# OTEC Maximum Net Power Output Using Carnot Cycle and Application to Simplify Heat Exchanger Selection

^{1}

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## Abstract

**:**

## 1. Introduction

_{2}emission due to energy generation and heat production was up to 13 540 million tons [1]. Efforts are to be made to lower this figure, especially to meet the Paris Agreement’s goal to limit global warming to below 2 °C [2]. Thus, it is necessary to develop renewable energies, which are not represented enough in today’s energy mix [3]. A drawback of most implemented renewable energies is their intermittency, therefore, they cannot be used for baseload energy demand without a storage system breakthrough. However, to generate electricity, ocean thermal energy conversion (OTEC) uses the difference between the surface seawater and the deep seawater temperature in tropical areas. As such areas present very low temperature change throughout the year, a steady power generation can be achieved. Moreover, OTEC has huge potential, as its resources are estimated at a maximum of 7 TW of net energy production [4]. In addition to power generation, it is possible with such a system to produce freshwater using the warm seawater. This water can be used to create pure hydrogen to store or transport the energy generated by the power plant. In addition to tackling climate change and providing clean energy, OTEC can contribute to reaching five other sustainable development goals defined in 2015 by the general assembly of the United Nations [5]. Indeed, OTEC produces deep seawater as a byproduct, which can be used for aquaculture and desalination. The implementation of such a power plant could play a major role in the economic growth of cities and countries, especially on islands. It would also promote industry, innovation, infrastructure, as well as providing employment for the construction and operation of the plant, auxiliaries, and other industries that make use of deep seawater.

- It addresses the lack of evaluation methods to efficiently select a heat exchanger for OTEC purposes and considers the trade-off between heat transfer performance and pressure drop.
- It is easily applicable for different heat exchangers as long as geometry, heat transfer coefficient correlation, and pressure drop correlation are provided.
- It is easily applicable to different seawater temperatures.

## 2. Description and Analysis

- The heat transfer coefficient of the working fluid is much greater than the seawater one as the working fluid undergoes a phase change [24].
- The thermal resistance due to fouling can be neglected.
- Changes in the water thermodynamic properties in the heat exchangers due to temperature variation can be neglected.

#### 2.1. Carnot Cycle: Concept and Equations

_{gross}, and the required pumping power to counter the seawater pressure drop in the heat exchanger, W

_{P}. The net power output, W

_{net}, of the Carnot cycle is therefore given by Equation (1) [7]:

_{e}) and the heat taken from the system (Q

_{c}) Equation (2) [25]:

_{p}, of the seawater. Subscript ws stands for warm source and cs stands for cold source. T

_{wsi}, T

_{csi}, T

_{wfe}, and T

_{wfc}are the temperature of the warm seawater at the inlet of the evaporator, the temperature of the cold seawater at the inlet of the condenser, the temperature of the working fluid in the evaporator, and the temperature of the working fluid in the condenser, respectively.

_{P}is defined as in [7]:

_{w}the seawater density, v the seawater mean velocity, and µ

_{w}the seawater dynamic viscosity.

_{w}is the seawater convective heat transfer coefficient, α

_{wf}is the working fluid heat transfer coefficient, t is the plate thickness, R

_{f}is the resistance due to fouling, and λ

_{p}is the thermal conductivity of the plate.

_{wf}is assumed to be much greater than α

_{w}. Therefore:

_{w}is calculated from the Nusselt number (Nu), defined as follows, with λ

_{w}being the water thermal conductivity and D the equivalent diameter:

_{net}, is deduced:

#### 2.2. Rankine Cycle: Concept and Equations

_{wf}is the mass flow rate of the working fluid. h

_{1}, h

_{2}, h

_{3}, and h

_{4}are the enthalpy values of the corresponding points in Figure 2 and with:

_{1}and h’

_{3}are the enthalpy values of the corresponding points in the case of an isentropic process, and η

_{T}and η

_{p}are the turbine and pump efficiencies, respectively. For the calculations, the turbine efficiency is equal to 0.85 and the pump efficiency is equal to 0.8.

