The importance of control considerations for heat exchanger network synthesis: a case study

THE IMPORTANCE OF CONTROL CONSIDERATIONS FOR HEAT EXCHANGER NETWORK SYNTHESIS: A CASE STUDY

S.G.Oliveira, F.S.Liporace, O.Q.F.Araújo and E.M.Queiroz

Departamento de Engenharia Química, Escola de Química, Bloco E, Sala 209, Centro de Tecnologia,

Universidade Federal do Rio de Janeiro, Cidade Universitária, CEP 21949-900,

Phone: [phone], Fax: [phone], Rio de Janeiro - RJ, Brazil,

E-mail: [e-mail]

(Received: May 2, 2000 ; Accepted: June 18, 2001)

Abstract - Cost optimization in the synthesis decision tree often leads to a reduced degree of freedom which degrades the process’s ability to reject disturbances as a consequence of low controllability. In fact, Heat Exchanger Networks (HENs) obtained by traditional synthesis procedures that ignore controllability aspects must be evaluated in this context a posteriori. The aim of this work was to develop a procedure that includes RGA and SVD measures of controllability, which are solely based on steady state information, thereby freeing the synthesis procedure of the cumbersome dynamic analysis. When a structure is defined during a traditional HEN synthesis procedure, a degree of freedom analysis is approached as a simulation problem. Next, an optimization is performed, since new variables are usually added to increase the degree of freedom of the HEN in order to render it controllable. A key point in the proposed procedure is the inference of controllability based on the proposed controllability measures, which also provide a control scheme by pairing controlled and manipulated variables during the process design. A HEN reported in the literature is used to illustrate the proposed procedure. The steady state simulator Aspen Plus and the dynamic simulator Aspen Dynamics (Aspentech, Inc.) were employed.

Keywords: control, heat exchanger network, cost optimization

INTRODUCTION

The control system of a chemical plant is often analyzed only after the plant design has been defined, and it has traditionally been evaluated by designer expertise, heuristic rules and/or trial-and-error methods. A dynamic process model is rarely used. Process synthesis and control system design are usually considered to be separate stages instead of an integrated procedure. Thus, in the synthesis phase, an easily controllable process alternative can be rejected in favor of a more economical alternative that might be extremely hard to control. Additionally, plants that are hard to control are normally not flexible and show no capacity to reject disturbances, features that can lead to extra control/operational costs.

It is noteworthy that controllability does not depend on controller type and is a property of the process that can only be altered by process modifications, which can involve changes in equipment (type, size, etc); location of sensors and actuators; addition of new equipment; addition of new process lines, such as bypasses; and redefinition of the control objective (for example, temperature targets in HENs) (Skogestad and Postlethwaite, 1996).

With a focus on process integration, the HEN synthesis problem has been widely studied during the last three decades. Only recently, however, studies dealing simultaneously with synthesis and control have been reported, but their approach is general(Luyben and Floudas, 1994; Luyben and Luyben, 1997). The need for better HEN control systems has been imposed by processes that deal with high quality products and with multiple steady-state operational conditions (flexible HEN). Further, it has been observed that an increase in process integration promotes interaction between operational parameters and results in a reduction in available degrees of freedom for achieving control objectives.

A flexible HEN must have the capability to operate under different operational conditions, i.e., more than one steady state condition. Many studies dealing with this subject are presented in the literature, such as Floudas and Grossmann (1986), Kotjabasakis and Linnhoff (1986), Ravagnani and Módenes (1996) and Oliveira et al. (1998). Such approaches deal exclusively with operational features of flexible HENs and do not account for dynamic and control aspects.

Controllability, a process feature associated with transient disturbances of a small amplitude around operating points (periods), is a fundamental requirement imposed on all HENs. Thus, during HEN synthesis interaction between design and control becomes essential. Nevertheless, this field has yet to be sufficiently explored. Controllability studies in HEN synthesis, including a study on a preliminary HEN control structure, are expected to improve the HEN’s operability. These aspects have been addressed by Marselle et al. (1982), Huang and Fan (1992), Mathisen (1994) and Oliveira et al. (1999).

