# Performance Optimization of a Condenser in Ocean Thermal Energy Conversion (OTEC) System Based on Constructal Theory and a Multi-Objective Genetic Algorithm

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## Abstract

**:**

## 1. Introduction

## 2. Plate Condenser and Its Performance

#### 2.1. Structure of Plate Condenser

#### 2.2. Assumptions of Model

- (1)
- The flow in the plate condenser is a stable state and homogeneous for flow direction.
- (2)
- The working fluid is a two-phase state at the inlet of the plate condenser.
- (3)
- Considering that the cold seawater is enough, the working fluid will be cooled to a saturated liquid state (SLS).
- (4)
- The pressure drops at the manifolds and ports are ignored because only the HTP structure is studied in this paper. The influences of the manifolds and ports on the pressure drops and overall performance of OTEC system will be studied in future work.

#### 2.3. Performance of Plate Condenser on Working Fluid Side

#### 2.4. Performance of Plate Condenser on Cold Seawater Side

#### 2.5. Overall Performance of Plate Condenser

## 3. Constructal Design for Plate Condenser with Conventional Optimization Method

#### 3.1. Optimization Objective of Constructal Design

#### 3.2. Optimization Procedure of Constructal Design

- (1)
- The single variable optimization is carried out. The relationship between the CF (${F}_{\mathrm{SP}}$) and HTP effective length (${L}_{\mathrm{eff}}$) is obtained with the given HTP width ($w$) and effective number (${N}_{\mathrm{eff}}$). The relationships between ${F}_{\mathrm{SP}}$ and $w$ as well as between ${F}_{\mathrm{SP}}$ and ${N}_{\mathrm{eff}}$ are obtained by applying a similar method.
- (2)
- The double variable optimization is carried out by releasing the constraint of $w$ on the basis of singly optimizing ${L}_{\mathrm{eff}}$. The relationships among ${F}_{\mathrm{SP}}$, ${L}_{\mathrm{eff}}$, and $w$ are obtained with the given ${N}_{\mathrm{eff}}$.
- (3)
- The three variable optimization is carried out by releasing constraint of ${N}_{\mathrm{eff}}$ on the basis of step 2. The relationships among ${F}_{\mathrm{SP}}$, ${L}_{\mathrm{eff}}$, $w$, and ${N}_{\mathrm{eff}}$ are obtained.
- (4)
- On the basis of step 3, the CORs of the plate condenser with different structural parameters and weighting coefficient are obtained. The subscripts “m” and “mm” mean the primary and twice minimizations, respectively, and the subscripts “opt” and “oo” mean the primary and twice optimizations, respectively.

#### 3.3. Results of Constructal Designs

#### 3.3.1. Single Variable Optimization

#### 3.3.2. Double Variable Optimization

#### 3.3.3. Three Variable Optimization

#### 3.3.4. Effects of Design Parameters on Optimization Results

## 4. Constructal Design for Plate Condenser with Multi-Objective Genetic Algorithm

#### 4.1. Optimization Procedure of Multi-Objective Genetic Algorithm

- (1)
- An initial population with $N$ scale is randomly generated, and the first offspring population is obtained through selection, crossover, and variation after non-dominated sorting.
- (2)
- The parent and offspring populations start to merge from the second generation. At the same time of performing the fast non-dominated sorting, the crowded distance of each individual in the non-dominated layer is calculated. A new parent population is formed by selecting suitable individuals based on the non-dominated relationship and the crowded distance of the individual.
- (3)
- A new offspring population is generated through selection, crossover, and variation of the parent population, and to circulate repeatedly until the ending conditions are satisfied. In this paper, the size of the population is set as 300, the evolution generation is set as 500, and “PlotFcns” is chosen as “gaplotpareto”.