_{2}and T

_{4}at Points 2 and 4 using the software REFPROP [27]. T

_{2}and T

_{4}can be evaluated from Equation (26):

_{2}and h

_{4}are calculated using the respective temperature and because the fluid is at a saturated state. It is possible to calculate h

_{3}using T’

_{3}= T

_{4}, s

_{2}= s’

_{3}, and Equation (26). For Point 1, s

_{4}= s’

_{1}and P

_{2}= P’

_{1}; P is the pressure of the fluid and s is its specific entropy. NTU is calculated as detailed for the Carnot cycle in Equations (15)–(22). Details for the calculations to obtain Equation (26) are given in Appendix A.

## 3. Optimization Process

_{net}, the objective function used with “minimize” was −w

_{net}.

#### 3.1. Carnot Cycle

#### 3.2. Rankine Cycle

_{ws}, Re

_{cs}, T

_{wso}, T

_{cso}, and m

_{wf}. As for the Carnot cycle, boundaries were used on the Reynolds number to restrain the water mean velocity between the 0.2 to 1.8 m/s range. Temperatures were taken to be between T

_{csi}and T

_{wsi}. m

_{wf}was restrained to be between 0.001 kg/s and a value that is equal to a water mean velocity of 1.8 m/s, as water mass flow rate is higher than ammonia mass flow rate in OTEC system.

## 4. Optimization Results

#### 4.1. Carnot Cycle Results

_{net}as a function of the Reynolds number for PHE 1, PHE 2, and PHE 3. Table 3 shows the heat transfer coefficient and pressure drop of the three heat exchangers.

_{ws,opt}and Re

_{cs,opt}found in these results show that, for a non-negligible range of Reynolds numbers, the net power output can be null or negative (dark blue areas on Figure 3). The flow rate should then be controlled rigorously to adjust the Reynolds numbers in the heat exchangers.

_{net,max}. These results show the importance of correctly selecting a heat exchanger, as w

_{net,max}is highly dependent on which one is used. Indeed, a huge difference was noticed between the results using PHE 3 and the results using the other two at their optimal operating point. PHE 1 led to a w

_{net,max}that was 158% higher than the one of PHE 3. As for PHE 2, the w

_{net,max}was 149% higher than the one achieved with PHE 3. This difference can be explained by the low heat transfer coefficient of PHE 3.

_{net,max}that was only 3.7% lower than PHE 1, even though its heat transfer coefficient was 35% lower than the one of PHE 1. This can be explained by a friction factor that was 60 to 66% lower in the case of PHE 2. In addition, PHE 2 was found to be less sensitive to a change in the Reynolds numbers. PHE 1, however, presented optimum Reynolds numbers that were 22% lower than those of PHE 2. From these observations, one can easily conceive a heat exchanger that would lead to a higher w

_{net,max}than PHE 1 and also present lower heat transfer coefficient if the pressure drop was low enough. This would be possible because a low pressure drop allows the use of higher Reynolds numbers to compensate for a low heat transfer coefficient. Such a heat exchanger, because of its low pressure drop, would be less affected by a change in the operating Reynolds numbers. A high pressure drop heat exchanger, however, would require relatively lower optimum Reynolds numbers, which implies a lower pumping power for water ducting and/or lower diameter pipes.

_{net,max}will change significantly with a change in the seawater temperature. Indeed a 20% drop in w

_{net,max}was observed for the three heat exchangers when a 2 °C change in the seawater source occurred. Yeh et al. found a decrease of 35% for the same temperature drop in the cold seawater, but warm seawater was fixed at 25 °C, instead of 30 °C in the present study [30]. Sinama et al. found a 40% decrease when the warm seawater temperature dropped from 28 °C to 25 °C against 32% in this study for the same seawater temperature values [17]. Additionally, for the same temperature change, Uehara and Ikegami showed a decrease of 44% in the net power output of the OTEC power plant [23]. In their paper, VanZwieten et al. showed a decrease of 20% and 16% when the temperature changed from 20.12 °C to 21.72 °C and from 20.12 °C to 21.37 °C, respectively, against a 17% drop in this study when a 1.5 °C change occurred [31]. For a change of 5 °C in the seawater source temperature, the calculated drop in w

_{net,max}was 45%. The difference in the decrease of the net power output with other studies can be explained by the fact that the authors considered water ducting as well as the working fluid circulation pump [17,23,30,31]. Therefore, when the temperature difference decreases, the gross power output greatly decreases, whereas losses due to pumping and pressure drop hardly change. The impact seems to be the same if the change consists of a decrease in the warm seawater source temperature or an increase of the cold seawater source temperature.

