# A Framework for Flexible and Cost-Efficient Retrofit Measures of Heat Exchanger Networks

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

#### 2.1. (Structural) Flexibility Analysis

#### 2.2. Multi-Period Design Problem and Critical Point Analysis

“The main idea is to identify points with the largest values of design variables at optimum objective function. This may be achieved by maximizing design variables one by one, while allowing uncertain parameters to obtain any value between the specified bounds, and simultaneously minimizing cost function.”[44] (p. 1607)

- The Karush–Kuhn–Tucker (KKT-) formulation;
- The two-level formulation; and
- The approximate one-level formulation.

## 3. Methodology

#### 3.1. HEN Retrofit Design Proposals for Single Operating Point—Generation of Superstructure

#### 3.2. Structural Feasibility Assessment—Reduction of Superstructure

#### 3.3. Determination of Critical Points

#### 3.4. Multi-Period Design Problem for Reduced Superstructure

#### 3.5. Feasibility Check

## 4. Illustrative Example

^{2}°C) was assumed.

#### 4.1. Structural Feasibility Assessment

#### 4.2. Economic Data and Identification of Critical Points

- Investment cost for a new HEX: $\mathrm{Inv}.\mathrm{cos}{\mathrm{t}}_{\mathrm{new}}$;
- Investment cost for increasing an existing HEX: $\mathrm{Inv}.\mathrm{cos}{\mathrm{t}}_{\mathrm{increase}}$;
- Operational cost for utility cooling: ${\mathrm{p}}_{\mathrm{CU}}$; and
- Operational cost for utility heating: ${\mathrm{p}}_{\mathrm{HU}}$.

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#### 4.3. Solution of the Multi-Period Design Problem and Final Feasibility Check

#### 4.4. Discussion of the Results of the Illustrative Example

## 5. Conclusions

- Development of a novel framework to achieve flexible and cost-efficient retrofit measures for HENs operating at multiple periods;
- Implementation of critical point analysis in a retrofitting framework to achieve flexible retrofit solutions of HENs;
- Possibility to employ well-proven, single-period retrofit design methods, e.g., advanced composite curves, Bridge analysis, etc., but also “experience-based” retrofit design proposals in a framework that guarantees flexibility and cost efficiency (with respect to investment and operating cost) for the entire operating period;
- Possibility to compare design proposals of different single-period retrofit design methods in a superstructure-based mathematical program ensuring flexibility and cost efficiency with respect to investment and operating cost;
- Automatization of the KKT-formulation, the two-level formulation, and the set-covering algorithm initially suggested by Pintarič and Kravanja [44] to identify critical points of different design proposals (see Supplementary Material); and
- Extension of the set covering algorithm suggested by Pintarič and Kravanja [44] in order to handle more complex network structures (see Appendix A).

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Discussion on the Determination of Critical Points

#### Appendix A.1. Two-Level Formulation

- The lower (control-design) level problem; and
- The upper (uncertainty) level problem.

Control Variables | Critical Points (Read as Follows) [T _{in,H1}, T_{in,H2}, T_{in,C1}, T_{in,C2}, Fcp_{H1}, Fcp_{H2}, Fcp_{C1}, Fcp_{C2}]with T in [°C] and Fcp in [kW/°C] | Total Annualized Cost [€/y] | Flexibility Index [-] |
---|---|---|---|

Q3, Q4 | [240.0, 190.0, 19.0, 165.0, 20.0, 20.0, 20.8, 28.0] [255.0, 190.0, 21.0, 165.0, 20.0, 30.0, 19.2, 28.0] | 221,400 | 0.69 |

Q3, Q5 | [240.0, 190.0, 21.0, 120.0, 10.0, 0, 20.8, 32.0] [255.0, 190.0, 21.0, 165.0, 20.0, 0, 20.8, 28.0] [0, 190.0, 21.0, 120.0, 20.0, 0, 20.8, 32.0] | 218,600 | 0.5 |

Q4, Q5 | [240.0, 190.0, 21.0, 120.0, 10.0, 30.0, 19.2, 32.0] [240.0, 190.0, 19.0, 120.0, 10.0, 20.0, 20.8, 32.0] | 250,000 | 1.00 |