The aim of the present study is to show the importance of taking into account preliminary control considerations in HEN synthesis. A procedure is proposed to verify the controllability of HEN using controllability measures based solely on steady-state information, which is simple and requires negligible computational efforts, compatible with the synthesis methods. From among the many measures available in the literature, the RGA method is chosen. Whenever this method indicates more than one process design solution, the SVD method is used. In this type of situation, SVD is applied to select the HEN structure from among the best RGA results. Note that the procedure proposed herein must be employed at each step of the synthesis algorithm to identify unfeasible structures that must be rejected or to receive modifications that will make it controllable and then feasible. As will be shown, these modifications to make the HEN controllable can affect its cost, which is the objective function in the synthesis algorithm.

The final structure of a HEN presented by Quesada and Grossmann (1993), which was synthesized with no controllability concerns, is used to introduce the proposed methodology. The original HEN had to be modified to impart controllability to the final HEN. Process changes include the addition of bypasses and modification of original heat transfer areas. Note that these modifications present the trade-off between increasing controllability and increasing the total cost of the final HEN.

THE ORIGINAL PROBLEM

Process streams data from the original HEN are presented in Table 1. Figure 1 shows the HEN structure proposed by Quesada and Grossmann (1993). Heat exchanger design and cost parameters are presented in Table 2.

HE(n) is the heat exchanger, Q(n) is the heat load, A(n) is the heat transfer area, TD(n) is the temperature difference, T(n) is the temperature, H(n) is the hot stream, C(n) is the cold stream, f1 is the split and m1 is the mixer.

Note that the target temperatures of streams C2 and C3 are not specified. It is assumed that these temperatures can be adjusted outside the HEN at their downstream targets.

Simulation

First, in order to validate the steady state model (built in Aspen Plus, Aspentech Inc.), a steady state simulation was performed and results were compared to those reported by Quesada and Grossmann (1993). A degree-of-freedom analysis of the HEN model indicated 17 variables and 16 equations; thus there is one degree of freedom. In order to solve the equation set in a sequential way, one of the 17 variables had to be chosen as a "tear variable." The split fraction f₁ (refer to Figure 1) was then chosen. In their study, Quesada and Grossmann (1993) adopted the heat load (Q1) of heat exchanger 1 (HE 1) as the "tear variable," which has as optimal value of Q1 = 978.1 kW, with a corresponding f₁ value (fraction of C1 deviated to HE1) equal to 0.9781. Some relevant steady-state simulation results obtained with f₁ = 0.9781 are presented in Table 3 and are in good agreement with those reported by Quesada and Grossmann (1993).

In the optimization procedure, the total annual cost of the HEN ($/year) was calculated by Equation (1). Global heat transfer coefficients were maintained constant and equal to the values shown in Table 2.

Optimization

The original HEN shows three temperature targets (outlet temperatures of C1, H1 and H2), i.e., three control objectives, and only one degree of freedom (one possible variable to manipulate, f₁). In order to be able to control the outputs in a multi-loop scheme at least as many manipulated inputs as controlled outputs are necessary (Skogestad and Postlethwaite, 1996). Such a network lacks functional controllability and modifications of the original structure are needed to attain control objectives. The variable that can be added without changing the main HEN structure, i.e., the connections between the heat exchangers, is the bypass. One possibility is to create bypasses on process streams H1, H2, C2 and C3 around HE 3 and HE 4, introducing four new manipulated variables, f₄, f₅, f₇ and f₈. Bypasses around HE 1 and HE 2 are not necessary as splitter f₁ already influences these two units. Figure 2 shows the modified HEN with the new process variables.

The new process variables require an optimization procedure in order to determine their operational conditions. The problem consists in minimizing the objective function represented by Equation (1), subject to the constraints resulting from

the specification of TDmin = 5.0 K in the original problem: T

₁³ 405.0 K; T

₂³ 405.0 K; T

₃£ 570.0 K; T

₄£ 713.0 K; T

₅ = 400.0 K; T

₆³ 370.0 K; T

₇³ 363.0 K.

Optimization results, also obtained with the Aspen Plus Simulator, show a heat transfer area of less than 1 m² for HE 3. As this is unusual, a new restriction was arbitrarily added: An ³ 1.0 m². Table 4 shows the final results. Note that all heat transfer areas have changed and that an increase in the HEN TAC occurred. Bypass costs were not accounted for in the objective function; nevertheless, their presence affected the heat transfer areas.

