# Finite-Time Thermodynamic Model for Evaluating Heat Engines in Ocean Thermal Energy Conversion

^{*}

## Abstract

**:**

## 1. Introduction

## 2. First Law of Thermodynamics

#### 2.1. Available Work with Ideal Conditions

_{W}drops and that of the cold seawater T

_{C}rises. The heat exchange process is idealized and the outlet temperature of the seawaters T

_{W}

_{,O}and T

_{C}

_{,O}will reach the same temperature as the heat engine T

_{H}and T

_{L}with isobaric changes in the heat sources by ideal heat exchangers. The heat transfer rates and—according to the law of energy conservation—the work output of the heat engine W can be calculated using the following equations:

_{p}(kJ/kgK) is the specific heat assumed constant in the temperature variation in the OTEC. The subscript terms W, C, HS, and O are the warm surface seawater, cold deep seawater, total heat source, and outlet, respectively. Here, the dependence of the specific heat on the temperature is assumed negligible since the temperature changes in seawater are sufficiently small. According to Figure 1 and Figure 2, the reversible heat engine received process is (1) isothermal expansion receiving the heat of Q

_{W}from warm seawater, (2) isenthalpic work output, (3) isothermal compression releasing the heat of Q

_{C}to cold seawater, and (4) isenthalpic compression.

_{th}is defined as:

_{C}

_{,O}or T

_{W}

_{,O}. The warm seawater outlet temperature and the cold seawater outlet temperature can be respectively expressed as:

_{W}

_{,O}or T

_{C}

_{,O}and maximized as $\partial W/\partial {T}_{W,O}=0$ or $\partial W/\partial {T}_{C,O}=0$ then the maximum work W

_{m}can be derived as follows [36]:

_{m}will be maximized when ${\left(m{c}_{p}\right)}_{W}={\left(m{c}_{p}\right)}_{C}$, whereas the thermal efficiency is maximized asymptotically when ∆T

_{W}= 0 as follows:

#### 2.2. Normalization of the Thermal Efficiency

_{0}is the equilibrium temperature [K]; the so-called “dead state” of OTEC where both heat sources finally reach the equilibrium temperature. Q

_{W}

_{,d}and Q

_{C}

_{,d}are the heat transfer rate of the heat sources. The heat capacity rate of the heat source C

_{HS}and the heat capacity rate ratio r are respectively defined as:

_{W}

_{,d}= Q

_{C,d}, hence:

_{W}

_{,d}and W

_{m}are obviously maximized when r = 0.5 with constant C

_{HS}.

_{W}

_{,loss}is the latter. Internal irreversibility indicates the irreversibility of the heat engine. The internal irreversibility is negligible in this study because the reversible heat engine is applied to a heat engine. The external thermal irreversibility is the external energy loss Q

_{W}

_{,loss}when the heat transfer rate of warm seawater Q

_{W}is subtracted from the transferable thermal energy of the heat source Q

_{W}

_{,d}.

_{W}

_{,loss}is equivalent to the heat leak in boilers and reactors. In conventional thermodynamics, the efficiency of a heat engine and the heat losses in boilers and reactors can generally be discussed separately because they are never related to the heat engine’s thermal efficiency. However, heat leak is non-negligible for the normalized thermal efficiency of OTEC because the transferable thermal energy is the available energy from the heat source, which is determined by the balance of heat capacity rates between the surface and deep seawater.

_{W}

_{,O}and T

_{C}

_{,O}, then the available temperature decreases as the seawater temperature change increases because the conventional thermal efficiency is the ratio between the work from the heat engine and the input thermal energy into the heat engine. Whereas the normalized thermal efficiency is the ratio of the work from the heat engine and the maximum available thermal energy from the heat source, it only depends on the work from the heat engine. Therefore, the maximization of the normalized thermal efficiency corresponds to the increase in the heat engine’s work output, which can effectively show the heat engine’s performance.

