(27)
∞
Rp(i) zv u(k,i) + ∑γ1,p(i) zv x (k,i) + ∑ i)1 i)1 2
2 1
∞
γ2,p(i) zv x (k,i) + qvC ,p(k) - qvH ,p(k) ∑ i)1 2 2
2
2
∀k ∈ {1, ..., ∞ }, ∀p ∈ {1, ..., n}
(28)
3.4. MEN Operator Model. This operator model consists of three equations. Equation 29 forces the s elements of w2(i), ∀j ∈ {1, ..., s}, to be (constant) multiples of one another.
0 ) R(i) w2(i)
∀i ∈ {1, ..., ∞}
(29)
The parameter R(i) specifies, according to the mass exchanger model and specified degrees of freedom D(i), the flow rates of all s - 1 inlets with respect to one of them, say, w12(i). The need for eq 29 was discussed during the introduction of sequence w2. Equation 30 models stream splitting operations from the states (v2, φ2) to the exchanger inlets (w2, τ2). When a state i is allowed to feed an exchanger j (or, equivalently, when exchanger j’s input is of the same composition and phase as the input of state i under consideration) the coefficient G2(i,j) is 1; otherwise it is 0. It is clear that the structure of the two-dimensional infinite matrix G2 depends on how both states i and exchangers j are ordered. Matrix G2 consists of 0’s and 1’s because w2(j), by definition, contains inlet flow rates. A similar equation written for the outlets, eq 31, consists of zero and nonzero entries because now the outlet flow rates are expressed as (constant) multiples of the inlets once the exchanger model is solved. Because a state can potentially feed a number of exchangers, splitting must necessarily occur. This splitting is diagrammatically shown as part of the MEN operator in Figure 2, although it could easily have been incorporated into the DN. ∞
v2(i) )
G2(i, j) w2(j) ∑ j)1
∀i ∈ {1, ..., ∞}
(30)
Equation 30 above does not involve vector multiplication; it is rather a sum of s-dimensional vectors w2, as G2(i,j) is merely a number (0 or 1). All like exchanger outlets having equal phase and composition are mixed to form a single stream corresponding to a particular state. It is these streams that exit the MEN operator, represented by sequence pairs (x2, γ2). All information about the states of the exiting exchanger streams, as well as the junction to which each will be directed before exiting the MEN, is embedded in matrix F2. As discussed in the Definitions subsection, specification of all mass exchanger degrees of freedom D, along
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with full knowledge of the exchanger inputs, means that exchanger outlet flow rates and compositions can be calculated independently. The output flow rates again relate to the single inlet flow rate to which the s - 1 inlet flow rates are also referred. (See eq 29.) If exchanger j’s outlet does not flow to state i, F2(i,j) ) 0. If exchanger j’s outlet enters state i, then F2(i,j) is a nonzero number. Matrix F2 combines, for exchanger outlets, the tasks performed by both G2 and R for exchanger inlets. ∞
x2(i) )
F2(i, j) w2(j) ∑ j)1
∀i ∈ {1, ..., ∞}
(31)