Obs.: f_i (i ¹1) represents the fraction of the stream that effectively flows through the exchanger.

After the addition of the new splitters, the HEN has five manipulated variables (f₁, f₄, f₅, f₇ and f₈). Hence, for the set of controlled variables (T_C1, T_H1, T_H2), there are ten possible sets of manipulated variables.

Table 4 shows that bypasses f₇ and f₈ should be completely open, i.e., saturated manipulated variables, while f₄ and f₅ should be partially open. As f₇ and f₈ are saturated, these values do not contribute to improving process controllability.

Control

After introducing bypasses and adapting the structure to their presence, the first step of the controllability analysis is a sensitivity study of the optimal results of the simulation stage.

Percent changes in the three controlled variables, i.e., the outlet temperatures of streams C1, H1 and H2 (T_C1, T_H1 and T_H2) with variations of -5.0 % in each input variable (manipulated variable) are shown in Table 5.

Results shown in Table 5 permit the building of the Gain Array (K). Its element K_ij is defined by Equation (2) (Ogunnaike and Ray, 1994):

where

CV_i is the controlled variable i,

CV_i,base is the controlled variable i in the base case (Table 4),

MV_j is the manipulated variable j and

MV_j,base is the manipulated variable j in the base case (Table 4). The resulting K array is

From K, it is possible to define a CV-MV pairing (Ogunnaike and Ray, 1994). Based on the need for simplicity and effectiveness, the Relative Gain Array (RGA) was selected. In some structures, RGA is not able to indicate the best pairing. In such cases, the Singular Value Decomposition (SVD) was used.

Although f₇ and f₈ are practically saturated, they remain in the pairing set for selection. If one (or both) of them is chosen, a new optimization should be conducted with constraints on its upper and lower bounds in order to allow an effective control action. It should be noted that this new optimization leads to new area values that can interfere with the previous CV-MV pairing.

Relative Gain Array (L )

Relative Gain Array L is obtained from K as follows:

Element L_ij is a measure of the interaction between control loops (cross effects). Match MV-CV is determined by choosing, for each j^th column (CV_j), the i^th row (MV_i) which has the element nearest to 1.

As the process matrix is not square, 3x3 square matrices were defined and their condition number were calculated. The results are shown in Table 6. The pairing indicated by this method is that with the lowest condition number (cn) (Ogunnaike and Ray, 1994).

The relative gain matrix with the lowest cn, representing the combination (f₁, f₄, f₈) in Table 6, is given by

Singular Value Decomposition (SVD)

SVD provides quantitative information about controllability, sensor location and controller pairings. Despite having been recommended for use only in scaled problems, in this study the use of SVD is proposed for cases where RGA fails to indicate a single best structure, i.e., more than one pairing presents similar RGA results. For such cases, SVD analysis is applied exclusively to the set of best pairings indicated by RGA analysis, which is a dimensionless measure of controllability. Furthermore, it is worth noting that the HEN output variables are almost exclusively temperatures and input variables are often flowrates._Therefore, problems of scaling are not as serious as those presented by distillation processes, for example, where output variables could be temperatures and molar fractions (ranging from 10^-2 to 10² orders of magnitude).

SVD of a matrix results in three other matrices: i) an orthogonal array whose columns are the singular left vectors (U), ii) an orthogonal array whose columns are the singular right vectors (V), and iii) a diagonal array whose scalars are the eigenvalues in decreasing order (S).

The best pairing (the choice CV-MV) is the one in which the sensor associated with the highest component of column vector U₁ (array associated with the controlled variables) is paired with the manipulated variable associated with the highest component of column vector V₁ (array associated with the manipulated variables). This procedure continues until every controlled variable (all columns of U) is paired with one manipulated variable. Condition number is calculated for matrix S (singular value array) (Ogunnaike and Ray, 1994).

The SVD matrix with the lowest cn, representing the combination (f₁, f₄, f₈) in Table 6, is given by

Table 6 shows all the manipulated variable sets and their respective condition numbers obtained with each procedure.

