Ind. Eng. Chem. Res. 2002, 41, 1511-1515
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PROCESS DESIGN AND CONTROL Energy Targeting in Heat Exchanger Network Synthesis Using Rigorous Physical Property Calculations Marcelo Castier* and Eduardo M. Queiroz Escola de Quı´mica, Universidade Federal do Rio de Janeiro, C.P. 68542, Rio de Janeiro, Rio de Janeiro, 21949-900, Brazil
Pinch points for heat exchanger network synthesis were determined using rigorously calculated thermodynamic properties, in contrast with the usual approach of assuming constant heat capacities (cP) and linear interpolations in enthalpy for phase changing streams. We discuss a more formal approach to the energy targeting problem, showing that its solution requires the use of a global minimization method, because of the possibility of multiple local minima in the objective function. We show three applications, two of them involving near-critical streams and the other containing several streams that undergo phase transitions. In all cases, we correctly detected pinch points that otherwise would have been wrongly located by targeting procedures based on the constant cP assumption and on a single linearization between the bubble and dew points for each stream. For this reason, the procedure presented here is an adequate alternative for the accurate determination of pinch points and utility targets when the physical properties have nonlinear behavior with respect to temperature. Introduction The pinch method for heat exchanger network design1-4 combines conceptual elegance with a systematic procedure that reduces the combinatorial burden often associated with process synthesis problems. For this reason, the method has gained wide acceptance and there is voluminous literature on its applications and extensions. The first step in the application of the method is the solution of the energy-targeting problem, i.e., determining the minimum rates of hot and cold utilities needed to accomplish all the heat transfer tasks for a set of process streams. This step can also be carried out in such a way that multiple level utility rates are determined, i.e., by finding out at which temperature range each portion of the hot and cold utilities should be provided. In this way, the use of utilities whose temperatures are unnecessarily high (or low) can be avoided. By use of such an extension of the original procedure, points where heat cannot be cascaded down to the next temperature interval in the first cascade (when no heat is supplied from a hot utility to the highest temperature interval) can be located, if they exist, for a given set of process streams. We will refer to these points as noncascadable points (NCPs). One or more of the NCPs will be a pinch point. Although the pinch method is not limited to simple thermodynamic modeling of the process streams, it is most often applied assuming that the molar heat capacity at constant pressure (cP) of streams is constant. However, it has long been recognized5 that the cP of a stream can vary considerably with temperature6 or a * To whom correspondence may be addressed: e-mail,
[email protected]; fax, +55-21-2542-6376; phone, +55-21-25627607.
stream may undergo a phase change in a given temperature (pure substances or azeotropes) or temperature interval (mixtures), such as a vapor-liquid transition. A possible approach is to split the original temperature intervals until the true hot and cold composite curves are satisfactorily approximated by a sequence of linear interpolations.7 In the case of phase change, it has been proposed3,8 to use the saturation conditions, such as bubble and/or dew temperatures in the case of mixtures, to define additional temperature intervals. These two methods have been successful in the solution of several energy-targeting problems but, depending on the specified tolerance for the approximation of the composite curves, may require the use of a large number of additional temperature intervals. In this paper, we formulate and solve the energytargeting problem using rigorous evaluations of thermodynamic properties and discuss features of this approach. In the next section, we use an example, only involving two streams containing pure components that behave as ideal gases, to show that the constant cP assumption can prevent the correct location of a pinch point even in such a simple situation. We then present a more formal procedure for the location of pinch points and show examples of its application. The procedure represents an alternative approach to interval linearizations when the constant cP assumption is inadequate. A Simple Example Let us consider a problem with only two process streams. Stream 1 contains pure nitrogen and needs to be cooled from 320 to 200 K. Stream 2 is a flow of pure oxygen that needs to be heated from 190 to 310 K. Both streams have the same flow rate of 100 g‚mol/s and
10.1021/ie000981o CCC: $22.00 © 2002 American Chemical Society Published on Web 02/23/2002
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Figure 1. Available heat versus temperature in the ideal gas streams example.