3.5. HEN + HEP Operator Model. The HEN + HEP operator is closely modeled on the author’s previous work on heat and power integration. An introduction to the concepts and modeling technique follows here. There are advantages in using heat pumps in heat exchange networks: they transfer heat against the temperature gradient, and they can reduce or eliminate the effect of the pinch pointsit is the pinch that primarily determines the overall heat use and efficiency of the network. Heat engines, on the other hand, can generate useful work and make the best use of a finite temperature difference between process streams. The model separates the actions of heat exchangers and heat engines/pumps by defining a heat exchange network (HEN) and a heat engine and pump network (HEP) that are optimally joined to satisfy process constraints. In general, a stream can participate in both networks. It is therefore split in two; this split is, in general, temperature-dependent, in the sense that each temperature interval can have a different split. Constraints ensure, however, that the sum is constant and equal to the stream specifications for all intervals. The HEN network uses pinch analysis, whereas the HEP network uses the authors’ novel formulation. The mathematical equivalence of heat pumps and heat engines allows a single second law statement for the whole HEP network. Similarly to a pinch diagram, the HEP model provides an optimal diagram that shows heat engine and pump operation. The net work generated or consumed is constant (at the optimum value), and the diagram points out all potential stream matches for heat pump and engine operation. For further details on this method, as well as background material on heat pumping, the reader is referred to ref 20 and, as an introduction, to ref 13. An HEN + HEP operator inlet stream splits (to participate in the HEN and HEP networks) and can recombine with itself only when the two segments have the same enthalpy (the composition does not change). The stream must be reconstituted before it leaves the operator and does not mix with other operator streams. As a result, all flow rates in to and out of the HEN + HEP operator remain constant. The ith HEN + HEP operator outlet will have the same flow rate as the ith inlet; this constraint translates into a mass balance for the operator
x1(i) ) w1(i)
∀i ∈ {1, ..., ∞}
(32)
The aforementioned description of the internal mixing/splitting operations necessitate the constraints in eqs 33-38 below. In conjunction with eq 32, these constraints imply that flow rates do not change across
the operator and specify the overall mass balance for the mixing or splitting operations. Equations must be written for both hot and cold streams. Variables with the superscripts HEN and HEP were defined earlier. H,HEN wH (j,i) + wH,HEP (j,i) 1 (j) ) w1 1 ∀i ∈ {1, ..., ∞}, ∀j ∈ {1, ..., ∞} (33)
(j,i) + wC,HEP (j,i) wC1 (j) ) wC,HEN 1 1 ∀i ∈ {1, ..., ∞}, ∀j ∈ {1, ..., ∞} (34) uH(j) ) uH,HEN(j,i) + uH,HEP(j,i) ∀i ∈ {1, ..., ∞}, ∀j ∈ {1, ..., ∞} (35) uC(j) ) uC,HEN(j,i) + uC,HEP(j,i) ∀i ∈ {1, ..., ∞}, ∀j ∈ {1, ..., ∞} (36) yH(j) ) yH,HEN(j,i) + yH,HEP(j,i) ∀i ∈ {1, ..., ∞}, ∀j ∈ {1, ..., ∞} (37) yC(j) ) yC,HEN(j,i) + yC,HEP(j,i) ∀i ∈ {1, ..., ∞}, ∀j ∈ {1, ..., ∞} (38) Next, the energy balance over each temperature interval is provided. The cascade variables are written as ∆i, and the heat input and output for mixing junctions is reflected by the quantities QH(i) and QC(i) respectively, introduced in eqs 20-25. Heat input to the operator (hot loads, heat removed from mixing junctions) increases the available heat, whereas heat removed from the operator (cold loads, heat added to mixing junctions) decreases the available heat. The index i denotes the temperature interval with which the heats are associated. By contrast, quantities Qhot and Qcold are the external utility consumption variables. We have that ∆1 ) Qhot. H H ∆i + QH y (i) + Qw1(i) + Qv2(i) + ∞
