4882
Ind. Eng. Chem. Res. 1997, 36, 4882-4893
Optimal Thermodynamic Approximation to Reversible Distillation by Means of Interheaters and Intercoolers Pio Aguirre,* Jose´ Espinosa,† Enrique Tarifa, and Nicola´ s Scenna INGAR (CONICET), Avellaneda 3657, 3000 Santa Fe, Repu´ blica Argentina
The purpose of this paper is to deal with the problem of heat and power integration on one side and the problem of minimizing heat exchange areas on the other side, in both conventional and nonconventional distillation columns. We consider the limiting case of columns operating at minimum reflux. The appropriate objective functions that one must consider are the entropy production rate and the total heat exchange area, respectively. This is done by means of optimal placement of a given number of interheaters (IHs) and intercoolers (ICs) in stripping and rectifying sections, respectively. To solve these problems, an appropriate thermodynamic model for both conventional and nonconventional distillative columns is formally presented. This model allows us to formulate an optimization problem involving thermodynamically reversible profiles in stripping and rectifying sections of the columns. Our approach differs from others previously reported in that multiple reversible profiles were identified for each section of the column which give rise to lower and upper bounds for the objective function of the minimization problem. In other words, we obtain two solutions for each column section: the first is a nonoptimal feasible one, and the second is an optimal but not necessarily feasible one. Finally, the comparison of our approach with a method based on pseudobinary reversible profiles is carried out. Optimizing with this curve, solutions will be generated with objective function values between our lower and upper bounds. Therefore, care would be taken in using a pseudobinary pinch point curve for the placement of intermediate heat-exchanger units especially when the difference between our upper and lower bounds for the objective function values is relatively great. Introduction Conventional distillation sequences exist in chemical processes wherein multicomponent feed mixtures are separated into products by two or more distillation columns. Column sequencing is still the subject of several research efforts in both synthesis and optimization areas (Hu et al., 1991; Koehler et al., 1992). Nonconventional modes of separations such as reactive and azeotropic distillation are less frequent but appear to have received more interest recently (Wahnschafft, 1992; Wahnschafft et al., 1992, 1994; Espinosa et al., 1995). The capital and operating costs involved in separation subsystems contribute in a great portion to the total cost of entire processes. Therefore, the development of methods to optimize sequences and operating conditions becomes an important research topic for both conventional and nonconventional distillative processes. Several strategies are available to supply sequences of conventional and azeotropic distillation that can be considered as generating optimal or near-optimal candidates (Westerberg and Stephanopoulos, 1975; Seader and Westerberg, 1977; Tedder and Rudd, 1978; Eliceche and Sargent, 1981; Kakhu and Flower, 1988; Floquet et al., 1988; Floudas and Paules, 1988; Koehler et al., 1992; Modi and Westerberg, 1992; Wahnschafft, 1992). It was also stated that subsystem synthesis and optimization must be carried out according to constraints imposed from the global process (Linnhoff et al., 1982; Linnhoff and Townsend, 1982; Townsend and Linnhoff, 1983). Energy integration depends upon whether power production (or consumption) is considered as a real * Author to whom all correspondence should be addressed. † Present address: Lehrstuhl A fu ¨ r Verfahrenstechnik der Technische Universita¨t Mu¨nchen, Boltzmannstrasse 15, 85747 Garching, Germany. S0888-5885(96)00811-1 CCC: $14.00
alternative or not. Cryogenic distillative separation processes and high-pressure distillation with dominant power load demand are good examples of one of the extremal points. In these cases, the synthesis strategy must be constructed, maintaining the goal of heat and power integration in order to reduce operating costs. Entropy generation is a “measure” that reflects the goal of heat and power integration. The purpose of this paper is to deal with the problem of heat and power integration in nonconventional distillation systems. The framework selected is the same as that in previous papers published in the field of dualpurpose desalination plants (Aguirre and Scenna, 1989) and thermal energy recovery systems (Irazoqui, 1986). Seawater desalination is a heat and power coupled problem. Since the entropy production and the heat exchange