## 3. Second Law of Thermodynamics

#### 3.1. Available Work Maximization

_{C}

_{,O}in Figure 6b. Therefore, the difference between the exergy and the actual power output is the loss as the system generates entropy. During the heat transfer process between seawater and the heat engine, the total entropy generation S

_{gen}

_{,HE}yielded by the temperature change in seawaters can be calculated by:

_{rev}will be one degree of freedom of T

_{W}

_{,O}or T

_{C}

_{,O}and the maximum available work output can be achieved by $\partial {W}_{rev,m}/\partial {T}_{W,O}=0$ or $\partial {W}_{rev,m}/\partial {T}_{C,O}=0$. Finally, the maximum available work output a and the optimum outlet temperature of warm and cold seawaters T

_{W}

_{,O,opt}, T

_{C}

_{,O,opt}will be:

_{W}

_{,O,opt}and T

_{C}

_{,O,opt}will be the same temperature so the heat source will reach an equilibrium state after energy conversion.

#### 3.2. Exergy and Entropy Generation

_{x}:

_{0}is defined as Equation (20) rather than the atmospheric temperature because the system is considered to have a finite quantity of heat sources and an adiabatic system as shown in Figure 1. Equations (28) and (31) are both the maximized work from the heat source thermal energy.

_{rev}

_{,m}as defined by Equations (28) and (31) as a function of the ratio of the heat source’s heat capacity rate. Figure 7 also shows the exergy as defined by Johnson [40] as follows:

_{rev}

_{,m}is the exergy of the thermal energy Ex

_{HS}in the ocean for OTEC. Here, the exergy efficiency η

_{th}

_{,Ex}is defined as:

_{W}

_{,loss}based on the equilibrium state and, thus, the results of entropy generation minimization will show the discrepancy due to the accompaniment of a dead state as the atmospheric temperature. However, the results in Figure 8 show that minimizing the entropy generation is equivalent to maximizing the system’s power output by introducing the equilibrium temperature state as the dead state.

#### 3.3. Ideal and Staging Carnot Heat Engines

_{Car}

_{,m,N}, optimum warm seawater heat transfer rate Q

_{W}

_{,opt,N}, and optimum cold seawater heat transfer rate Q

_{C}

_{,opt,N}can be theoretically formulated as follows:

_{W}

_{,opt,N}and T

_{C}

_{,opt,N}can be calculated as follows:

_{W}

_{,O,opt,N}and T

_{C}

_{,O,opt,N}form an asymptotic curve in relation to the dead state temperature as expressed in Equation (29). As this temperature approaches an increase in the number of stages, the Q

_{loss}

_{,W}and S

_{gen}

_{,HS}decrease. This is because the effective temperature of staging heat engine will increase, although the temperature difference between the outlet of heat sources decrease as the stage increases. The optimized staging is the series of the heat engine that utilizes heat effectively and the thermal efficiency of the total staging heat engine corresponds to Equation (12).

## 4. Performance Evaluation of OTEC Heat Engines

_{rev}

_{,m}) will be maximized when r = 0.502–0.504. In general, the flow rate balance allows free design in an OTEC system; therefore, the potential of OTEC will be ${E}_{x,HS,r=0.5}=\frac{{C}_{HS}}{2}\Delta {T}_{HS}$. Then, the upper bound of the available work for a simple Carnot heat engine is half of the exergy.

_{W}

_{,∆P}and ∆S

_{C}

_{,∆P}in Equation (25) are non-negligible.