#### 4.2. Results of Constructal Design

## 5. Conclusions

- (1)
- There is a primary optimal HTP effective length (${L}_{\mathrm{eff},\mathrm{opt}}=1.350\mathrm{m}$), a primary optimal HTP width (${w}_{\mathrm{opt}}=1.325\mathrm{m}$), and a primary optimal HTP effective number (${N}_{\mathrm{eff},\mathrm{opt}}=110$) to make ${F}_{\mathrm{SP}}$ respectively reach 0.922, 0.997, and 0.997. ${L}_{\mathrm{eff}}$ has a more significant effect than $w$ and ${N}_{\mathrm{eff}}$, and it can be chosen as the main design parameter to improve the performance of the plate condenser.
- (2)
- Continuing to optimize $w$ on the basis of singly optimizing ${L}_{\mathrm{eff}}$ can partly improve the comprehensive performance of the plate condenser. The twice minimum CF (${F}_{\mathrm{SP},\mathrm{mm}}$) after simultaneously optimizing ${L}_{\mathrm{eff}}$ and $w$ is 0.901, which is 2.3% less than ${F}_{\mathrm{SP},\mathrm{m}}$ after singly optimizing ${L}_{\mathrm{eff}}$. The twice optimal HTP effective length (${L}_{\mathrm{eff},\mathrm{oo}}$) and ${w}_{\mathrm{opt}}$ are $1.15\mathrm{m}$ and $1.55\mathrm{m}$, respectively.
- (3)
- Further optimizing the HTP effective number (${N}_{\mathrm{eff}}$) on the basis of twice optimization cannot significantly improve the comprehensive performance of the plate condenser. The corrugation angle ($\beta $), corrugation wavelength ($\mathsf{\Lambda}$), effective volume (${V}_{\mathrm{eff}}$), and weighting coefficient (${a}_{0}$) have different influences on the optimal performance and optimal construct. ${F}_{\mathrm{SP},\mathrm{mm}}$ gradually augments with the increases of $\beta $, $\mathsf{\Lambda}$, and ${a}_{0}$, and gradually diminishes with the increase of ${V}_{\mathrm{eff}}$.
- (4)
- Pareto optimal set can provide better choices for the performance optimizations of the plate condenser.
- (5)
- ${L}_{\mathrm{eff}}$ and $w$ are two important parameters of the plate condenser. Single, double, and three variable optimizations, as well as the Pareto optimal set, all provide the optimal design values of the plate condenser, and they can be an important basis and criteria for designers to design the plate condensers.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$A$ | heat transfer area, ${\mathrm{m}}^{2}$ |

${A}_{0}$ | heat transfer area of single heat transfer plate, ${\mathrm{m}}^{2}$ |

${A}_{\mathrm{pro}}$ | projected area, ${\mathrm{m}}^{2}$ |

${A}_{\mathrm{s}}$ | cross-sectional area of single flow channel, ${\mathrm{m}}^{2}$ |

$b$ | plate spacing between two adjacent plates, $\mathrm{m}$ |

${c}_{p}$ | specific heat at constant pressure, $\mathrm{J}/\left(\mathrm{kg}\cdot \mathrm{K}\right)$ |

${d}_{\mathrm{h}}$ | hydraulic diameter, $\mathrm{m}$ |

${F}_{\mathrm{SP}}$ | composite function |

$f$ | friction factor |

$G$ | mass flow rate per cross-sectional area, $kg/\left({m}^{2}\cdot s\right)$ |