_{net,max}, the change in the operating points and the increase of the negative power output operating conditions make the flow rate monitoring even more important if the system is to be installed where the water condition changes throughout the year. In addition, for an installation where the seawater temperature does not vary, calculations need to be adjusted to the actual condition to find the optimal operating point.

#### 4.2. Rankine Cycle Results

_{net,max}values that were, as expected, lower than those using the Carnot cycle, as shown in Table 4.

_{net,max}ranging from 90 W/m

^{2}to 471 W/m

^{2}for PHE 3 at the lowest temperature difference and PHE 1 at the highest temperature difference, respectively. As a comparison, the study performed by Uehara et al. presented a net power output of 153.9 W/m

^{2}at a temperature difference, ΔT, of 20 °C and a net power output of 236 W/m

^{2}at a ΔT of 23 °C [23]. For the same ΔT, the performances of the two heat exchangers they considered were between those of PHE 3 and PHE 2, giving a w

_{net,max}of 90 W/m

^{2}and 240 W/m

^{2}, respectively, at a ΔT of 20 °C, and 143 W/m

^{2}and 360 W/m

^{2}, respectively, at a ΔT of 23 °C. These values are rather low but are still in the range of what was found in the current study. Moreover, it should be noted that, in their paper, they considered the pumping power required for water ducting, working fluid pumps, and the heat transfer coefficient of the working fluid, which explains the lower power output.

^{2}and a gross power output of 278 W/m

^{2}for a ΔT of 24 °C in their optimization using plate heat exchangers [16]. This was lower than the 293 W/m

^{2}found with PHE 2 at a ΔT of 21.5 °C. In their work, the authors considered water ducting, ammonia pumping, and the ammonia heat transfer coefficient, although only the latter contributed to decreasing the gross power output.

_{net,max}occurred with a decrease of two degrees in the temperature difference, and a 47%–50% decrease was observed with a decrease of five degrees in the temperature difference, depending on the heat exchanger. These changes were the same as those observed with the Carnot cycle. For this cycle too, a change in the water temperature will not affect which heat exchanger is the most suitable even if the gap between each heat exchanger performance can vary along with the temperature difference. Moreover, although it is not as clear as for the Carnot cycle calculation, the Reynolds numbers of the optimized operating points tend to increase with the temperature difference. This confirms that calculations are needed for each application and its specific operating condition to find the optimal operating point. It also confirms that the flow rate should be monitored and adapted in case a change of temperature occurs. The difference between the Reynolds numbers for PHE 1 and PHE 2 remains close to the results for the Carnot cycle, with values 21% to 35% lower for PHE 1.

_{net,max}achieved using a Rankine cycle over the one achieved using the Carnot cycle was found to be fairly constant, with figures ranging between 0.73 and 0.78 for all heat exchangers and for all the computed ΔT. It is therefore possible, for the selection of the heat exchangers, to base the calculation on the Carnot cycle only.