Control Variables | Critical Points (Read as Follows) [T _{in,H1}, T_{in,H2}, T_{in,C1}, T_{in,C2}, Fcp_{H1}, Fcp_{H2}, Fcp_{C1}, Fcp_{C2}]with T in [°C] and Fcp in [kW/°C] | Total Annualized Cost [€/y] | Flexibility Index [-] | Convergence between Upper and Lower Problem |
---|---|---|---|---|

Q4, Q5, Q6 | [240.0, 190.0, 19.0, 120.0, 20.0, 30.0, 20.8, 32.0] [240.0, 190.0, 19.0, 120.0, 20.0, 20.0, 20.8, 32.0] [255.0, 190.0, 21.0, 120.0, 20.0, 0, 19.2, 32.0] | 227,200 | 0.92 | Not converged (30 iterations) for some design variables |

Q4, Q6, T8 | [240.0, 190.0, 19.0, 120.0, 20.0, 20.0, 20.8, 32.0] [255.0, 190.0, 19.0, 120.0, 20.0, 20.0, 19.2, 32.0] | 227,200 | 0.92 | converged |

Q6, T2, T8 | [240.0, 190.0, 21.0, 120.0, 20.0, 0, 20.8, 32.0] [255.0, 190.0, 21.0, 120.0, 20.0, 0, 19.2, 32.0] | 227,200 | 0.92 | converged |

T1, T4, T7 | [240.0, 210.0, 19.0, 165.0, 10.0, 30.0, 20.8, 28.0] [255.0, 210.0, 19.0, 165.0, 20.0, 30.0, 20.8, 28.0] | 227,200 | 0.92 | converged |

- Ratio between the number of process streams and the number of process-to-process HEX: The higher this number, the less complex the HEN; and
- Number of process-to-process HEX connected to a single process stream: The higher this number, the more complex the HEN.

#### Appendix A.2. A Karush–Kuhn–Tucker (KKT-)Formulation

Proposal Number | Critical Points (Read as Follows) [T _{in,H1}, T_{in,H2}, T_{in,C1}, T_{in,C2}, Fcp_{H1}, Fcp_{H2}, Fcp_{C1}, Fcp_{C2}]with T in [°C] and Fcp in [kW/°C] | Total Annualized Cost [€/y] | Flexibility Index [-] |
---|---|---|---|

MER | [240.0, 190.0, 21.0, 120.0, 20.0, 0, 20.8, 32.0] [255.0, 0, 21.0, 165.0, 20.0, 30.0, 19.2, 28.0] [240.0, 210.0, 21.0, 120.0, 0, 30.0, 19.2, 32.0] [0, 190.0, 19.0, 120.0, 10.0, 20.0, 20.8, 32.0] | 202,000 | 0.75 |

_{H2}in the first identified critical point of the set shown in Table A3). These “0” elements result from the calculation of the marginal values of the uncertain parameters, meaning that the corresponding uncertain parameter was discovered to have no influence on at least one of the design variables when solving the KKT formulation (compare [44]). When solving the design problem, the uncertain parameters corresponding to these “0” elements are defined as variables for the respective critical point, i.e., these uncertain parameters can obtain any value within the uncertainty span at the solution of the design problem. As the identified sets were incomplete, it seemed that the influence of one or several uncertain parameters was not correctly detected. This was investigated by extending the set covering algorithm. In addition to the three steps reported in the appendix of [44], a fourth step was added. In this step, each “0” element present in the set of critical points, obtained after the first three steps, was replaced with all possible combinations with respect to the other points in the set and the set covering formulation (AP6 in [44]) is solved.

#### Appendix A.3. Adding Critical Points via Flexibility Assessment

## Appendix B

T_{in,H1} [°C] | T_{in,H2} [°C] | T_{in,HC1} [°C] | T_{in,C2} [°C] | Fcp_{H1} [kW/°C] | Fcp_{H2} [kW/°C] | Fcp_{C1} [kW/°C] | Fcp_{C2} [kW/°C] | W_{s} [-] |
---|---|---|---|---|---|---|---|---|