$h$ | enthalpy, $\mathrm{J}/\mathrm{kg}$ |

$K$ | total heat transfer coefficient, $W/\left({m}^{2}\cdot K\right)$ |

$L$ | length, $\mathrm{m}$ |

$m$ | mass, $\mathrm{kg}$ |

$N$ | number of heat transfer plates or flow channels |

$Nu$ | Nusselt number |

$n$ | quantity of condensation sections |

$P$ | pumping power, $\mathrm{W}$ |

$p$ | pressure, $\mathrm{Pa}$ |

$Pr$ | Prandtl number |

$Q$ | quantity of heat transfer, $\mathrm{J}$ |

$R$ | fouling resistance, $\left({m}^{2}\cdot K\right)/W$ |

$Re$ | Reynolds number |

${S}_{\mathrm{g}}$ | entropy generation, $\mathrm{J}/\mathrm{K}$ |

$T$ | temperature, $\mathrm{K}$ |

$t$ | corrugation pitch, $\mathrm{mm}$ |

$V$ | volume, ${\mathrm{m}}^{3}$ |

$w$ | width, $\mathrm{m}$ |

$X$ | dimensionless corrugation parameter of heat transfer plate |

$x$ | vapor quality |

Greek letters | |

$\alpha $ | surface heat transfer coefficient, $\mathrm{W}/\left({\mathrm{m}}^{2}\cdot \mathrm{K}\right)$ |

$\beta $ | corrugation angle, ${}^{\mathrm{o}}$ |

$\delta $ | thickness, $\mathrm{m}$ |

$\varphi $ | surface enlargement factor of heat transfer plate |

${\eta}_{\mathrm{p}}$ | pump efficiency |

$\lambda $ | thermal conductivity, $\mathrm{W}/\left(\mathrm{m}\cdot \mathrm{K}\right)$ |

$\rho $ | density, $kg/{m}^{3}$ |

$\mu $ | dynamic viscosity, $\mathrm{Pa}\cdot \mathrm{s}$ |

$\mathsf{\Lambda}$ | corrugation wavelength, $\mathrm{mm}$ |

$\Delta T$ | logarithmic mean temperature difference, $\mathrm{K}$ |

$\Delta p$ | pressure drop, $\mathrm{Pa}$ |

Subscripts | |

ave | average value |

c | cold seawater |

cond | condenser |

eq | equivalent value |

eff | effective value |

$i$ | sequence number of each small condensation section |

in | inlet |

int | initial value |

ios | isolated system |

l | saturated liquid state |

m | primary minimum value |

mm | twice minimum value |

opt | primary optimal value |

oo | twice optimal value |

out | outlet |

p | heat transfer plate |

sum | total |

v | saturated vapor state |

wf | working fluid |

$1,2,3,4,5$ | cyclic state points |

Superscript | |

$\cdot $ | rate, ${\mathrm{s}}^{-1}$ |

**Abbreviations**

CF | composite function |

COR | constructal optimization result |

EGR | entropy generation rate |

HE | heat exchanger |

HRSG | heat recovery steam generator |

HTC | heat transfer coefficient |

HTP | heat transfer plate |

HTR | heat transfer rate |

MFR | mass flow rate |

OTEC | ocean thermal energy conversion |

SLS | saturated liquid state |

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**Figure 9.**Relationships of ${F}_{\mathrm{SP},\mathrm{m}}$ and ${L}_{\mathrm{eff},\mathrm{opt}}$ versus $w$.

**Figure 10.**Relationships of ${F}_{\mathrm{SP},\mathrm{mm}}$, ${L}_{\mathrm{eff},\mathrm{oo}}$, and ${w}_{\mathrm{opt}}$ versus ${N}_{\mathrm{eff}}$.

**Figure 12.**Relationships of ${F}_{\mathrm{SP},\mathrm{mm}}$, ${L}_{\mathrm{eff},\mathrm{oo}}$, and ${w}_{\mathrm{opt}}$ versus $\beta $.

**Figure 13.**Relationships of ${F}_{\mathrm{SP},\mathrm{mm}}$, ${L}_{\mathrm{eff},\mathrm{oo}}$, and ${w}_{\mathrm{opt}}$ versus $\mathsf{\Lambda}$.

**Figure 14.**Relationships of ${F}_{\mathrm{SP},\mathrm{mm}}$, ${L}_{\mathrm{eff},\mathrm{oo}}$, and ${w}_{\mathrm{opt}}$ versus ${V}_{\mathrm{eff}}$.

**Figure 15.**Relationships of ${F}_{\mathrm{SP},\mathrm{mm}}$, ${L}_{\mathrm{eff},\mathrm{oo}}$, and ${w}_{\mathrm{opt}}$ versus ${a}_{0}$.