## 5. Conclusions

- For the sole purpose of a heat exchanger comparison, calculations based on the Carnot cycle for any source temperature were sufficient, as the cycle and temperature difference do not have an impact on the choice of the heat exchanger even though they do change the power output and optimized operating conditions. The Rankine cycle calculations presented a maximum net power output 23–27% lower than for the Carnot cycle that dropped ~10% each time a temperature difference decrease of 1 °C was observed. The evolution of the power output as a function of the temperature difference was found to follow the same trends as found in other studies.
- The maximum net power output was found to highly depend on the chosen heat exchangers. For the highest temperature difference, the most suitable heat exchangers among the three considered led to a maximum power output 158% and 165% higher than the worst heat exchanger for the Carnot and Rankine cycles, respectively.
- Due to the trade-off that exists between the heat transfer coefficient and the pressure drop, the heat exchanger presenting the highest heat transfer coefficient is not necessarily the one that will lead to the highest maximum net power output. In this study, for the Carnot cycle, PHE 2 competed with PHE 1 as it led to a maximum net power output that was only 3.7% lower than the one of PHE 1, even though its heat transfer coefficient was 35% lower.
- Heat exchangers with a high pressure drop and those with a low pressure drop have been found to have their own advantages and drawbacks. High pressure drop heat exchangers require lower Reynolds numbers, and therefore a smaller pumping power and/or a smaller pipe diameter are needed. Low pressure drop heat exchangers are less sensitive to a change in the Reynolds numbers, which can be useful in case a change in the operating conditions is needed. This is even more important as the results showed that negative net power output can be reached for low enough Reynolds numbers.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Nomenclature | ||

A | [m^{2}] | Heat transfer surface area |

As | [m^{2}] | Heat exchanger surface area |

B | [K/W] | $\frac{plate\text{}thickness}{{\lambda}_{plate}A}$ |

Cp | [J/(kg·K)] | Specific heat |

C | [W/K] | $\dot{m}Cp$ heat capacity |

D | [m] | Equivalent diameter |

f | [-] | Friction factor |

h | [J/kg] | Specific enthalpy |

L | [m] | Length of the plate |

m | [kg/s] | Mass flow rate |

Nu | [-] | Nusselt number |

P | [kPa] | Pressure |

Pr | [-] | Prandtl number |

Q | [W] | Heat transfer rate |

Re | [-] | Reynolds number |

R_{f} | [m^{2}·K/W] | Resistance due to fouling |

S | [m^{2}] | Total cross-sectional surface area |

s | [J/(kg·K)] | Specific entropy |

T | [K] | Temperature |

U | [W/(m^{2}·K)] | Overall heat transfer coefficient |

v | [m/s] | Mean velocity |

Wi | [m] | Width of the plate |

W | [W] | Power output |

w | [W/m^{2}] | Power output per unit of surface area |

x | [-] | Vapor quality |

α | [W/(m^{2}·K)] | Heat transfer coefficient |

ΔT | [K] | Temperature difference |

λ | [W/(m·K)] | Thermal conductivity |

Λ | [-] | Lagrange multiplier |

μ | [Pa·s] | Dynamic viscosity |

ρ | [kg/m^{3}] | Density |

τ | [Pa] | heat exchanger wall shear stress |

Subscripts | |

cs | Cold source |

csi | Cold source inlet |

cso | Cold source outlet |

gross | Gross |

in | Inlet |

max | Maximum |

net | Net |

opt | Optimum |

out | Outlet |

p | Plate |

w | Water |

ws | Warm source |

wsi | Warm source inlet |

wso | Warm source outlet |

wf | Working fluid |

## Appendix A

_{2}and T

_{4}are the temperatures on the corresponding points in Figure 2.

_{3}= T

_{4}, assuming the fluid is at a saturated state at the inlet and outlet of the condenser, Equation (A3) becomes:

_{cs}and ε

_{cs}are defined by [25]:

_{4}can be deduced from Equation (A7) as:

_{1}= T

_{2}is made, and then the calculations are the same. Equation (A10) is used:

_{ws}and ε

_{ws}are defined the same way:

_{2}can be expressed as:

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**Figure 1.**(

**a**) Model of the ocean thermal energy conversion (OTEC) system and (

**b**) temperature–entropy diagram of a Carnot cycle.

**Figure 2.**(

**a**) Description of the Rankine cycle and (

**b**) temperature–entropy diagram of a Carnot cycle.

**Figure 3.**(

**a**) Maximum net power output of an OTEC power plant using Plate heat exchanger (PHE) 1 as both evaporator and condenser as a function of Reynolds, (

**b**) maximum net power output of an OTEC power plant using PHE 2 as both evaporator and condenser as a function of Reynolds, and (

**c**) maximum net power output of an OTEC power plant using PHE 3 as both evaporator and condenser as a function of Reynolds.