H H,HEN λH(j,i)[h(τH (j,i) + ∑ 1 (j),T(i)) - h(γ1 (j),T(i))]w1 j)1 ∞
κH(j,i)[h(RH(j),T(i)) - h(R′H(j),T(i))]uH,HEN(j,i) + ∑ j)1 ∞
µH(j,i)[h(β′H(j),T(i)) - h(βH(j),T(i))]yH,HEN(j,i) ) ∑ j)1 ∞
λC(j,i)[h(γC1 (j),T(i)) - h(τC1 (j),T(i))]wC,HEN (j,i) + ∑ 1 j)1 ∞
κC(j,i)[h(R′C(j),T(i)) - h(RC(j),T(i))]uC,HEN(j,i) + ∑ j)1 ∞
µC(j,i)[h(βC(j),T(i)) - h(β′C(j),T(i))]yC,HEN(j,i) + ∑ j)1 C (i) + QvC2(i) + ∆i+1 QCy (i) + Qw 1
∀i ∈ {1, ∞} (39)
The quantity h(ζ(j),T(i)) gives the enthalpy of stream j in subsequence ζ at temperature T(i). An equation to obtain this quantity is eq 68, although any more
Ind. Eng. Chem. Res., Vol. 43, No. 24, 2004 7833
complex model can be used without affecting the optimal nature of IDEAS. This is true because all enthalpy calculations occur independently of optimization calculations. If the stream does not exist in temperature interval i [T(i) is greater than the stream’s dew point or smaller than its bubble point], the corresponding binary variable in eq 39 will be 0. The cascade equations above use the enthalpy function because the saturated enthalpy values ζn(j) correspond to bubble- and dewpoint temperatures and not to other temperatures in the interval list. Whereas Qhot is clearly ∆1, Qcold, the cold (external) utility consumption, is not given by a cascade variable. An overall balance expresses the desired quantity ∞
Qcold )
∞
Rn(j) u(j) - ∑βn(j) y(j) + Qhot + W ∑ j)1 j)1
(40)
Equation 40 is an energy balance over the whole system (distillation column), not just the HEN + HEP operator or the HEP subnetwork. The quantity W is the net system work, which is defined later. Turning again to the HEN subnetwork, the second law for the pinch states that all available heat is positive, or that heat is transferred from hot to cold streams only; this fact is simply given by
∀i ∈ {1, ∞}
∆i g 0
∞
∑ i)1
∞
QH y (i) +
∑ i)1
QvH (i) + ∑ i)1
∞ ∞
κH(j,i)[s(RH(j),T(i)) - s(R′(j),T(i))]uH,HEP(j,i) + ∑ ∑ i)1 j)1 ∞ ∞
µH(j,i)[s(β′H(j),T(i)) - s(βH(j),T(i))]yH,HEP(j,i) ∑ ∑ i)1 j)1 ∞ ∞
λC(j,i)[s(γC1 (j),T(i)) - s(τC1 (j),T(i))]wC,HEP (j,i) ∑ ∑ 1 i)1 j)1 ∞ ∞
κC(j,i)[s(R′C(j),T(i)) - s(RC(j),T(i))]uC,HEP(j,i) ∑ ∑ i)1 j)1 ∞ ∞
µC(j,i)[s(βC(j),T(i)) - s(β′C(j),T(i))]yC,HEP(j,i) ) 0 ∑ ∑ i)1 j)1
(43)
The final expression gives the value of the work W generated/consumed for the HEP subnetwork and the distillation network as a whole. ∞ ∞
W)
2
(j,i) + ∑ ∑λH(j,i)[h(τH1 (j),T(i)) - h(γH1 (j),T(i))]wH,HEN 1 i)1 j)1 ∞ ∞
∑ ∑κH(j,i)[h(RH(j),T(i)) - h(R′H(j),T(i))]uH,HEN(j,i) + i)1 j)1 ∞ ∞
∑ ∑µ i)1 j)1
H H,HEP λH(j,i)[s(τH (j,i) + ∑ ∑ 1 (j),T(i)) - s(γ1 (j),T(i))]w1 i)1 j)1
∞
H Qw (i) + 1
∞ ∞
H
∞ ∞
(41)
The first law over the same HEN subnetwork is an energy balance
Qhot +
thermodynamic model. An example is eq 72; thermodynamic model complexity does not matter because entropy is independently calculated. In the following equation, heat loads do not appear because they enter only the HEN subnetwork.