_{HS}is calculated in the heat source condition by Equation (28); the maximum work W

_{m}, which depends on the heat engine, is estimated by Equations (22), (28) and (36); the thermal efficiency η

_{th}uses the conventional first law efficiency (Equation (4)); Equation (23) defined the normalized thermal efficiency η

_{th}

_{,Nor}; Equation (33) expresses the exergy efficiency. Case 2 and Case 3 have the same η

_{th}; however, the hex of Case 2 is much higher than Case 3 and the η

_{th}of Case 4 is higher than for Case 1; however, Case 1 has the higher hex. With the same heat source temperature condition, the comparison between Case 1 and Case 5 shows that Case 5 has a much higher hex. If the system applies the higher W

_{m}/E

_{xHS}heat engine, the hex tends to increase accordingly except in Case 9. In Case 9, using the Kalina cycle, even though it has the highest W

_{m}/E

_{xHS}, the hex is smaller than most Rankine cycle cases. The result in Table 2 is estimated under various different assumptions, including the performance of heat exchangers, turbines, and working fluid pump; however, the object function for the optimization must be selected adequately to maximize the efficiency and the energy conversion performance will be expressed by the hex and η

_{th}

_{,Nor}. Although economic analysis is also required in the optimization of the OTEC system design, the proposed evaluation method is effective for evaluating the practical OTEC design and optimization. To evaluate the system performance, the normalized thermal efficiency (Equation (23), and the exergy efficiency (Equation (33)) are applicable for evaluating the system if the W in each equation uses a net power.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

C | (kW/K) | Heat capacity |

c_{p} | (kJ/(kg K)) | Specific heat |

Ex | (kW) | Exergy |

H | (kJ) | Enthalpy |

m | (kg/s) | Mass flow rate |

Q | (kW) | Heat flow rate |

R | (-) | Ratio of warm seawater heat-source Heat capacity |

S | (kJ/(kg K)) | Entropy |

T | (K) | Temperature |

ΔT | (K) | Temperature difference |

ΔT_{HS} | (K) | ${\left(\sqrt{{T}_{W}}-\sqrt{{T}_{C}}\right)}^{2}$ |

Greek Symbols | ||

η | (%) | Efficiency |

Subscripts | ||

C | Cold deep seawater | |

CA | Curzon-Ahlborn | |

Car | Carnot cycle | |

d | Transferable | |

H | High-temperature working fluid temperature on heat engine | |

HS | Heat source | |

L | Low-temperature working fluid temperature on heat engine | |

loss | Lost thermal energy (defines as the difference from the equilibrium state) | |

N | Number of stage | |

Nor | Normalized | |

m | Maximized | |

O | Outlet | |

opt | Optimum | |

rev | Reversible | |

th | Thermal | |

W | Warm surface seawater |

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**Figure 3.**Concept of the relationship between the surface seawater temperature change and thermal efficiency η

_{th}, heat flow rate Q

_{H}, and power output W.

**Figure 4.**The concept of energy distribution entering the heat engine and the remaining thermal energy of heat sources. (

**a**) The conceptual model of the heat engine and heat loss and (

**b**) the conceptual T–E diagram.

**Figure 5.**Thermal efficiencies and power output as functions of the surface seawater temperature change in a heat engine. T

_{W}= 303.15 K; T

_{C}= 278.15 K; C

_{HS}= 1 kW/K; r = 0.5.

**Figure 6.**Conceptual diagrams of an ideal reversible heat engine. (

**a**) T–s diagram, and (

**b**) T–E diagram.

**Figure 7.**Exergy and maximum available work as a function of heat capacity rate ratio. T

_{W}= 303.15 K; T

_{C}= 278.15 K; C

_{HS}= 1 kW/K.

**Figure 8.**Relationship between the exergy efficiencies and entropy generation in heat sources as a function of the temperature change in surface seawater. T

_{C}= 278.15 K; C

_{HS}= 1 kW/K; r = 0.5. The open circles show the maximum points and closed circles show the minimum points.

**Figure 9.**The staging Carnot cycle (

**a**) exergy efficiency and entropy generation of heat source, and (

**b**) heat-source discharge temperatures as functions of the number of stages. T

_{W}= 303.15 K; T

_{C}= 278.15 K; C

_{HS}= 1 kW/K; r = 0.5.