area are competitive items, an optimal costbased design can be found. We proposed a method to obtain optimal thermodynamic solutions: For a given amount of total heat exchange area, power, and distillate water demands, find the optimal distribution of the heat exchange area among the constitutive parts that optimizes the total entropy production. This minimum tells us that the power production capability of the system reaches a maximum for the given investment measured as the value of the total heat exchange area involved. We want to extend to nonconventional distillation problems the general methodology developed previously. In so doing, the first step consists of the design of a column showing an “optimal thermodynamic approximation to reversible distillation”. This column should in further steps be integrated with the process streams including also the remaining columns. Our goal is to find the column that performs a prefixed separation and shows the minimum theoretical power demand (or equivalently the maximum power production capability) under the constraint of a given © 1997 American Chemical Society
Ind. Eng. Chem. Res., Vol. 36, No. 11, 1997 4883
number of heat-transfer units. The number of heat exchangers play the same role as the total heat exchange area in the desalination problem. By means of an appropriate placement of interheaters (IHs) in the stripping section and intercoolers (ICs) in the rectifying section of a column, an approximation to the thermodynamically reversible operation of the column can be reached when the total number of heat-transfer units is given. We consider in this paper that the only way to enhance the thermodynamic performance of this column is by means of an adequate distribution of the heat loads along the column and the total heat load itself. This is done by using IHs and ICs and minimizing at the same time the reflux ratio. At minimum reflux conditions columns with infinite numbers of trays will result. To solve this problem, an appropriate thermodynamic model, that can be used for both conventional and nonconventional distillative columns, is formally presented. This model allows us to formulate an optimization problem involving thermodynamically reversible profiles in stripping and rectifiyng sections of the columns. Since multiple reversible profiles can be calculated for each column product in multicomponent mixtures, different solutions to the optimization problem can be obtained according to the profiles considered. However, not all these profiles will produce feasible column design. We can identify two interesting solutions for each section of the column: one of the approximations to the reversible operation of the column gives a feasible column; the second approximation gives the best solution but feasibility cannot be assured for the column. Since the optimization problem is one of minimization of entropy production, these two solutions provide values for the objective function which can be termed upper and lower bounds. Similar models were also developed by other authors (Kayihan, 1980; Naka et al., 1980; Terranova and Westerberg, 1989; Taprap and Ishida, 1996; Agrawal and Fidkowski, 1996). However, our approach differs in that for multicomponent systems we make use of all possible reversible profiles for each section of the column instead of only one pseudobinary reversible profile. Unlike the other approaches, the problem is explicitly solved for the case when the total number of units is prefixed. As we stated above, the power production capability can be also used to optimally integrate the column to power cycles, to heat pumps, to process streams, and to other distillative columns. Furthermore, by relaxing the power-producing capability in order to minimize the heat exchange area, a new solution can be derived (Irazoqui, 1986). This new solution corresponds to the case of minimizing the heat exchange area involved in the heating and cooling devices of the distillation column when the hot and cold utility temperatures are given. Another reliable method for the purpose of heatexchanger integration in distillation processes has been recently proposed by Dhole and Linnhoff (1993), which holds for complicated nonideal multicomponent distillation systems. The method is based on a practical near-minimum thermodynamic condition that accounts for inefficiencies through an actual column simulation. The method given by Dhole and Linnhoff is used in this paper for columns operating at minimum reflux. By applying their method, we obtain a temperature vs heat load curve (or equivalently vs Lmin/Vmin) that lies between the two reversible profiles of our method. Optimizing with this curve, solutions will be generated
Figure 1. Thermodynamic column model.