H
H
(j,i)[h(β′ (j),T(i)) - h(β (j),T(i))]y
H,HEN
(j,i) -
∑ ∑λH(j,i)[h(τH1 (j),T(i)) i)1 j)1 ∞ ∞
κH(j,i)[h(RH(j),T(i)) ∑ ∑ i)1 j)1 ∞ ∞
∞ ∞
∞ ∞
∑ ∑κC(j,i)[h(R′C(j),T(i)) - h(RC(j),T(i))]uC,HEN(j,i) i)1 j)1
∞
∞
∞
C QCy (i) - ∑Qw (i) - ∑QvC (i) - Qcold ) 0 ∑ i)1 i)1 i)1
h(βH(j),T(i))]yH,HEP(j,i) -
λC(j,i)[h(γC1 (j),T(i)) ∑ ∑ i)1 j)1 ∞ ∞
(j,i) h(τC1 (j),T(i))]wC,HEP 1
κC(j,i)[h(R′C(j),T(i)) ∑ ∑ i)1 j)1
∞ ∞
∑ ∑µC(j,i)[h(βC(j),T(i)) - h(β′C(j),T(i))]yC,HEN(j,i) i)1 j)1
h(R′H(j),T(i))]uH,HEP(j,i) +
µH(j,i)[h(β′H(j),T(i)) ∑ ∑ i)1 j)1 ∞ ∞
(j,i) ∑ ∑λC(j,i)[h(γC1 (j),T(i)) - h(τC1 (j),T(i))]wC,HEN 1 i)1 j)1
H,HEP (j,i) + h(γH 1 (j),T(i))]w1
∞ ∞
h(RC(j),T(i))]uC,HEP(j,i) -
µC(j,i)[h(βC(j),T(i)) ∑ ∑ i)1 j)1
(42)
h(β′C(j),T(i))]yC,HEP(j,i) (44)
where, again, the enthalpy function h is given by any appropriate equation. The heat and power integration model requires an entropy balance for the HEP subnetwork to ensure feasibility of the heat engine and pump operation. The expression for entropy s (the symbol is easily distinguished from the number of mass exchanger input streams by the context) is given by any appropriate
Variable bounds for the whole model are as follows
1
2
u g 0, y g 0, w1 g 0, v2 g 0, x1 g 0, x2 g 0, z g 0, Qhot g 0, Qcold g 0, ∆ g 0, QH y g 0, H C g 0, Qw g 0, QvH2 g 0, QvC2 g 0 QCy g 0, Qw 1 1
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Figure 4. Oxygen/nitrogen separation specifications for the distillation example.
3.6. Objective. The objective of this optimizationbased IDEAS model is the total operating process cost, which takes the form
inf
C H u,y,x1,x2,w1,w2,z,QH y ,Qy ,Qw1
c1Qhot + c2Qcold + c3W (45)
parameter
value
feed N2 vapor feed mole fraction N2 vapor outlet mole fraction N2 liquid outlet mole fraction pressure (Pa) RN2/O2 AO2 (first Antoine coefficient) BO2 (second Antoine coefficient) CO2 (third Antoine coefficient) heat capacity of liquid N2 heat capacity of vapor N2 heat capacity of liquid O2 heat capacity of vapor O2 λN2 λO2 Tref hot utility cost CH cold utility cost CC work utility cost CW
10 kg‚mol/s 0.7 0.9 0.1 101130 4.173 8.82082 319.013 -6.45 57.43 kJ/(kg‚mol K) 29.26 kJ/(kg‚mol K) 53.12 kJ/(kg‚mol K) 29.66 kJ/(kg‚mol K) 6200 kJ/kg‚mol 6200 kJ/kg‚mol 70 K .105 $/kJ 1.030 $/kJ 2.070 $/kJ
C ,QH ,QC ,Q ,Q Qw ,W,∆ 1 v2 v2 hot cold
3.7. Local Mixing Formulation. All streams entering the DN can, in general, split to feed any or all streams exiting the DN. Furthermore, all mixing junctions (junctions where mixing occurs, i.e., at the righthand side and top of the DN) must satisfy mass, enthalpy, and composition balances. Such an arrangement is called “global mixing” because no restrictions exist on which internal DN streams mix to form the outlet junctions. A modified “local mixing” formulation can also be used. In local mixing, one is guided by a well-known principle in process design stating that streams of widely different properties (temperature in a heat integration design or composition in a mass integration system) should not be mixed in the interest of reducing thermodynamic irreversibility. For example, mixing a high-temperature stream with a low-temperature stream to produce an intermediate-temperature stream is thermodynamically inefficient, because the two original streams could have been utilized as hot and cold streams, respectively, bringing their final temperatures closer to the eventual intermediate one. Advantage should be taken of the useful temperature extreme of each stream. In fact, heat pumping essentially provides a manner to do just that, in conjunction with heat exchangers. It is for this reason that heat pumping can produce a network with a lower process cost. A formulation employing local mixing, therefore, allows mixing only of streams similar in composition and enthalpy to provide for thermodynamically efficient process operations. 4. Case Study This case study focuses on the separation task depicted in Figure 4, which is the cryogenic separation of a 70/30% nitrogen/oxygen mixture using distillation. This external inlet stream is to be separated into two outlet streams of known and consistent flow rates and compositions. Data for the considered case are given in Table 4. Because the model assumes a minimum approach temperature of 0 °C, the optimum solution is driven to feature mixing of streams with similar compositions (and therefore similar temperatures) to improve thermodynamic efficiency. It is for this reason that a local mixing formulation is used in this example.