behave as ideal gases, and therefore, their enthalpies are insensitive to pressure. The ideal gas cP expression and the corresponding parameters from Reid et al.9 were used to compute the enthalpies. A minimum temperature difference of 10 K was assumed. With this set of specifications, there is a single temperature interval in the usual procedure for energy targeting. Assuming a constant cP for each stream, equal to the average value of the ideal gas cP in the corresponding temperature interval, the available heat in the interval is equal to 2450 W, as shown by the dotted line in Figure 1. Since this value is positive, the conclusion is that only cold utility would be needed at a rate of 2450 W. However, if we plot (solid line in Figure 1) the available heat as a function of temperature (using the hot stream temperature scale), we observe a minimum value inside the temperature interval. Moreover, this minimum value is negative, indicating an enthalpy deficit, thereby characterizing a NCP. By use of the hot stream temperature scale, the NCP occurs at 282.7 K, with a corresponding deficit of 570 W that needs to be provided by a hot utility. As would be expected, a second pass through the cascade, adding the hot utility load, shows that the NCP coincides with the pinch point. Accordingly, the need for cold utility also increases by 570 W to a total of 3020 W. Therefore, we observe that a pinch occurs inside the original temperature interval, and both hot and cold utilities are needed to perform the heat transfer tasks to satisfy the imposed minimum temperature difference. Then we note that it is necessary to split the original temperature interval at the NCP in order to obtain the proper solution to this problem. Formulation The usual approach for the determination of pinch points first requires the identification of temperature intervals based on the supply and target temperatures of each stream. If a stream undergoes phase changes, such as vapor-liquid transitions, the saturation (bubble and/or dew) temperatures may be used to define additional temperature intervals. The heat available in the highest temperature intervals is used to supply the need for heating in the lowest temperature intervals. If this cumulative heat load that is cascaded becomes negative
at the lower temperature limit of an interval, this indicates an energy deficit that cannot be cascaded downward and needs to be compensated by the use of a hot utility. At this stage, if one wishes to calculate the necessary utility loads at different temperature levels, this point should be considered as a NCP. It is then necessary to zero the cumulative heat load and restart the heat cascading procedure downward to the lowest temperature intervals. However, the example of the previous section illustrates the fact that pinch points may occur inside the intervals defined by the supply and target stream temperatures if constant cP values are not assumed. A similar discussion can be found in the work of ElHalwagi et al.,10 but the objective of their procedure was to find the temperatures and flow rates that minimize the cost function of a heat-induced separation network. To locate NCPs, and then pinch points, we used a procedure that preserves the idea of cascading the available heat from the highest to the lowest temperature intervals but includes new aspects because of the nonlinear behavior expected from rigorous calculations of thermodynamic properties. We start by locating the temperature intervals only based on the supply and target stream temperatures, taking into account the specified value of the minimum temperature difference (∆Tmin). The intervals are numbered from the highest to the lowest temperature levels, with indexes from 1 to NI. An interval j is delimited by the temperatures Tj and Tj+1 (Tj > Tj+1) using the hot stream temperature scale. The corresponding values in the cold stream temperature scale are Tj* and Tj+1* , given by Tj* ) Tj - ∆Tmin and Tj+1* ) Tj+1 - ∆Tmin. The heat available in interval j is denoted by Qj. The cumulative heat available in the beginning of interval j is denoted by Cj-1. The initial value for this variable is C0 ) 0. In the beginning of the main calculation loop, we assign j r 1 and proceed as follows: Step 1. While j < (NI - 1) Step 1.1. Define
Q(T) )
∑
(Hi(Tj) - Hi(T)) -
i∈{hot}j
∑
(Hi(Tj *) -
i∈{cold}j
Hi(T *)) for Tj g T g Tj+1 where {hot}j and {cold}j denote the hot and cold streams present in interval j, respectively. To evaluate the stream enthalpies, we tested their phase stability11 and used an isothermal flash algorithm12 if more than one phase was present. The variable Q(T) is the available heat at a temperature T in interval j. Step 1.2. Minimize Q(T), for Tj g T g Tj+1. As will be illustrated in one of the examples, Q(T) may have more than one minimum. For this reason, we used the Global Line Search algorithm, as implemented by Neumaier13 in the MATLAB software, to bracket the minimum points. The final refinement of each minimum was obtained using the intrinsic function fmin available in MATLAB. Steps 1.3 and 1.4 discuss the possible outcomes of this minimization. Step 1.3). If Q(T) is a monotonically increasing or decreasing function of temperature in interval j, or if its only stationary point in the interval is a maximum or saddle, then the minimum value is