Figure 2. Thermodynamic system model to compute the minimum theoretical power demand.
with objective function values between our lower and upper bounds. Thermodynamic Problem Formulation The problem to be addressed in this paper is stated as follows: The conditions of a multicomponent feed stream and the conditions of the bottom and distillate product streams entering and leaving a distillative column are given. The column operates at constant pressure, and no assumptions are made about both vapor-liquid equilibrium and enthalpy models. The streams leaving each step in the column are not necessarily at equilibrium; that is to say, the plate efficiency can be less than or equal to 1. The objective is to approximate the optimal thermodynamic performance of the column that satisfies the constraints imposed by an appropriate model. We consider in this paper that the only way to enhance the thermodynamic performance of this column is by means of an adequate distribution of the heat loads along the column and the total heat load itself. This is done by using IHs and ICs and minimizing at the same time the reflux ratio. Figures 1 and 2 depict the model to be dealt with to render possible the obtention of a representative value for the minimum theoretical power demand of a given arrangement as well as the comparison between several alternatives of placement of ICs and IHs. The column in Figure 1 is regarded as the sum of three dependent parts: the rectifiyng section showing an unknown heat capacity flow rate W(T) which is considered as a hot stream (HS), the stripping section with an also unknown heat capacity flow rate w(t) which is, in turn, considered
4884 Ind. Eng. Chem. Res., Vol. 36, No. 11, 1997
as a cool stream (CS), and finally an adiabatic region I. From the point of view of heat and power integration, the surroundings of the system shown in Figure 1 interact only through the heat exchangers with the column. Each IC allows transfer from the rectifying section of the column to that surrounding the heat amount:
QD i )
∫W(T) dTD
i
(1)
Let us consider a column operating with only one reboiler and one condenser, that is, an adiabatic column. Assuming that all columns entering and leaving streams are at vapor-liquid equilibrium, then the distillate and bottom temperatures are fixed. Condensing and reboiling are carried out at constant temperature, and they are the bubble point for the liquid mixtures (Terranova and Westerberg, 1989). The energy balance for this column and the entropy production rate are
HD + HB - HF ) QB - QD σ ) SD + SB - SF -
QB QD + >0 TB1 TD1
(2)
Optimal Thermodynamic Approximation to the Reversible Distillation with IHs and ICs: Minimum Entropy Production Rate The appropriate objective function to find the minimum power consumption according to the constraints imposed is the total entropy generation rate derived from the above-stated model:
σ)
(3)
Since the feed and product streams are fixed, only QB, and QD and their distributions along the stripping and rectifying sections of the column can be modified in order to enhance the thermodynamic performance of the column. The minimum theoretical power demand depends upon the difficulty of the separations and on the mass flow rates entering and leaving the column. The thermodynamic model depicted in Figure 2 is useful to understand this theoretical value. As Aguirre (1987) demonstrated, since the separation task is given, the minimum power consumption required to perform the prefixed separation using only one distillative column can be calculated if the auxiliary temperature T* is fixed and if W(T) and w(t) are known. By means of reversible power generating cycles (RPGC) and reversible heat pumps (RHP) and including large enough thermal baths, the changes in the system and in the environments to carry out the desired transformation in the feed stream are the net input of τ* units of power and the output of Q* units of heat from the thermal bath with temperature T*. The other constitutive parts of the system shown in Figure 2 remain unchanged. The power and heat flows are not necessarily in the sketched directions. However, for two different alternatives in the heat load distribution, it can be demonstrated that the following equalities hold for the power and heat flows according to Figure 2:
T* δσ ) δQ* ) δτ*
than one column, the entropy production rate could be further diminished especially into region I. There is, however, an exceptional case for which the entropy production rate could drop to zero in only one column coupled to RPGCs and RHPs as in Figure 2: this is just the case of the so-called “reversible or preferred” separation (Petlyuk et al., 1965; Koehler et al., 1991; Stichlmair et al., 1993). Restricting the problem to one separation column, any change in net power input needed to perform the desired separation is proportional to the entropy production rate change.