The present IDEAS model can accommodate any mass exchanger model (exhibiting various degrees of freedom), as well as any thermodynamic stream property model, from heat-capacity-based to UNIFAC. The individual mass exchanger solution is computed independently of the IDEAS optimization and enters through matrices R, G2, F2, and D. Mass exchange and thermodynamic calculations can be highly nonlinear, but the IDEAS formulation remains linear. 4.1. Mass Exchanger Model. The mass exchanger model in eqs 48-53 is employed. Because this is a twoout out component system and yout 1 is specified, y2 ) 1 - y1 , out out and x1 and x2 derive from the definition of Rij in the following way
xout 2
)
yout 2 out R12(1 - yout 2 ) + y2
out xout 1 ) 1 - x2
(46) (47)
Turning to the mass exchanger model, all coefficients for the total, component, and enthalpy balances (eqs 48, 50, and 52, respectively) have been defined. A threeequation, three-variable linear system remains to provide the unknown flow rates Lin/Vin, Lout/Vin, and Vout/ Vin. For this two-component problem and a fixed yout 1 , no iterations are necessary. Outlet temperatures and enthalpies are calculated according to eqs 63, 64 and 65-67, respectively. 4.2. McCabe-Thiele Design. A typical distillation column design is first made using the McCabe-Thiele method, with the assumption of constant molal overflow. For the data in Table 4, the hot and cold utilities necessary are 11.74 and 27.24 MJ/s, respectively. The total operating cost for this system is $29,300/s. A pinch (HEN) diagram can be drawn for this system; there is no heat integration possible, however. The condenser, from which heat is removed, has a low temperature because the low-boiling (more volatile) product reaches the top of the column. The reboiler, which requires heat input, is at a higher temperature because the highboiling product reaches the bottom of the column. The low-temperature heat produced by the condenser cannot be used by the reboiler.
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Figure 5. McCabe-Thiele distillation design and its associated pinch diagram.
The McCabe-Thiele design and a corresponding pinch diagram are shown in Figure 5. Several important parameters can be seen in both the equipment design and the pinch diagram. For the latter, there is a vertical pinch region spanning the temperature range 80.4-87.5 K. The cost for the same McCabe-Thiele design is smaller when a heat pump operates between the condenser and reboiler. There is temperature change (from the bubble point to the dew point and vice versa) with an associated energy change (input/output) for the condenser and reboiler units. This defines a heat capacity, and both the condenser and reboiler are modeled as (finite heat capacity) streams. The analytical expression that gives optimum heat pumping conditions applies here, as presented by the authors in ref 13. Given the problem specifications, minimum operating cost necessitates that all of the cold curve (the reboiler heat) must be supplied by heat pumping. This partly satisfies the heat removal requirement on the condenser, but still, 16.60 MJ/s of cold utility must be consumed. The work requirement for this process is 1.10 MJ/s. There is no hot utility requirement. The total operating cost for this optimal heat-pump-integrated McCabe-Thiele design is $19,400/s, which represents a cost reduction of 34%. Figure 6 shows the unit operation structure for the new design, which features a heat pump, no reboiler, and a reduced condenser utility consumption. Below is a temperature-enthalpy diagram showing in dashed lines the part of the network that is heat pumped and in solid lines the part that
Figure 6. McCabe-Thiele distillation design with optimal heat pump integration and its associated temperature-enthalpy diagram.