**Figure 4.**(

**a**) Maximum net power output per square meter as a function of the temperature difference between warm and cold water, (

**b**) Reynolds number of the warm water inside the evaporator as a function of the temperature difference between warm and cold water, and (

**c**) Reynolds number of the cold water inside the condenser as a function of the temperature difference between warm and cold water.

**Figure 5.**(

**a**) Maximum net power output per square meter as a function of the temperature difference between warm and cold water, (

**b**) Reynolds number of the warm water inside the evaporator as a function of the temperature difference between warm and cold water, and (

**c**) Reynolds number of the cold water inside the condenser as a function of the temperature difference between warm and cold water.

Heat Exchanger | PHE 1 | PHE 2 | PHE 3 |
---|---|---|---|

Length L (mm) | 960 | 718 | 1765 |

Width Wi (mm) | 576 | 325 | 605 |

Thickness t (mm) | 0.7 | 0.5 | 0.6 |

Space between plates δ (mm) | 4.00 | 3.95 | 2.68 |

Equivalent diameter D (mm) | 8.00 | 7.90 | 5.36 |

Material | SUS316 | Titanium | Titanium |

Thermal conductivity λ_{p} (W/(m·K)) | 16.3 | 21 | 21 |

Pattern | Herringbone (72°) | Herringbone (30°) | Fluting and drainage |

Number of plates | 120 | 20 | 52 |

Total heat transfer area A (m^{2}) | 100.3 | 3.96 | 40.6 |

Total cross surface area S (m^{2}) | 0.140 | 0.012 | 0.041 |

Heat Exchanger | d | γ | n | β | ξ |
---|---|---|---|---|---|

PHE 1 | 0.111 | 0.8 | 1/3 | 1.4863 | −0.0540 |

PHE 2 | 0.058 | 0.8 | 1/3 | 6.5059 | −0.3292 |

PHE 3 | 0.051 | 0.8 | 1/3 | 0.7371 | −0.1274 |

Heat Exchanger | α_{ws} (W/m^{2}·K) | α_{cs} (W/m^{2}·K) | f_{ws} (−) | f_{cs} (−) |
---|---|---|---|---|

PHE 1 | 20 569 | 14 450 | 0.913 | 0.945 |

PHE 2 | 13 259 | 9 334 | 0.307 | 0.379 |

PHE 3 | 11 191 | 7 843 | 0.242 | 0.263 |

PHE 1 | PHE 2 | PHE 3 | ||||
---|---|---|---|---|---|---|

Carnot | Rankine | Carnot | Rankine | Carnot | Rankine | |

w_{net} (W/m^{2}) | 613 | 471 | 590 | 451 | 237 | 178 |

Re_{ws} | 8361 | 7876 | 10,690 | 10,121 | 6255 | 5753 |

V_{ws} (m/s) | 0.837 | 0.788 | 1.08 | 1.03 | 0.934 | 0.859 |

Re_{cs} | 4383 | 4381 | 5618 | 5762 | 3274 | 3277 |

V_{cs} (m/s) | 0.832 | 0.831 | 1.08 | 1.11 | 0.927 | 0.928 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Fontaine, K.; Yasunaga, T.; Ikegami, Y.
OTEC Maximum Net Power Output Using Carnot Cycle and Application to Simplify Heat Exchanger Selection. *Entropy* **2019**, *21*, 1143.
https://doi.org/10.3390/e21121143

**AMA Style**

Fontaine K, Yasunaga T, Ikegami Y.
OTEC Maximum Net Power Output Using Carnot Cycle and Application to Simplify Heat Exchanger Selection. *Entropy*. 2019; 21(12):1143.
https://doi.org/10.3390/e21121143

**Chicago/Turabian Style**

Fontaine, Kevin, Takeshi Yasunaga, and Yasuyuki Ikegami.
2019. "OTEC Maximum Net Power Output Using Carnot Cycle and Application to Simplify Heat Exchanger Selection" *Entropy* 21, no. 12: 1143.
https://doi.org/10.3390/e21121143