∫0Q
dQD(T) T
D
∫0Q
dQB(t) + Φ0 t
B
(5)
where Φ0 is a constant (entropy change flow between the products and the feed) and QB(t) and QD(T) are the heat distribution functions along the stripping and rectifying sections, respectively. QB and QD are the total heat loads that must be exchanged in the regions below and above the feed to perform the specified separation. From the thermodynamic point of view, the problem is to find the heat load distribution along the column which makes the functional σ a minimum. In the next sections, we will impose further simplifying assumptions regarding the types of possible functions to be consider for the heat load distribution. Since we are interested in using a prefixed total number of IHs and ICs to enhance the thermodynmic performance and according to the usual assumption that in each heat exchanger the heat capacity flow rate is high enough to cause a constant boiling temperature, the above equation reduces to N D QD
σ)
∑ i)1 T
NB QB
i
i
∑ i)1 t
∑ i)1
(6)
+ Φ0
i
ND
s.t.
i
NB
QDi ) QDmin
and
QB ) QB ∑ i)1 i
min
(7a,b)
and
QBmin - QDmin ) HB + HD - HF
(8)
(4)
where δσ is the change in entropy production rate into the column (owing to the irreversibilities in the stripping and rectifying sections) due to a modification in the heat load distribution, δτ* is the increase in the net input power, and δQ* is the increase in the net output heat flux from the bath with temperature T*. The entropy production in region I cannot be independently modified because the separation is carried out in only one column. If we were able to do the same separation with more
QDmin (QBmin) is the heat demand in the rectifying (stripping) section and corresponds to the condition of minimum reflux. Since the conditions of minimum reflux can be very accurately determined, then the problem of finding the appropriate heat distribution has to be solved once the total numbers of heat exchangers NB and ND are given. In the next section and according to the constraints imposed, we will obtain upper and lower bounds for the value of the objective function. They are useful to select a feasible initial scheme of IHs
Ind. Eng. Chem. Res., Vol. 36, No. 11, 1997 4885
a
characterized by
S[1]: xi > 0 {i e nD]; otherwise xi ) 0 S[2]: xi > 0 {i e nD ∧ i ) nD + 1}; otherwise xi ) 0 S[3]: xi > 0 {i e nD ∧ i ) nD + 2}; otherwise xi ) 0 S[j]: xi > 0 {i < nD ∧ i ) nD + (j - 1)}; otherwise xi ) 0 S[nC - (nD - 1)]: xi > 0 {i e nD ∧ i ) nC]; otherwise xi ) 0 (9)
b
Figure 3. Multiple reversible profiles for multicomponent ideal mixtures: (a) ternary mixture; (b) quaternary mixture.
and ICs placement and could be further implemented by rigorous simulation of a column operating at reflux close to the minimum. These bounds for the objective function are related to the calculation of all the reversible profiles corresponding to each column section; therefore, a brief description of such profiles is needed. Upper and Lower Bounds for the Optimization Problem of IHs and ICs Placement. When a bottom (distillate) multicomponent composition is given, multiple reversible distillation profiles can be calculated (Koehler et al., 1991; Wahnschafft et al., 1992; Espinosa et al., 1995). These profiles show the following characteristics: At each point the vapor and liquid compositions satisfy the mass balances for the column section and they are at thermodynamic equilibrium. To move from one point to another point on these profiles, the vapor (and hence the liquid) flow rate must be modified. There exist multiple reversible profiles for multicomponent ideal mixtures as is depicted in Figure 3a for a ternary system and in Figure 3b for a quaternary mixture. Given a specified product composition, each reversible profile differs from the others in the number of components present (Franklin and Forsyth, 1953). Each one of the different solutions S[j] (with j ) 1, nC - nD + 1) for a given distillate containing nD species is
where nC and nD represent the number of components in the feed and in the distillate, respectively. In line with this assertion, the distillate containing two components of the quaternary system shows the first pinch profile S[1] on the edge of compositions 1-2, the second pinch profile S[2] on the 1-2-3 composition space, and the third pinch profile S[3] on the 1-2-4 composition space. Any adiabatic profile departing from the same product will pass successively close to pinch points located on these composition profiles as dipicted in Figure 3b. In highly nonideal mixtures, additional disjointed reversible paths that do not pass through the product composition can be expected (Koehler et al., 1993; Espinosa et al., 1995). Ultimately, Poellmann and Blass (1994) and Aguirre and Espinosa (1996) have presented methods to solve reversible profiles even for those cases where highly nonideal mixtures are considered. We will explain how to compute the upper and lower bounds for the objective function, focusing our attention on a ternary mixture; the conclusions derived are, however, valid for any mixture. First, note that, for a given reflux or reboil ratio, adiabatic profiles computed with different plate efficiencies will show different paths. Fortunately, all of these will end at the same point on the corresponding pinch point curve. Hence, the pinch point profiles are plate model independent. In Figure 3a adiabatic stripping and rectifying profiles at minimum reflux are depicted. Two additional different reflux ratios were also used to generate adiabatic profiles departing from the distillate. These profiles, independent of the plate model, pass close to the corresponding binary pinch on the edge 1-2 and finally end at the ternary pinch on the reversible profile on the right. Instead of the adiabatic column, we can develop a rectifying section using three ICs, giving rise to a new profile that still makes the column feasible. In fact, the new rectifying profile intersects the adiabatic stripping profile at the feed pinch which corresponds to the minimum reflux condition. The heat load distribution as a function of the temperature was obtained by trial and error in order to maintain the column feasibility. This procedure can be iteratively applied in order to minimize the entropy generation rate according to eqs 6-8. Upper and lower bounds for this optimization problem can, however, be computed in order to generate an optimal thermodynamic approximation in terms of heat load distribution. The reversible profiles corresponding to the binary and ternary pinches show a characteristic heat load distribution, as is depicted in Figure 4. From the heat load distribution of the ternary pinch profile, a lower bound for the minimization of the entropy production rate is obtained. If we were able to use infinitely many ICs, we could in the limit approach the
4886 Ind. Eng. Chem. Res., Vol. 36, No. 11, 1997
Figure 4. Characteristic heat load distribution: system isopropyl alcohol-1-propanol-1-butanol.
ternary pinch profile but we will loses column feasibility. The entropy production rate, however, would reach a “global minimum” satisfying only the formal constraints but not the column feasibility. In fact, the ternary reversible profile did not intersect the adiabatic stripping profile. For a given finite number of ICs in the rectifying section placed on the ternary reversible curve, column feasibility cannot be assured in general. The value of the objective function for the minimization problem computed with the placement of ICs on the ternary reversible curve reaches a lower bound. Solutions with the same constraints showing lower objective function values cannot be found. On the other hand, the heat distribution corresponding to the binary pinch is an upper bound for the optimal objective function value. In this case, even when the number of ICs were infinitely large, the resulting profile will look like the adiabatic profile of Figure 3a. The composition profile will depart from the distillate composition and, after infinitely many steps at which the streams are always at binary equilibrium (edge 1-2), the last portion of the profile will proceed into the concentration simplex and will coincide with the adiabatic profile. The design of such a column is always feasible. Using a finite number of ICs placed on the binary heat load curve, column feasibility is assured. The objective function value, for this case the entropy production rate, is greater than the solution obtained with the ternary heat load curve. Note that from the optimization point of view we termed this solution an upper bound. However, feasible designs with objective function values greater than this upper bound for the minimization problems can be encountered. In fact, placing ICs at lower temperatures than that corresponding to the binary pinch curve for the same heat loads, a feasible solution is obtained with a higher value of entropy production rate. In this example a point on the ternary reversible profile is normally termed as nonactive pinch, whereas one on the binary pinch profile is called active pinch; on the other hand, there is only one interesting stripping pinch profile: the ternary profile due to the bottom stream contains all the components present in the feed to the column. This situation depends upon the products specification, and it changes according to whether the direct or the indirect separation is performed.
