# A Preliminary Investigation of the Effect of Ocean Thermal Energy Conversion (OTEC) Effluent Discharge Options on Global OTEC Resources

## Abstract

**:**

## 1. Introduction

^{−1}(one terawatt is 10

^{12}W, and one sverdrup is 10

^{6}m

^{3}s

^{−1}); this also corresponds to a net power ratio of about 45% (i.e., power produced over nominal power). What mainly differentiates the ‘bulk’ output from all models and scenarios so far is the magnitude of the overall OTEC deep cold seawater flow rate at which OTEC net power would peak. In one-dimensional models, it basically takes much smaller OTEC seawater flow rates to modify the existing seawater temperature profiles. These points can be appreciated in Figure 1, where the output of the most basic steady-state 1-D model has been adapted ‘locally’ to every 1 degree by 1 degree (latitude-longitude) cell of the OTEC region [4]; the precise definition of the OTEC region, as well as the OTEC warm-to-cold seawater flow rate ratio (1.5 here, instead of 2) and net power formula were adapted for comparison to the most recent 3-D study [10].

## 2. One-Dimensional Steady-State Model of the Water Column with OTEC

_{m}(z = L). The OTEC process was represented by two sinks, in the mixed layer (warm seawater intake) and at z = z

_{cw}(cold seawater intake), as well as by a source at z = z

_{mix}(mixed effluent discharge). The present study extends the model to cases of separate OTEC evaporator and condenser effluent discharges, at z = z

_{evap}and z = z

_{cond}, respectively. This is schematically illustrated in Figure 2. In mathematical terms, sources (depicted as circles with a dot) and sinks (depicted as circles with a cross) are Dirac distributions centered at specific vertical locations; they are singular, but with the property that a spatial integration (across such singularities) generates known step functions. Thus, the vertical domain can be split in four regions (labeled in the right-hand-side of Figure 2) with specific vertical mass flows, normalized by ρA

_{OTEC}, and heat fluxes, normalized by ρc

_{p}A

_{OTEC}. The OTEC implementation area A

_{OTEC}is a passive parameter in this 1-D model, with an order-of-magnitude value of 100 million square kilometers (10

^{14}m

^{2}); ρ and c

_{p}are representative values of the density and specific heat of seawater, e.g., 1025 kg m

^{−3}and 4000 J kg

^{−1}K

^{−1}. Owing to the very low thermodynamic efficiencies of OTEC cycles, an assumption that the overall OTEC process does not remove heat from the ocean is also made (lifting such assumption can be shown to result in negligible differences). Therefore, we have w

_{ww}δT

_{evap}≈ w

_{cw}δT

_{cond}, where the positive temperature differences δT

_{evap}and δT

_{cond}are respectively defined by the seawater temperature of the evaporator effluent θ

_{evap}= T − δT

_{evap}, and by the seawater temperature of the condenser effluent θ

_{cond}= θ

_{cw}+ δT

_{cond}. Finally, please note that for the mixed layer to be in steady-state equilibrium, a heat flux wT

_{p}(not shown in Figure 2), where T

_{p}is polar water temperature, must be extracted at the domain’s presumed margins to mimic deep water formation.

_{cw}. A similar issue does not arise for the mixed-layer temperature T in a one-dimensional model under the assumption that the overall OTEC process does not remove heat from the ocean. In other words, there is only one possible steady-state mixed-layer temperature, although transient cooling does occur in time-domain calculations [5]; note that a three-dimensional model would allow permanent surface cooling in the OTEC region if surface warming occurs elsewhere [7,8,9,10]. The chosen expression for the OTEC seawater condenser warming δT

_{cond}corresponds to the simplified OTEC temperature ladder shown in Nihous [5], Figure 2.

_{p}at z = 0 and θ = T at z = L, in addition to three temperature continuity conditions at z

_{cw}, z

_{cond}and z

_{evap}. The published algorithm for mixed OTEC effluent discharge is recovered when z

_{cond}= z

_{evap}= z

_{mix}[4]; Region 2 in Figure 2 merely vanishes and Equation (2) need not be considered.

_{cw}across the OTEC cold seawater sink. Next, in the absence of OTEC operations (w

_{ww}= w

_{cw}= 0), the steady-state mixed-layer heat balance can be written:

_{ww}is zero (no OTEC) or not, i.e., dθ/dz(L) = w(T − T

_{p})/K.