In the present case study, the RGA analysis results in two sets with the same lowest condition number (in bold) and is, therefore, inconclusive. Then, this is a case where the pairing should be defined with the aid of SVD results. The final results are thus T_H1 - f₁, T_H2 – f₈ and T_C1 - f₄. That is, the outlet temperature of HE 1 (T_H1) will be controlled by bypass f1, the outlet temperature of HE 2 (T_H2) will be controlled by bypass f8 and the outlet temperature of mixer m1 (T_C1) will be controlled by bypass f₄. The P & I diagram is shown in Figure 3.

Once the bypasses to be used as control were defined, a new optimization was recommended, imposing upper and lower bounds for bypasses f₁, f₄ and f₈. No bounds on bypasses f₅ and f₇ were imposed for optimization, as they were not used in the control scheme.

It was verified that, for any upper bound value, the optimum f₈ always lies on this bound. Furthermore, the higher this bound is, the smaller the TAC. Therefore, an analysis had to be conducted to determine the upper bound value in order to minimize the TAC and also to allow bypass f₈ to control its target (T_H2). A similar situation occurred for bypass f₄, but with the lower bound. A steady-state sensitivity analysis with the new variable values showed that a span of 0.1 for both bypasses would be sufficient to respond to a 5.0% variation in the cold stream C1 mass flowrate. Therefore, the bounds assumed for f_i values in the second optimization procedure were 0.1 £ f₄£ 0.5; 0.5 £ f₈£ 0.9 and 0.1 £ f₁£ 0.9. The new optimization results are given in Table 7.

A comparison of the results in Tables 4 and 7 indicates changes in some heat exchanger areas and bypass fractions. These changes led to variations in the results obtained by the sensitivity analysis, upon which the control study and CV-MV pairing were based. Therefore, at this point, a new control study had to be performed, following the same procedure in order to verify whether the CV-MV pairing was to be maintained. If the answer were no, a new optimization, preceded by an analysis and definition of the constraints on the new control bypasses, would be needed. This procedure had to be repeated until maintenance of the CV-MV pairing after a new optimization was achieved.

In the present study, a new sensitivity analysis was conducted using the data in Table 7. Its results gave the same CV-MV pairing, as can be seen in Table 8, which shows the new RGA and SVD results. Therefore, the CV-MV pairing shown in Figure 3 was used.

With the pairing selected, the structure is defined as controllable. This procedure is to be conducted at every step of a HEN synthesis in order to account for the controllability aspects.

In the following section, a study that was aimed at showing that the structure obtained was indeed controllable is discussed. An additional procedure such as this could be used in flexible HEN synthesis to analyze the feasibility of transitions from one period to another. As a by-product, this analysis indicates a control system and its preliminary tuning parameters. Moreover, the procedure provides information that could make it possible to account for controller costs in HEN synthesis.

DYNAMIC SIMULATION AND CONTROL PARAMETERS SETUP

The results shown in Table 7 were used to perform a HEN dynamic study using the Aspen Dynamics (Aspentech, Inc.). When modeling the heat exchangers, the following approach was adopted:

(a) heat exchangers were assumed to be in a countercurrent configuration;

(b) the model was based on finite differences, with both shell and tube sides assumed to be plug flow that were approached as a series of well-mixed tanks ("cells");

(c) the number of cells (N) was defined after various successive simulations. Results indicated that the value N = 10 could be assumed for all heat exchangers.

The proportional gain (Kc) and integral time constant (ti) for each of the three controllers were determined based on the consideration that the process response could be approximated by a first-order response with a time delay. Therefore, the parameters to be identified from dynamic responses to variations in manipulated variables were process gain (k), time delay (q) and time constant (t). With these values and using the Cohen-Coon correlations (Ogunnaike and Ray, 1994), Kc and ti were estimated for each controller. Note that there are other procedures to define the control system. Again, the choice was based on simplicity.

Figures 4, 5 and 6 show the control variable dynamic response for a ± 5.0% variation in its respective manipulated variables. From these figures, parameters k, q and t were obtained for each CV-CM pair and the results are shown in Table 9.

Values for Kc and ti given in Table 9 were unsatisfactory in closed loop due to interactions between loops (Luyben, 1990). To overcome this problem, Luyben (1990) proposed a detuning factor for the proportional action in order to reduce interaction between loops. Thus, a detuning factor, ranging between 10 and 500, was used in the dynamic simulation. Figure 7 shows the dynamic behavior of process-controlled variables, and Figure 8 shows the respective behaviors of manipulated variables for a perturbation of +5.0% in the flowrate of stream C1.