Summarizing, for the ternary mixture of Figure 3a, the binary and ternary pinch profiles supply the limiting heat load distributions that can be used to compute upper and lower bounds for the optimization problem of placement of ICs in the rectifying section. The conditions analyzed in this example can be easily extended to multicomponent mixtures and to all possible separations. The only pieces of information needed to obtain the lower and upper bounds are the reversible profiles of the products and the information of on which pinch curve the last active pinch point lies and on which pinch curve the first nonactive pinch point lies. In order to determine the last active and the first nonactive pinch point curves, the method developed by Poellmann et al. (1994) has to be applied. In cases in which the last active pinch is the last overall, the lower bound represents the upper bound too and this is the solution in the case for the stripping section in the ternary mixture of Figure 3a. The Mathematical Model. In this section, the appropriate placement of ICs (IHs) over the heat load distribution curves corresponding to reversible profiles is developed. The problem is stated as follows: Given a reversible pinch profile, the minimum heat load corresponding to the minimum reflux condition, and the number of ICs to be placed, find the heat load and the temperature at which each IC must work in order to minimize the total entropy generation rate. The problem in eq 1 can be rewritten as a minimization problem for the rectifying section: N D QD
Min
∑ i)1 T
i
(10a)
i
ND
s.t.
QD ) QD ∑ i)1 i
(10b)
min
where the ICs are numbered from the distillate toward the feed. Let us define the following variables: n-1
Q h Dn )
QD ∑ i)1
i
(11)
Q*D(T) is the cumulative heat load corresponding to the reversible profile for a given temperature T. Assuming T1 to be the distillate temperature, if we assign QD1 units of heat to the first IC, then the temperature T2 at which the second IC must be placed can be, at least, that obtained from the equality QD1 ) Q*D(T2). This equality for the nth temperature is n-1
Q h Dn )
QD ) Q* ∑ D(Tn) i)1 i
(12)
It must be noted that, for a given cumulative heat load, more than one temperature (three strictly) could be computed from a given reversible profile for both cases, namely, active and nonactive tangent pinches (Koehler et al., 1991). Both situations correspond to the presence of extrema in the function: cumulative heat load versus temperature. For this case, the minimum temperature for the rectifying section and the maximum temperature for the stripping will be adopted. We will return to this point at the end of this section.
Ind. Eng. Chem. Res., Vol. 36, No. 11, 1997 4887
Equations 11 and 12 allow us to relate the heat load distribution among constant-temperature operating devices to the heat load distribution in reversible profiles. When four ICs are able to be used, the problem is
σ ) Φ1 +
[
QD1 T1
QD3
QD2 +
T2(QD1)
+
T3(QD1 + QD2)
+
QD4 T4(QD1 + QD2 + QD3)
]
QD1 + QD2 + QD3 + QD4 ) QDmin T1 ) constant;
Φ1 ) constant
(13)
and eliminating the first restriction and QD1 we have
σ ) Φ2 +
[
-(QD2 + QD3 + QD4)
QD2
T1
+
T2(QD2,QD3,QD4) QD3
QD4 +
T3(QD3,QD4) T1 ) constant;
Figure 5. Rectified heat load distribution: system acetonechloroform-benzene.
+
]
T4(QD4)
Φ2 ) constant
2
QD2 ∂T2 ∂σ 1 )∂QD3 T1 (T )2 ∂QD3 2
|
)0
QD ,QD
2
|
QD4 ∂T4 (T )2 ∂QD4 4
|
) 0 (16) QD ,QD
|
) QD ,QD 3
4
∂T3 ∂QD3
|
∂T2 ∂QD3
|
) QD ,QD 2
)
QD ,QD 2
4
∂T4 ∂QD4
|
4
|
∂T3 ∂QD4
∂T2 ∂QD4
2
4
|
QD ,QD 2
3
2
3
)QD ,QD
QD ,QD 2
3
∂T4 ∂Q h D4
(21)
2
D3
4
D4
3
∂T2 ∂Q h D2
∂T3 ∂Q h D3 (18)
{ } ∂Q h Di
∂Ti ∂Q h Di
) 0 (17)
)-
)QD ,QD
|
QD4 ∂T4 T3 - T4 + )0 T3 T4 (T )2 ∂Q h
)
Taking into account that the summation over the heat loads must equal QDmin jointly with eq 11, the following equalities should be in order:
∂T2 ∂QD2
(20)
3
Note that until now, we have made the two following assumptions: given a heat load in the first IC, then a temperature T2 can be computed, and the derivative of the temperature with respect to the cumulative heat load can be computed as
3
3
QD3 ∂T3 T2 - T3 + )0 T2 T3 (T )2 ∂Q h
D2
4
QD3 ∂T3 1 + (T )2 ∂QD4 T4
QD ,QD 2
3
3
4
-
(19)
2
QD ,QD
∂T3 ∂QD3 QD2 ∂T2 ∂σ 1 )∂QD4 T1 (T )2 ∂QD4
(15)
QD3 1 T3 (T )2
+ 2
|
QD2 ∂T2 T1 - T2 + )0 T1 T2 (T )2 ∂Q h
(14)