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either at Tj or at Tj+1. The available heat in the interval is Qj ) Q(Tj+1) and the cumulative heat load is calculated as Cj ) Cj-1 + Qj. Here, there are two possibilities: (i) If Cj g 0, assign j r j + 1, and proceed to the next interval in step 1. (ii) If Cj < 0, store Tj+1 as a NCP temperature and Cj as the hot utility required between the previous and the new NCP. After this, assign Cj r 0 and j r j + 1 and proceed to the next interval in step 1. Step 1.4. If Q(T) has at least one minimum inside interval j, let M be the number of minima and Tm denote the temperatures (in the hot stream scale, organized in decreasing order), in which they occur. Initially, assign m r 0 and then perform the following loop whose objective is to check if any of the minimum points would turn the cumulative heat load into a negative value, characterizing a NCP:
• While m < M • Assign m r m + 1, and calculate C• ) Cj-1 + Q (Tm). • If C• < 0, • Store Tm as a NCP temperature and C• as the hot utility required between the previous and the new NCP. • Move temperatures from j + 1 to NI one position forward in the list and insert Tm in position j + 1. Increase the number of intervals by 1, i.e., NI r NI + 1. • Assign Cj r 0 and j r j + 1. • Proceed to the next temperature interval in Step 1. •End If • End While In this case, none of the minimum points gives a NCP and the following assignments should be made: • Qj r Q(Tj+1), Cj r Cj-1 + Qj. Then proceed to the next temperature interval in step 1. Step 2. End While (opened in step 1). The described procedure locates NCPs either at the boundaries or inside the original temperature intervals. When the latter happens, the original temperature intervals are split at the NCP temperatures. Therefore, the number of intervals after applying the procedure may be larger than its initial value. The total hot utility load is calculated by adding the heat requirements at the NCPs. To determine the pinch point, one can assume that the total hot utility load is transferred to the highest temperature interval and cascaded downward to the lowest temperature levels. The points where the cumulative heat load becomes equal to zero are the pinch points, but in general, there is only one pinch point. Results and Discussion We present three examples with a relatively small number of streams and intervals, selected to emphasize
Figure 2. Composite curves for the n-butane stream + n-pentane stream problem (example 1).
the particular aspects of the procedure used here, and contrast them with those of the usual energy-targeting algorithms. The effect of increasing the number of temperature intervals would be an increase in the global line searches that would be required. Even though more global optimization problems would have to be solved, each of them would continue to be in a single variable (temperature). The effect of increasing the number of streams would be an increase of the number of isothermal flash calculations necessary for each evaluation of the objective function. Possibly, none of these two aspects would prevent the use of the procedure in large problems of practical interest. In all examples, critical properties, acentric factors, and coefficients for an ideal gas cP polynomial in temperature were taken from Reid et al.9 Stream enthalpies at each condition were calculated using an isothermal flash procedure12 that employs the global phase stability test11 to determine the number of phases present at equilibrium. Fugacity coefficients and enthalpies were evaluated using the Soave-RedlichKwong equation of state14 with binary interaction parameters (kij) set equal to zero. A minimum temperature difference of 10 K was specified in all cases. In all examples, we report the NCP and pinch temperatures using the hot streams temperature scale. Example 1. In this example, there are two streams, each of them containing a pure fluid at its critical pressure. Stream 1 is a n-butane (Tc ) 425.2 K, Pc ) 38.0 bar) flow that needs to be cooled from 500 to 400 K, and stream 2 contains n-pentane (Tc ) 469.7 K, Pc ) 33.7 bar) that needs to be heated from 390 to 490 K. Both streams have the same flow rate of 100 g‚mol/s. If we assume constant cP for these streams, the available heat is negative, indicating a pinch at the lowest temperature, i.e., 400 K in the hot stream temperature scale. This would lead to the conclusion that only hot utility is required. However, a minimization of the available heat in the interval provides a NCP at 434.6 K in the hot stream temperature scale. The single NCP is also the pinch point, and both hot (157 621 W) and cold (77 311 W) utilities are necessary. These results can be observed in Figure 2, which shows the hot and cold composite curves calculated using rigorously calculated thermodynamic properties. It should be noted
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Table 1. Stream Properties for Example 3a stream
flow rate (g‚mol/s)
component 1
component 2
x1
P (bar)
source
target
T (K) bubble
1 2 3 4 5
120 350 100 100 150
n-heptane n-butane ethane ethane ethane
n-octadecane n-heptane n-heptane n-heptane n-heptane
0.90 0.05 0.50 0.10 0.10
3 4 2 8 2
505.0 415.0 370.2 355.0 210.0
415.0 370.2 224.9 495.0 400.0
418.25 412.62 224.85 358.99 277.45
a
dew 538.26 423.58 370.33 453.88 392.06
Key: x1, mole fraction of component 1; T, temperature; P, pressure.