_{p}) − w

_{ww}δT

_{evap}}/K.

_{mix}= L), both Regions 1 and 2 vanish, while Equations (1) and (2) are no longer applicable. Only Regions 3 and 4 are left in Figure 2, and only Equations (3) and (4) need to be solved. The steady-state heat balance of the mixed layer is now written:

_{p}) − w

_{cw}(T − θ

_{cw})}/K. Mathematically, Equation (8) is equivalent to the mixed-layer heat balance for scenarios of artificial upwelling (of deep water) into the mixed layer [11]1, since the OTEC warm seawater source and sink contributions here cancel out.

## 3. Results

^{2}yr

^{−1}and w = 4 m yr

^{−1}; solutions in each Region labeled in Figure 2 then involve simple exponentials of z and constants (or in the special case w

_{cw}= w in Region 3, a linear function of z). In addition, we set T

_{p}= 0 °C and T = 25 °C, and select an OTEC seawater flow-rate ratio w

_{ww}/w

_{cw}of 2. The mixed layer is 75 m thick over a water column of 4000 m, and the OTEC deep cold seawater intake is maintained at a water depth of 1000 m (z

_{cw}= 3075 m). Variable parameters are the OTEC deep cold seawater flow rate w

_{cw}, as well as the evaporator and condenser effluent discharge depths (coordinates z

_{evap}and z

_{cond}in general). Once the temperature profile for a given OTEC scenario is known, OTEC power is determined from the following formula used in earlier work [4,5]:

_{tg}is 0.85, and T in the denominator of the bracketed expression is the absolute steady-state temperature of the mixed layer, i.e., 298.15 K.

#### 3.1. Separate Discharges versus Mixed Discharge under Baseline Scenarios

_{mix}at a water depth where the mixed OTEC effluent would be neutrally buoyant without OTEC (i.e., initially) [4]. Given the initial temperature profile obtained in this one-dimensional model, and since there is no consideration of salinity (which would also affect buoyancy), it corresponds here to a water depth of 253 m (z

_{mix}= 3822 m, for an initial mixed effluent temperature of 18.33 °C). This approach defines baseline OTEC effluent discharge scenarios in what follows. Accordingly, in the case of separate discharges, both evaporator and condenser effluent discharges are then assumed to be initially neutrally buoyant. This corresponds to water depths of 136 m (z

_{evap}= 3939 m, for an evaporator effluent temperature of 22.5 °C) and 602 m (z

_{cond}= 3473 m, for an initial condenser effluent temperature of 10 °C), respectively.

_{cw}= A

_{OTEC}w

_{cw}, in sverdrups. The dotted line indicates the condition when the advective drawdown induced in Region 3 by the OTEC deep seawater intake, w

_{cw}, is exactly equal to the background upward advection rate w. Baseline effluent discharge scenarios correspond to the blue and red curves. It is striking that separate discharges allow a maximum OTEC net power of 4.3 TW while a mixed discharge corresponds to a value of 2.7 TW. These peak values correspond to OTEC cold seawater flow rates w

_{cw}(Q

_{cw}) of 7.5 m yr

^{−1}(23.8 Sv) and 4.5 m yr

^{−1}(14.3 Sv), respectively. This confirms the nearly constant OTEC seawater flow intensity at peak net power production, of the order of 0.20 TW Sv

^{−1}. It indicates, in turn, that the warming of the deep cold seawater intake temperature is nearly the same in all instances of maximum OTEC net power production. This can be seen in Figure 4, where the blue and red temperature profiles are very similar below and across the deep-water intake depth of 1000 m. Other curves demonstrate that less degradation of the temperature profile occurs with separate discharges at given OTEC flow rates. In other words, less heat penetrates the oceanic water column across the ocean-atmosphere interface during the transient phase. This heat can be quantified by the integral $\rho {c}_{p}{A}_{OTEC}{\displaystyle {\int}_{0}^{L}\left\{\theta (t=\infty ,z)-\theta (t=0,z)\right\}}dz$: separate OTEC effluent discharges correspond to about 60% only of the value obtained for mixed discharge.

#### 3.2. Mixed Discharge at Variable Depth

_{cw}(Q

_{cw}) of 3.5 m yr

^{−1}(11.1 Sv).