In order to obtain better knowledge of the process dynamics, other disturbances were imposed. These perturbations were of +5% in the flowrates of streams H1 and H2. Results on the dynamic behavior of controlled and manipulated variables are given in Figures 9, 10, 11 and 12.

Figure 7 indicates that disturbances in stream C1 caused the largest impact on the network’s dynamic behavior. This fact can be observed by the intensive oscillation of controlled variables around their set points. Figure 8 shows the behavior of the manipulated variables due to the disturbance in stream C1 in order to keep the controlled variables at their set points.

Figures 9 and 11 show the impact of disturbances in the flowrates of streams H1 and H2, respectively, on controlled variable behavior for the control loop in operation. Both disturbances result in peaks in the values of the controlled, variables followed by a smooth return to their set points. The dynamic behavior of the corresponding manipulated variables is shown in Figures 10 and 12.

CONCLUSIONS

First it was shown that the HEN proposed by Quesada and Grossmann (1993) did not present the proper number of manipulated variables to control the target temperatures, i.e., the HEN is not functionally controllable. Using the procedure proposed herein, new splitters were added to the structure, thereby increasing the number of manipulated variables. Therefore, a new design of the network’s heat exchangers was required, which led to changes in the values of the original heat transfer areas, thus showing the influence of control considerations on the final synthesis result.

In the next step of the proposed procedure, a controllability analysis identified bypasses that were more effective in the HEN control. Then the HEN was redesigned, without modifying its main structure by a new optimization imposing upper and lower bounds on these control splitters (selected by controllability analysis) in order to avoid their saturation during dynamic upsets. If this new optimization procedure leads to new heat transfer areas, an iterative procedure is needed to assure the selected MV-CV pairing is correct. Next, a dynamic simulation of the closed-loop system was developed and performed, showing that the proposed control loops were able to absorb variations in inlet conditions of the process streams, thereby validating the procedure proposed herein.

In the proposed procedure, considerations on the control system are to be made after each network structure definition during the synthesis. These considerations increase the TACs even without accounting for the control system (hardware, sensors and valves) costs. This suggests the possibility that these considerations will affect the phase of network structure definition (synthesis). Structures previously rejected, but showing some advantages in terms of process control, could lead to a lower TAC. The results shown in Figures 7 and 8 could be an indication that the structure adapted to this HEN is not the best choice. In the procedure presented herein, the definition of f values was based exclusively on steady-state information and were tested in a dynamic simulation.

Summing up, the procedure proposed herein is simple and effective enough to be used in HEN synthesis. It makes it possible to take into account control-related costs, which are usually neglected in traditional HEN synthesis procedures. In the case study, it is shown that even when neglecting direct control costs, the addition of bypasses, which increases the number of manipulated variables making control possible, causes changes in the heat transfer areas and then in the network TAC.

NOMENCLATURE

Latin Letters

A heat transfer area (m2) C cold stream CI cost coefficient per heat transfer area (US$/m²) CV controlled variable f split H hot stream HE heat exchanger K gain array m mixer MCp specific heat flowrate (kW/K) MV manipulated variable N number of cells Q heat load (kW) RGA Relative Gain Array S singular value array SVD Singular Value Decomposition T temperature (K) TAC total annual cost (US$/year) TDmin minimum temperature difference (K) TD temperature difference (K) Ti inlet temperature (K) To outlet temperature (K) U array associated with the controlled variables Ui global heat transfer coefficient (kW/Km²) V array associated with the manipulated variables

Greek Letters

k process gain (K/s) Kc proportional gain (s/K) L relative gain array q time delay (s) t time constant (s) ti integral time constant (s)

ACKNOWLEDGEMENTS

The authors would like to acknowledge the financial support received from FAPERJ (Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro), CAPES (Coordenadoria de Aperfeiçoamento de Pessoal de Nível Superior) and CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico).

REFERENCES

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Publication Dates

• Publication in this collection
  30 July 2001
• Date of issue
  June 2001

History

• Accepted
  18 June 2001
• Received
  02 May 2000