Figure 7. Minimum operating cost as a function of the number of states (NS) used: HEN only.
consumes hot and cold utilities. This case is relatively simple, and no HEP diagram is given. 4.3. IDEAS Design. The problem is next solved within the IDEAS framework, employing only an HEN operator, for which all heat engine/pumping operations are disabled. The minimum cost is $26,000/s with hot and cold utility use of 9.25 and 24.34 MJ/s, respectively. This represents an 11% improvement over the standard McCabe-Thiele design. The cost just quoted is the converged value as the number of states NS increases. The convergence plot is shown in Figure 7. The pinch diagram for this optimal solution is shown in Figure 8. The hot and cold utility use is too small to include in the figure. Notice the overlap of the hot and
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Figure 8. Pinch diagram for the HEN-only minimum utility cost solution.
Figure 10. HEN (pinch) diagram for the example: HEN + HEP.
Figure 11. HEP entropy diagram for the example: HEN + HEP. Figure 9. Minimum operating cost as a function of the number of states (NS) used: HEN + HEP.
cold composite streams. Pinch analysis by itself does not feature such a pinch region but typically just a pinch point. The freedom of internal process streams to change flow rates within the IDEAS framework accounts for the pinch region of Figure 8. Overlap of the hot and cold curves is first seen here in conjunction with optimal heat and power integration.20 One expects, then, that the IDEAS optimum with the HEN + HEP operator will have a similarly behaved HEN diagram with no nonoverlapping regions and an even better performance objective. When the example is solved with the HEN + HEP operator fully active, the minimum utility cost drops by about 40% compared to the HEN optimum. The new objective is $15,300/s. The cold utility use is 15.02 MJ/ s, and there is no hot utility use. The net work is -0.066 MJ/s, which means that work is produced. The convergence curve for this problem is shown in Figure 9. The HEN (pinch) diagram for this case is presented in Figure 10. One notices that it essentially consists of the cold utility (15.02 MJ/s) acting as a sink for a process stream. A smaller heat exchange section where about 0.3 MJ/s of heat is transferred exists at about T ) 80.4 K. The hot stream for that segment consists of process streams, whereas the cold is at a constant temperature and is due to the heat added to the external vapor mixing junction. In fact, the rest of the pinch diagram, which consists of small segments at various temperature levels, also features heat added/removed to/from various mixing junctions. These heats are very small because the streams mixing together have similar compositions
and the mixture does not change phase. Their values are on the order of 10-4 MJ/s or smaller. In practice, such small heat loads are transparent to the process and should not even be supplied because other considerations, such as having a thermodynamic state slightly above or below saturation to avoid two-phase flow, result in much larger process changes. Incidentally, such a requirement can easily be incorporated into this IDEAS model by adjusting the user-specified enthalpies associated with junctions and internal flows of the DN. The HEN pinch diagram for the HEN + HEP example essentially consists of the cold utility consumption; its maximum abscissa of 16 MJ/s is orders of magnitude smaller than the heat transported in the pinch diagram of the HEN solution, Figure 8. This difference needs to be evaluated in conjunction with the amount of heat transferred in the HEP section before any conclusions can be drawn. The HEP entropy diagram for the active HEN + HEP operator is shown in Figure 11. For the most part, heat engine operation can be seen except in the hightemperature portion, where the curves overlap. The lengths of the two curves on the horizontal axis are equal, as they should be. Figure 12 contains the HEP enthalpy diagram for the process, which shows that 1,300 MJ/s of heat is transferred in the HEP subnetwork. This is about an order of magnitude smaller than the corresponding value of 10,000 MJ/s for the HEN solution, translating to smaller capital costs and a smaller overall process. The difference in horizontal length of the two curves is the net work. Notice that most of the activity in this design is concentrated in the HEP subnetwork. This behavior is
Ind. Eng. Chem. Res., Vol. 43, No. 24, 2004 7837
Figure 12. HEP enthalpy diagram for the example: HEN + HEP.