The conditions of minimum over the variables QD2, QD3, and QD4 led to the following equations:
QD2 ∂T2 ∂σ 1 1 )+ ∂QD2 T1 T2 (T )2 ∂QD2
Therefore, the conditions for a local optimum are
-1
∂Ti
(22)
As we mentioned above, in the cases of tangential pinches, the function
Q h Dn ) Q* D(Tn)
(23)
shows one local maximum and one local minimum, and therefore for a given heat load, depending on the region, three temperatures can be computed. For this case, the function will be rectified in order to obtain a function
T ) T(Q h ) ) T(Q*)
(24)
In Figure 5 a scheme of such a case and the rectified function proposed are depicted. The partial derivatives are easily obtained by a numerical derivative of the heat load profile of the pinch curve. This can be done because there exists the rectified function (24). The equation system (19)-(21) can be solved using the following algorithm: Given an admissible error value , QDmin, and T1 and for four ICs: (1) Assume a value for T4. (2) Compute QD4 by means of the rectified function (24) and QDmin.
4888 Ind. Eng. Chem. Res., Vol. 36, No. 11, 1997
(3) Compute T3 with eq 21. (4) Compute QD3 by means of the rectified function (24) and the values for QDmin and QD4. (5) Compute T2 with eq 20. (6) Compute QD2 by means of the rectified function (24) and the values for QDmin, QD4, and QD3. (7) Compute T1new with eq 19. (8) If |T1new - T1| < , stop; otherweise, T4 must be modified and then go to step 1. Several robust numerical methods are able to compute reversible profiles; however, we resort to our own algorithm (Aguirre and Espinosa, 1996). Since the basic procedure was homotopy continuation, we obtain a small number of pinch points on the reversible profile that are located in regions such that the curvature of the pinch point path in the composition simplex exceeds a certain degree. The allowable degree of curvature is a parameter which can be tuned in order to obtain smooth changes. Therefore, linear interpolation between adjacent points in the composition space on the reversible profile is satisfactory. The algorithm proposed for the optimal placement of ICs and IHs shows local convergence and, of course, whenever multimodal functions are involved, we obtain a local minimum. To find out the global solution, either a homotopy continuation approach or an exhaustive search has to be developed. In one of the examples given below such cases of multimodal functions were found. To deal with the complete column, the above algorithm is used for 1, 2, 3, ..., Nmax ICs and for 1, 2, 3, ..., Nmax IHs. Then, if the total number of heat exchangers NT (NT > 2) is given, we can find the appropriate combination of IHs and ICs which make the total entropy production rate a minimum by simply ordering the results obtained for each column section. This algorithm was used to compute the examples developed in the next sections. Before continuing with the optimal thermodynamic placement of IHs and ICs, let us develop another interesting case that arises when the temperatures of the hot and cold utilities are given and corresponds to minimization the total heat exchange area. In these systems no power cycles or heat pumps are involved and the entropy production rate plays no more the important role as in the case of coupled heat and power problems. Minimum Heat Exchange Area The problem can be stated as follows: Given a distillative column operating adiabatically at minimum reflux and taking heat from a hot utility with temperature Th and rejecting heat to a cold utility with temperature Tc, determine the conditions for the placement of a fixed total number of IHs and ICs in such a way that the total heat exchange area becomes a minimum. The temperature difference for heat exchange will increase when more IHs or ICs are used, and then the total area will decrease. This is similar to the optimal thermodynamic approximation developed above. In fact, we can get the upper and lower bounds as previously described and the mathematical model which differs only slightly from the above-stated system. Instead of the entropy production rate, we use the total heat exchange area as the objective function. The operating temperatures in the heat exchangers and the global heat-transfer coefficient are assumed constant. The upper bound corresponds to the case of placement of the heat exchanger using the last active pinch point curve as the T ) T(Q*) function, whereas the lower