243.65 | 208.10 | 19.59 | 155.49 | 15.44 | 23.23 | 20.14 | 28.60 | 1/11 |

249.72 | 202.83 | 20.29 | 156.26 | 16.14 | 21.22 | 19.87 | 29.51 | 1/11 |

246.41 | 202.73 | 20.34 | 162.06 | 16.50 | 28.27 | 20.49 | 29.59 | 1/11 |

248.81 | 194.63 | 20.04 | 133.91 | 16.40 | 25.15 | 20.23 | 31.24 | 1/11 |

249.82 | 202.35 | 20.76 | 136.65 | 12.00 | 27.33 | 19.72 | 31.40 | 1/11 |

247.16 | 200.40 | 19.21 | 124.26 | 14.96 | 25.44 | 19.88 | 30.61 | 1/11 |

248.18 | 200.50 | 20.56 | 158.58 | 19.83 | 24.80 | 20.51 | 28.78 | 1/11 |

242.30 | 206.05 | 19.89 | 157.40 | 11.83 | 22.99 | 19.61 | 30.97 | 1/11 |

244.50 | 207.36 | 19.13 | 160.11 | 14.09 | 25.36 | 19.99 | 30.75 | 1/11 |

248.71 | 204.38 | 20.02 | 140.01 | 13.03 | 20.93 | 20.18 | 30.28 | 1/11 |

253.53 | 191.07 | 19.39 | 127.44 | 12.99 | 27.18 | 20.29 | 30.82 | 1/11 |

Proposal Number | Critical Points (Read as Follows) [T _{in,H1}, T_{in,H2}, T_{in,C1}, T_{in,C2}, Fcp_{H1}, Fcp_{H2}, Fcp_{C1}, Fcp_{C2}]with T in [°C] and Fcp in [kW/°C] |
---|---|