Figure 3. Composite curves for the three-stream problem (example 2).
Figure 4. Composite curves for the five-stream problem (example 3).
that the usual procedure of creating new temperature intervals based on bubble and/or dew points would not work in this example because there is no phase change. However, in the near-critical region, heat capacities change very rapidly even in the absence of phase transitions, and therefore, the assumption of constant cP can lead to results that are not only quantitatively but also qualitatively wrong. Example 2. This problem is an extension of example 1, containing three streams, each of them containing a pure fluid at its critical pressure. Stream 1 is a flow of n-butane (Tc ) 425.2 K, Pc ) 38.0 bar, 45 g‚mol/s). Stream 2 contains isopentane (Tc ) 460.4 K, Pc ) 33.9 bar, 35 g‚mol/s). These two streams need to be cooled from 500 to 400 K. Stream 3 contains n-pentane (Tc ) 469.7 K, Pc ) 33.7 bar, 100 g‚mol/s) that needs to be heated from 390 to 490 K. With these specifications, there is only one original interval in this problem. The procedure located two NCPs that occur at 461.8 and 429.6 K, with hot utility requirements respectively equal to 134 532 and 2174 W. Therefore, the total hot utility load is equal to 136 706 W and the pinch point is located at the lowest NCP temperature (429.6 K). The calculated cold utility requirement is equal to 19 702 W. In this problem, the distance between the hot and cold composite curves (Figure 3) passes through two minima inside the single original temperature interval, and the use of a global optimization method was essential for the correct location of the NCPs and of the pinch point. Example 3. In this example, there are five streams, and each of them is a binary mixture. Table 1 presents the problem specifications. Although not a problem specification, we also present in Table 1 the bubble and dew temperatures of the streams in order to provide
information about their number of phases in each region of the composite curves. Our calculations indicate that there are two NCPs in this problem. The higher temperature NCP occurs at 444.3 K and requires 82 795 W of hot utility. The other NCP is at 241.5 K, demanding 17 849 W of hot utility. Therefore, there is a total requirement of 100 644 W of hot utilities and the pinch point is at 241.5 K. The calculated requirement of cold utility in this problem is equal to 20 391 W. The hot and cold composite curves are shown in Figure 4. In this problem, the NCP temperatures occur inside the original temperature intervals. Furthermore, they coincide neither with the bubble nor with the dew temperatures of any of the process streams (Table 1), which are often used to define new temperature intervals.3,5 This example also illustrates the convenience of formulating the energytargeting problem as an optimization problem for accurately locating the pinch temperatures and utility heat loads. Conclusions In this paper, pinch points for heat exchanger network synthesis were determined using rigorously calculated thermodynamic properties, in contrast with the usual approach of assuming constant heat capacities, and linear interpolations in enthalpy for phase changing streams. We discussed a more formal approach to the energy-targeting problem, which uses a global minimization method, because of the possibility of multiple local minima in the objective function. The procedure was used to solve three problems, two of them involving near-critical streams and the other containing several hydrocarbon mixture streams that undergo phase tran-