bound corresponds to the case of placement of the heat exchangers using the first nonactive pinch point curve as the T ) T(Q*) function. Each section of the column is solved separately, and then by ordering the optimal individual values obtained, the solution for the complete column is easily derived. In the case of four IHs the minimum conditions for the heat exchange area are
T1 - T2 ∂T2 Q2 + )0 Th - T1 Th - T2 ∂Q h
(25)
∂T3 T2 - T3 Q3 + )0 Th - T2 Th - T3 ∂Q h
(26)
∂T4 T3 - T4 Q4 + )0 Th - T3 Th - T4 ∂Q h
(27)
B2
B3
B4
The above-stated algorithm is also useful in solving this system. Results and Discussion Entropy Production. Before presenting a demonstrative example, we will discuss some a priori conditions at optimality for the sake of simplicity for the case of adding only one heat exchange unit. The problem of optimal thermodynamic placement of IHs and ICs can be restated as follows: Given the operating conditions of an adiabatic distillative column at minimum reflux, find the temperature and heat load at which a new heat exchanger unit must work satisfying the optimum criterion of entropy production rate. To deal with this column, the above-developed algorithms are used for Nmax ICs ) 1 in the rectifying section and for Nmax IHs ) 1 in the stripping section. Then, since the total number of heat exchangers is NT ) 3, we do select the heat exchanger that minimizes the entropy production rate. Note that nothing about on which column section the new heat exchanger will be placed was said; it is a consequence of the optimization procedure. The first new heat exchanger will operate in the rectifying section or in the stripping section depending upon on the shape of the heat load distribution profiles obtained from the pinch point curves for each side of the column. If the heat load distribution profiles were symmetric, one can expect that more impact is obtained on the entropy production rate by placing the first new heat exchanger in the rectifying section. This is so because the heat load at the minimum temperature of the column generates the maximum term in the entropy expression; then any change in the heat load at the minimum temperatrue gives rise to the maximum change in the entropy production. If, on the other hand, we analyze the problem of minimizing the total heat exchange area, for symmetric heat load profiles, the maximum impact on the objective function will be given by the minimum temperature difference with the utility considered. In both problems, minimum entropy production rate and minimum heat exchange area, it can be demonstrated that the first heat exchanger added to the adiabatic column will perform the maximum change in the objective function overall. Any further increase in one heat exchanger unit will lead to smaller changes in the objective function value. By means of an example we want to show the impact in the total power consumption involved in using ICs or IHs. The simplest one is that of low-temperature
Ind. Eng. Chem. Res., Vol. 36, No. 11, 1997 4889
Figure 6. Use of the heat pumps scheme in low-temperature distillation.
distillation. The largest energy costs in these separation processes are associated with reboilers and condensers cooled with a refrigerant. A refrigerant must be supplied to the partial or total condenser to condense the overhead vapor to obtain the desired reflux. This refrigerant is obviously obtained in a heat pump separated from the distillative column. The energy requirements of low-temperature distillation can be reduced by resorting to multieffect distillation or using a heat pump to “pump” heat from the condenser to the reboiler using the same distillation products as the cycle fluid. There are several proposed processes according to these principles. Our objective is not to find the best but only to analyze the impact of optimal placement of ICs and IHs. In fact, further improvements are reached by placing ICs and IHs that reduce the temperature difference for the heat pump cycle, diminishing the net power input required. In Figure 6 the use of heat pumps according to the most common scheme in low-temperature distillation is depicted. The first one is the conventional process carried out at low pressure (