#### 3.3. Condenser-Effluent Discharge at Variable Depth (with Evaporator-Effluent Discharge within the Mixed Layer)

## Acknowledgments

## Conflicts of Interest

## Appendix A

A_{OTEC} | nominal area favorable for OTEC (m^{2}) |

c_{p} | specific enthalpy of seawater (J kg^{−1} K^{−1}) |

h_{m} | mixed layer thickness (m) |

K | vertical thermal diffusion coefficient (m^{2} s^{−1}) |

L | height of oceanic water column below mixed layer (m) |

P | overall OTEC net power (W) |

T | mixed layer temperature (K) |

T_{p} | polar water temperature (K) |

w | background vertical upward advection rate (m s^{−1}) |

w_{cw} | OTEC deep cold seawater withdrawal rate (m s^{−1}) |

w_{ww} | OTEC surface warm seawater withdrawal rate (m s^{−1}) |

z | vertical coordinate measured from the seafloor (m) |

z_{cond} | vertical coordinate of the OTEC condenser effluent discharge (m) |

z_{cw} | vertical coordinate of the OTEC deep cold seawater intake (m) |

z_{evap} | vertical coordinate of the OTEC evaporator effluent discharge (m) |

z_{mix} | vertical coordinate of the OTEC mixed effluent discharge, when z_{evap} = z_{cond} (m) |

δT_{cond} | seawater temperature rise in OTEC condenser (K) |

δT_{evap} | seawater temperature drop in OTEC evaporator (K) |

ε_{tg} | nominal efficiency of OTEC turbo-generator (K) |

ρ | nominal density of seawater (kg m^{−3}) |

θ | seawater temperature (K) |

θ_{cond} | OTEC condenser effluent temperature (K) |

θ_{cw} | OTEC deep cold seawater withdrawal temperature (K) |

θ_{evap} | OTEC evaporator effluent temperature (K) |

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1 | There are typographic errors in [11], Equation (7): within the bracket, T should replace T _{0} and −Q_{up}T is missing. |

**Figure 1.**Overall OTEC net power as a function of nominal OTEC net power (i.e., with unchanging seawater temperatures) using two models of vastly different complexity; 1-D results have been adapted to inputs and conventions of the 3-D study.

**Figure 2.**Schematic description of steady-state fluxes in one-dimensional model of the water column with OTEC and separate effluent discharges. OTEC seawater intakes are shown as circles with crosses, OTEC seawater effluent discharges as circles with a dot. Not shown is a heat flux wT

_{p}leaving the mixed layer ‘horizontally’ that would represent deep water formation far from the OTEC region.

**Figure 3.**Steady-state OTEC net power as a function of overall cold seawater flow rate for baseline mixed (blue) and separate (red) effluent discharge scenarios, as well as for a mixed-effluent discharge into the mixed layer (black).

**Figure 4.**Seawater temperature profiles without OTEC (Initial Profile), and with OTEC under baseline (initial neutral buoyancy) effluent discharge scenarios: Separate Discharges (SD) or Mixed Discharge (MD); ‘maximum’ refers to the cold seawater flow w

_{cw}at which OTEC net power peaks.

**Figure 5.**Maximum OTEC net power as a function of mixed-effluent discharge depth for mixed-effluent discharge scenarios; the red line indicates the so-called baseline protocol (initial neutral buoyancy).

**Figure 6.**Maximum OTEC net power as a function of condenser-effluent discharge depth, with the evaporator effluent discharge within the mixed layer; the red line indicates the so-called baseline protocol (initial neutral buoyancy).

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**MDPI and ACS Style**

Nihous, G.
A Preliminary Investigation of the Effect of Ocean Thermal Energy Conversion (OTEC) Effluent Discharge Options on Global OTEC Resources. *J. Mar. Sci. Eng.* **2018**, *6*, 25.
https://doi.org/10.3390/jmse6010025

**AMA Style**

Nihous G.
A Preliminary Investigation of the Effect of Ocean Thermal Energy Conversion (OTEC) Effluent Discharge Options on Global OTEC Resources. *Journal of Marine Science and Engineering*. 2018; 6(1):25.
https://doi.org/10.3390/jmse6010025

**Chicago/Turabian Style**

Nihous, Gérard.
2018. "A Preliminary Investigation of the Effect of Ocean Thermal Energy Conversion (OTEC) Effluent Discharge Options on Global OTEC Resources" *Journal of Marine Science and Engineering* 6, no. 1: 25.
https://doi.org/10.3390/jmse6010025