Parameters | Notations | Values | Variation Ranges | Units |
---|---|---|---|---|

Initial pumping power due to friction loss | ${P}_{\mathrm{sum},\mathrm{int}}$ | 6728.42 | - | $\mathrm{W}$ |

Initial EGR in heat transfer process | ${\dot{S}}_{\mathrm{g},\mathrm{int}}$ | 30.85 | - | $\mathrm{W}/\mathrm{K}$ |

Total HTR of the plate condenser | ${\dot{Q}}_{\mathrm{cond}}$ | 1.25 × 10^{6} | - | $\mathrm{W}$ |

Evaporation pressure of working fluid in the evaporator | ${p}_{\mathrm{eva}}$ | 991.85 | - | $\mathrm{kPa}$ |

Temperature of cold seawater at the inlet of the plate condenser | ${T}_{\mathrm{c},\mathrm{in}}$ | 277.15 | - | $\mathrm{K}$ |

Efficiency of working fluid pump | ${\eta}_{\mathrm{p},\mathrm{wf}}$ | 0.8 | - | - |

Efficiency of cold seawater pump | ${\eta}_{\mathrm{p},\mathrm{c}}$ | 0.8 | - | - |

Quantity of condensation sections | $n$ | 20 | - | - |

Fouling resistance of the HTP on working fluid side | ${R}_{\mathrm{wf}}$ | 0.7 × 10^{−5} | - | ${\mathrm{m}}^{2}\cdot \mathrm{K}/\mathrm{W}$ |

Fouling resistance of the HTP on the cold seawater side | ${R}_{\mathrm{c}}$ | 1.7 × 10^{−5} | - | ${\mathrm{m}}^{2}\cdot \mathrm{K}/\mathrm{W}$ |

Weighting coefficient | ${a}_{0}$ | 0.75 | 0.45~0.95 | - |

MFR of cold seawater | ${\dot{m}}_{\mathrm{c}}$ | 103.00 | 93~133 | $\mathrm{kg}/\mathrm{s}$ |

Effective volume of the plate condenser | ${V}_{\mathrm{eff}}$ | 1.00 | 0.8~1.2 | ${\mathrm{m}}^{3}$ |

Effective length of the HTP | ${L}_{\mathrm{eff}}$ | 1.80 | 0.75~2.00 | $\mathrm{m}$ |

Width of the HTP | $w$ | 1.20 | 0.75~2.00 | $\mathrm{m}$ |

Effective number of the HTP | ${N}_{\mathrm{eff}}$ | 100 | 70~180 | - |

Corrugation wavelength of the HTP | $\mathsf{\Lambda}$ | 10 | 7~13 | $\mathrm{mm}$ |

Corrugation angle of the HTP | $\beta $ | 45 | 30~60 | ° |

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## Share and Cite

**MDPI and ACS Style**

Wu, Z.; Feng, H.; Chen, L.; Ge, Y. Performance Optimization of a Condenser in Ocean Thermal Energy Conversion (OTEC) System Based on Constructal Theory and a Multi-Objective Genetic Algorithm. *Entropy* **2020**, *22*, 641.
https://doi.org/10.3390/e22060641

**AMA Style**

Wu Z, Feng H, Chen L, Ge Y. Performance Optimization of a Condenser in Ocean Thermal Energy Conversion (OTEC) System Based on Constructal Theory and a Multi-Objective Genetic Algorithm. *Entropy*. 2020; 22(6):641.
https://doi.org/10.3390/e22060641

**Chicago/Turabian Style**

Wu, Zhixiang, Huijun Feng, Lingen Chen, and Yanlin Ge. 2020. "Performance Optimization of a Condenser in Ocean Thermal Energy Conversion (OTEC) System Based on Constructal Theory and a Multi-Objective Genetic Algorithm" *Entropy* 22, no. 6: 641.
https://doi.org/10.3390/e22060641