generally observed and is due to the fact that a large amount of heat can be processed with a relatively small net work cost or even a net work benefit. As a consequence, one of the two utilities is zero, indicating that the HEP subnetwork reaches the limits of its capacity to optimally utilize the available hot/cold heat. Figure 13 is a diagram of the optimal HEN + HEP network given by Figures 10-12 that also contains details on the optimum numerical solution. It is an alternative representation of the DN/HEN + HEP/MEN model on which the formulation was based (Figure 2). One advantage of using this diagram is the easily visualizable junction-to-junction flow structure, as compared to the DN operator, where it is difficult to discern the origin and destination of internal flows. Also, in Figure 13, the MEN and HEN + HEP operator functions are better elucidated; all exchanger inlet and outlet compositions are shown, and streams active in the HEN + HEP are the ones connecting the liquid to the vapor junctions. The particular solution shown in the diagram is for 31 (vapor and liquid) junctions, and as it stands, the solution has a utility cost 4.5% larger than the minimum. Each junction can feed (i) the junction of same phase above or below it (local mixing), (ii) the junction of equal composition but different phase (this represents a pass through the HEN + HEP operator), (iii) an exchanger that accepts feed corresponding in composition and phase to the junction, and (iv) an external outlet At the same time, the junction is fed by (i) the junction of same phase above or below it (local mixing), (ii) the junction of equal composition but different phase (this represents a pass through the HEN + HEP operator), (iii) an exchanger that produces feed corresponding in composition and phase to the junction, and (iv) an external inlet The key of Figure 13 describes the action of the 18 exchangers. On the left-hand side (vapor), the lowcomposition inlet is the one on top. The vapor outlet is enriched in that component (nitrogen) as that is the more volatile of the two (RN2/O2 ) 4.173). The opposite occurs in the liquid side (on the right), where the exchanger outlet (top) diminishes in the more volatile component. Table 5 compares the optimal costs of all methods discussed in this example and shows that the IDEAS approach with heat and power integration produces the lowest operating cost, almost 50% lower than that of the traditional McCabe-Thiele design.
Figure 13. Diagram of the 31-junction IDEAS network. Table 5. Utility Consumption and Optimal Costs for a Variety of Design Methodologies, Elucidating the Utility of the IDEAS with Heat and Power Integration Approach McCabeThiele
IDEAS HEN
W (MJ/s) 0.00 0.00 QH (MJ/s) 11.74 9.25 QC (MJ/s) 27.24 24.34 cost ($/s) 29,300 26,000
McCabe-Thiele IDEAS with optimal HP HEN + HEP 1.10 0.00 16.60 19,400
-0.07 0.00 15.02 15,300
5. Conclusions This work introduces the infinite-dimensional statespace (IDEAS) approach to process synthesis for heatand power-integrated distillation networks. Heat exchanger and pump/engine action is optimally determined through the globally optimum model for heat and power integration developed in ref 20. It is demonstrated that the globally minimum utility cost formulation (encompassing all process structures) is given by a linear program. The IDEAS optimal heat and power integration method accomplishes the cryogenic separa-
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tion of a 70/30% gaseous nitrogen/oxygen mixture with a minimum utility cost that is 50% lower than that of the classical McCabe-Thiele design. Acknowledgment The authors gratefully acknowledge the financial support of the National Science Foundation, under Grants GER 9554570, CTS 9528653, CTS 9876489, and CTS 0301931. The authors also thank James Drake and Stevan Wilson for useful discussions during the course of the project. Nomenclature Sets of Infinite Sequences ∞ l1 ) absolute sum if elements are finite, ∑i)1 |R(i)| < ∞ l∞ ) absolute value of any element is finite, supi)1,...,∞ |a(i)|