1) | [240.0, 190.0, 21.0, 120.0, 10.0, 20.0, 20.8, 32.0] |

[255.0, 210.0, 19.0, 120.0, 10.0, 30.0, 20.8, 32.0] | |

[240.0, 190.0, 19.0, 120.0, 10.0, 20.0, 20.8, 32.0] | |

2) | [240.0, 190.0, 19.0, 120.0, 10.0, 20.0, 20.8, 32.0] |

[240.0, 190.0, 19.0, 120.0, 20.0, 20.0, 20.8, 32.0] | |

[255.0, 190.0, 19.0, 165.0, 20.0, 20.0, 19.2, 28.0] | |

[255.0, 190.0, 19.0, 165.0, 0, 20.0, 20.8, 28.0] | |

3) | [240.0, 190.0, 21.0, 120.0, 10.0, 20.0, 20.8, 28.0] |

[240.0, 190.0, 19.0, 120.0, 10.0, 20.0, 20.8, 28.0] | |

[255.0, 190.0, 0, 120.0, 20.0, 20.0, 20.8, 28.0] | |

4) | [240.0, 190.0, 21.0, 120.0, 20.0, 20.0, 20.8, 32.0] |

[240.0, 190.0, 19.0, 0, 10.0, 20.0, 20.8, 0] | |

[255.0, 190.0, 0, 165.0, 20.0, 20.0, 19.2, 28.0] | |

MER) | [255.0, 190.0, 21.0, 165.0, 20.0, 30.0, 19.2, 28.0] |

[255.0, 190.0, 19.0, 120.0, 10.0, 20.0, 20.8, 32.0] | |

[240.0, 190.0, 21.0, 120.0, 20.0, 20.0, 20.8, 32.0] | |

[240.0, 210.0, 21.0, 120.0, 10.0, 30.0, 19.2, 32.0] | |

[255.0, 210.0, 21.0, 165.0, 20.0, 30.0, 19.2, 28.0] | |

[240.0, 190.0, 21.0, 120.0, 20.0, 30.0, 20.8, 32.0] | |

[240.0, 210.0, 21.0, 120.0, 20.0, 30.0, 19.2, 32.0] | |

[240.0, 190.0, 19.0, 120.0, 10.0, 20.0, 20.8, 32.0] |

HEX | ΔUA [kW/°C] | ΔA [m^{2}] |
---|---|---|

1 | 13.58 | 25.97 |

2 | 1.02 | 1.95 |

7 | 77.81 | 148.77 |

8 | 9.94 | 19.01 |

HEX | ΔUA [kW/°C] | ΔA [m^{2}] |
---|---|---|

1 | 0.0 | 0.0 |

2 | 26.5 | 50.67 |

7 | 26.59 | 50.84 |

8 | 19.35 | 37.0 |

HEX | ΔUA [kW/°C] | ΔA [m^{2}] |
---|---|---|

1 | 74.87 | 143.16 |

2 | 39.71 | 75.93 |

7 | 57.59 | 110.11 |

HEX | ΔUA [kW/°C] | ΔA [m^{2}] |
---|---|---|

1 | 74.87 | 143.16 |

2 | 20.04 | 38.32 |

7 | 24.69 | 47.21 |

HEX | ΔUA [kW/°C] | ΔA [m^{2}] |
---|---|---|

1 | 26.71 | 51.07 |

2 | 103.96 | 198.78 |

7 | 33.22 | 63.51 |

8 | 9.57 | 18.30 |

9 | 89.36 | 170.85 |

Parameter | Value |
---|---|

Initial temperature | 10 K |

Final temperature | 2 × 10^{−20} K |

Annealing of temperature | 0.974*T |

Stop criterion | No improvement after 2000 iterations |

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**Figure 1.**(Hyper-)rectangle with respect to expected variations and maximum scaled (hyper-)rectangle inscribed within the feasible region in the space of the varying inlet temperatures of a heat exchanger network [38].

**Figure 2.**Retrofitting framework to achieve flexible and cost-efficient retrofit measures of heat exchanger networks (HENs).

**Table 1.**Hot utility demand for the initial heat exchanger network (HEN) and the different retrofit proposals as well as the potential savings for average operating conditions.

Proposal Number | Hot Utility Demand [kW] (at Average Conditions) | Hot Utility Savings [kW] (at Average Conditions) |
---|---|---|

Initial | 2700 | - |

1) | 1450 | 1250 |

2) | 1900 | 800 |

3) | 2000 | 700 |

4) | 2000 | 700 |

5) | 1450 | 1250 |

MER) | 750 | 1950 |

HEX | UA Value [kW/°C] | A [m^{2}] |
---|---|---|

1 | 33.68 | 64.4 |

2 | 12.71 | 24.3 |

Stream | ΔT^{+} [°C] | ΔT^{−} [°C] | ΔFcp^{+} [kW/°C] | ΔFcp^{−} [kW/°C] |
---|---|---|---|---|

H1 | 5 | 10 | 5 | 5 |

H2 | 10 | 10 | 5 | 5 |

C1 | 1 | 1 | 0.8 | 0.8 |

C2 | 25 | 20 | 2 | 2 |

Proposal Number | Structural Flexibility Index |
---|---|

1) | 1.47 |

2) | 1.47 |

3) | 1.0 |

4) | 1.0 |

5) | 0.07 |

MER) | 1.18 |

Utility | Price [€/MWh] |
---|---|

Cooling (${\mathrm{p}}_{\mathrm{C}\mathrm{U}}$) | 1.7 × 10^{−3} |

Heating (${\mathrm{p}}_{\mathrm{H}\mathrm{U}}$): | 15 |

**Table 6.**Total annualized cost, annual net savings, and flexibility index of the different retrofit design proposals in the reduced superstructure.

Proposal Number | Total Annualized Cost [€/y] | Net Savings [€/y] | Flexibility Index [-] |
---|---|---|---|

1) | 228,000 | 86,600 | 1.00 |

2) | 237,300 | 77,300 | 1.19 |

3) | 250,000 | 64,600 | 1.00 |

4) | 262,900 | 51,700 | 1.00 |

MER) | 223,100 | 91,500 | 1.00 |

**Table 7.**Total annualized (capital) cost and flexibility index of the different retrofit design proposals in the reduced superstructure without considering the representative operating points.

Proposal Number | Total Annualized (Capital) Cost [€/y] | Flexibility Index [-] |
---|---|---|

1) | 19,600 | 1.00 |

2) | 19,600 | 1.00 |

3) | 37,200 | 1.00 |

4) | 37,200 | 1.00 |

MER) | 95,600 | 1.00 |

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**MDPI and ACS Style**

Langner, C.; Svensson, E.; Harvey, S.
A Framework for Flexible and Cost-Efficient Retrofit Measures of Heat Exchanger Networks. *Energies* **2020**, *13*, 1472.
https://doi.org/10.3390/en13061472

**AMA Style**

Langner C, Svensson E, Harvey S.
A Framework for Flexible and Cost-Efficient Retrofit Measures of Heat Exchanger Networks. *Energies*. 2020; 13(6):1472.
https://doi.org/10.3390/en13061472

**Chicago/Turabian Style**

Langner, Christian, Elin Svensson, and Simon Harvey.
2020. "A Framework for Flexible and Cost-Efficient Retrofit Measures of Heat Exchanger Networks" *Energies* 13, no. 6: 1472.
https://doi.org/10.3390/en13061472