Ideal Heat Engines | Carnot Cycle | Ideal Cycle |
---|---|---|

Practical heat engines | Rankine and Claud | Uehara and Kalina |

Maximum work | $r\left(1-r\right){C}_{HS}\Delta {T}_{HS}$ | ${C}_{HS}{T}_{W}\left[r+\left(1-r\right)\left(\frac{{T}_{C}}{{T}_{W}}\right)-{\left(\frac{{T}_{C}}{{T}_{W}}\right)}^{1-r}\right]$ |

Potential (r = 0.5) | $\frac{{C}_{HS}\Delta {T}_{HS}}{4}$ | $\frac{{C}_{HS}\Delta {T}_{HS}}{2}$ |

Normalized thermal efficiency | $\frac{\Delta {T}_{HS}}{{T}_{W}-{T}_{C}}$ | $\frac{r+\left(1-r\right)\left(\frac{{T}_{C}}{{T}_{W}}\right)-{\left(\frac{{T}_{C}}{{T}_{W}}\right)}^{1-r}}{r\left(1-r\right)\left[1-\left(\frac{{T}_{C}}{{T}_{W}}\right)\right]}$ |

Exergy efficiency | $\frac{r\left(1-r\right)\Delta {T}_{HS}}{{T}_{W}\left[r+\left(1-r\right)\left(\frac{{T}_{C}}{{T}_{W}}\right)-{\left(\frac{{T}_{C}}{{T}_{W}}\right)}^{1-r}\right]}$ | 1 |

Case | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

Heat engine (Stages) | Rankine | Rankine | Rankine | Rankine | Rankine | Rankine (Two) | Rankine (Two) | Rankine (Three) | Kalina | Kalina |

W (kW) | 3877 | 5750 | 15,733 | 20 | 5000 | 6350 | 6079 | 6425 | 20 | 6420 |

T_{W} (°C) | 28 | 29 | 28 | 26 | 28 | 29 | 28 | 28 | 26 | 29 |

T_{C} (°C) | 4 | 6 | 5 | 5 | 4 | 6 | 4 | 4 | 5 | 6 |

C_{HS} (kW/K) | 69192 | 69856 | 347476 | 505 | 66891 | 69856 | 66891 | 66891 | 505 | 69856 |

r | 0.51 | 0.50 | 0.59 | 0.51 | 0.51 | 0.50 | 0.51 | 0.51 | 0.51 | 0.50 |

W_{m}/E_{xHS} | 50% | 50% | 50% | 50% | 50% | 67% | 67% | 75% | 91% | 90% |

η_{th} | 2.2% | 3.2% | 3.2% | 2.4% | 3.1% | 3.2% | 3.7% | 3.9% | 2.4% | 3.2% |

η_{th}_{,Nor} | 0.93% | 1.43% | 0.81% | 0.76% | 1.25% | 1.58% | 1.52% | 1.60% | 0.76% | 1.60% |

η_{EX} | 22.5% | 36.2% | 20.4% | 20.7% | 30.0% | 39.9% | 36.5% | 38.6% | 20.7% | 40.4% |

Reference | [44] | [7] | [11] | [45] | [8] | [7] | [8] | [8] | [45] | [7] |

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**MDPI and ACS Style**

Yasunaga, T.; Ikegami, Y.
Finite-Time Thermodynamic Model for Evaluating Heat Engines in Ocean Thermal Energy Conversion. *Entropy* **2020**, *22*, 211.
https://doi.org/10.3390/e22020211

**AMA Style**

Yasunaga T, Ikegami Y.
Finite-Time Thermodynamic Model for Evaluating Heat Engines in Ocean Thermal Energy Conversion. *Entropy*. 2020; 22(2):211.
https://doi.org/10.3390/e22020211

**Chicago/Turabian Style**

Yasunaga, Takeshi, and Yasuyuki Ikegami.
2020. "Finite-Time Thermodynamic Model for Evaluating Heat Engines in Ocean Thermal Energy Conversion" *Entropy* 22, no. 2: 211.
https://doi.org/10.3390/e22020211