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sitions In all cases, we correctly detected pinch points that otherwise would have been wrongly located by targeting procedures based on the constant cP assumption and on a single linearization between the bubble and dew points for each stream. For this reason, the procedure presented here is an adequate alternative for the accurate determination of pinch points and utility targets when the physical properties have nonlinear behavior with respect to temperature. A comparison of the computational effort of the proposed procedure and that of extensive piecewise linearizations in order to achieve the same specified tolerance for the pinch location in large problems remains to be investigated. Nomenclature Cj-1 ) cumulative heat available in the beginning of interval j cP ) molar heat capacity at constant pressure Hi ) total enthalpy of stream i M ) number of minima in the available heat load of interval j NI ) number of temperature intervals P ) pressure Pc ) critical pressure Qj ) heat available in interval j T ) temperature Tc ) critical temperature Tj ) upper temperature limit of interval j (hot stream scale) Tj* ) upper temperature limit of interval j (cold stream scale) Tj+1 ) lower temperature limit of interval j (hot stream scale) Tj+1* ) lower temperature limit of interval j (cold stream scale) x1 ) mole fraction of component 1 Greek Letter ∆Tmin ) minimum temperature difference
Acknowledgment The authors acknowledge Professor Arnold Neumaier (University of Vienna, Austria) for his assistance on the use of the Global Line Search algorithm. Financial
support for this work was provided by FAPERJ, CNPq/ Brazil, and PRONEX (Grant no. 124/96). Literature Cited (1) Linnhoff, B.; Hindmarsh, E. The Pinch Design Method for Heat Exchanger Networks. Chem. Eng. Sci. 1983, 38, 745. (2) Linnhoff, B. Pinch AnalysissA State-of-the-Art Overview. Chem. Eng. Res. Des. 1993, 71, 503. (3) Douglas, J. M. Conceptual Design of Chemical Processes; McGraw-Hill International Edition: Singapore, 1988. (4) Biegler, L. T.; Grossmann, I. E.; Westerberg, A. W. Systematic Methods of Chemical Process Design; Prentice Hall: Upper Saddle River, NJ, 1997. (5) Linnhoff, B.; Townsend, D. W.; Boland, D.; Hewitt, G. F.; Thomas, B. E. A.; Guy, A. R.; Marsland, R. H. A User Guide on Process Integration for the Efficient Use of Energy; IChemE, Rugby, U.K., 1982. (6) Kemp, I. C. Some Aspects of the Practical Application of Pinch Technology Methods. Chem. Eng. Res. Des. 1991, 69, 471. (7) Westphalen, D. L.; Wolf Maciel, M. R. Pinch Analysis Based on Rigorous Physical Properties. Braz. J. Chem. Eng. 1999, 16, 279. (8) Liporace, F. S. Sı´ntese de Redes de Trocadores de Calor: Proposta de um Procedimento Automa´ tico e Estudo da Influeˆ ncia da Presenc¸ a de Correntes que Mudam de Fase, D.Sc. Thesis, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil, 2000. (9) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids, 4th ed.; McGraw-Hill: New York, 1987. (10) El-Halwagi, M. M.; Srinivas, B. K.; Dunn, R. F. Synthesis of Optimal Heat-Induced Separation Networks, Chem. Eng. Sci. 1995, 50, 81. (11) Michelsen, M. L. The Isothermal Flash Problem. Part I. Stability. Fluid Phase Equilib. 1982, 9, 1. (12) Michelsen, M. L. The Isothermal Flash Problem. Part II. Phase Split Calculation. Fluid Phase Equilib. 1982, 9, 21. (13) Neumaier, A. Global Line Search: GLSsUnivariate Local or Global Optimization (http://solon.cma.univie.ac.at/∼neum/ software/ls), University of Vienna, Austria, 2000. (14) Soave, G. Equilibrium Constants from a Modified RedlichKwong Equation of State. Chem. Eng. Sci. 1972, 42, 381.
Received for review November 20, 2000 Revised manuscript received October 23, 2001 Accepted November 5, 2001 IE000981O
