A Preliminary Investigation of the Effect of Ocean Thermal Energy Conversion (OTEC) Effluent Discharge Options on Global OTEC Resources – part II by Gerard Nihous

.livello difficile
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ARGOMENTO: ENERGIE ALTERNATIVE
PERIODO: XXI SECOLO
AREA: RICERCA
parole chiave: OTEC

One-Dimensional Steady-State Model of the Water Column with OTEC
2.0 Section
As mentioned in Section 1, a vertical one-dimensional steady-state model of the water column was previously used to evaluate large-scale OTEC resources [4]. Symbols used in describing this model and its present extension are listed in Appendix A. A diffusion-advection equation for seawater temperature was solved from the seafloor (z = 0) to the base of a mixed layer of thickness h_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_pA_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 (10¹⁴ m²); 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_cwδT_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.

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 wTp leaving the mixed layer ‘horizontally’ that would represent deep water formation far from the OTEC region.

The steady-state temperature profile in the water column can be determined from the following steady-state heat flux equations:

−Kdθdz+(w+www)θ = wTp+wwwT zevap≤z≤L, 

(1)

−Kdθdz+wθ = wTp+wcw3www(T−θcw)8(www+wcw) zcond≤z≤zevap,

(2)

−Kdθdz+(w−wcw)θ = wTp−wcwθcw zcw≤z≤zcond,

(3)

−Kdθdz+wθ = wTp  0≤z≤zcw

(4)

These first-order ordinary differential equations require only one boundary condition each, and can easily be solved, especially if we take the vertical diffusion coefficient K and background upward advection velocity w to be constant, although such an expedient choice is not necessary [11]. Since large-scale OTEC processes do affect ocean temperatures in the water column, however, one more unknown appears in the specified right-hand-sides, i.e., the cold seawater intake temperature θ_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.

The system is algebraically closed with the boundary conditions (−K dθ/dz + wθ) = wT_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.

Before presenting results in Section 3, a few comments about this one-dimensional model are proposed to better understand the steady-state mixed-layer heat balance under various OTEC seawater discharge scenarios, while a few expected features of the seawater temperature profile may provide calculation check points. Firstly, by comparing Equations (3) and (4), we note that the z-derivative of θ is continuous at z = z_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:

{−Kdθdz(L)+wθ(L)} − wTp = 0,

(5)

where the last term in the left-hand-side corresponds to deep-water formation, and the upward heat flux from the water column (between brackets) is defined by Equation (1). With OTEC operations, and if no effluent is discharged into the mixed-layer, the steady-state mixed-layer heat balance becomes:

−wwwT+{−Kdθdz(L)+(w+www)θ(L)} − wTp = 0,

(6)

where the first term in the left-hand-side represents the OTEC warm seawater intake sink, and the upward heat flux from the water column (between brackets) is defined by Equation (1) as before. Comparing Equations (5) and (6) reveals that the z-derivative of θ at z = L remains the same whether w_ww is zero (no OTEC) or not, i.e., dθ/dz(L) = w(T − T_p)/K.

If the OTEC evaporator effluent is discharged in the mixed layer, Region 1 in Figure 2 vanishes and Equation (1) is no longer applicable. The steady-state mixed-layer heat balance is altered as follows:

www(T−δTevap)−wwwT+{−Kdθdz(L)+wθ(L)} − wTp = 0,

(7)

where the first term in the left-hand-side represents the OTEC evaporator effluent source, and the upward heat flux from the water column (between brackets) is now defined by Equation (2). In this case, the z-derivative of θ at z = L decreases to limit diffusive heat losses from the mixed layer and compensate for the new source of cooler water; we have dθ/dz(L) = {w(T − T_p) − w_wwδT_evap}/K.

Finally, in the most practical case of a mixed-effluent discharge within the mixed layer (i.e., z_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:

(wwwT+wcwθcw)−wwwT+{−Kdθdz(L)+(w−wcw)θ(L)} − wTp = 0,

(8)

where the first term in the left-hand-side represents the net OTEC mixed-effluent source, and the upward heat flux from the water column (between brackets) is defined by Equation (3). In this case, the z-derivative of θ at z = L further decreases to limit diffusive heat losses from the mixed layer and compensate for a net input of much cooler water; we have dθ/dz(L) = {w(T − T_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.

Results
3.0 Section
Equations (1) through (4) are solved subject to their boundary conditions. Since the focus here is to explore different protocols for handling OTEC seawater effluents, several input choices in the model are kept the same as in Nihous [4]. In particular, constant values of the background diffusion and advection parameters are selected, i.e., K = 2300 m² 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]:

Pnet = ρcpεtgAOTEC{(T−θcw)2−0.3(T−θcw|wcw= 0)28T}wcw,

(9)

where the nominal turbo-generator efficiency ε_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

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).

The baseline mixed effluent discharge scenario previously adopted consisted of choosing z_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.

Figure 3 shows OTEC net power calculated from Equation (9), in terawatts, as a function of the overall OTEC deep cold seawater volume flow rate Q_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 ρcpAOTEC∫L0{θ(t=∞,z)−θ(t=0,z)}dz: separate OTEC effluent discharges correspond to about 60% only of the value obtained for mixed discharge.

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 wcw at which OTEC net power peaks.

3.2. Mixed Discharge at Variable Depth
Next, mixed effluent discharge scenarios were considered at different depths, from the mixed layer downward. As noted earlier, a mixed discharge within the mixed layer may be the most practical choice from an engineering perspective, while it mathematically corresponds to an upwelling of deep cold seawater into the mixed layer.

Figure 5 shows the dependence of maximum OTEC net power as a function of mixed-effluent discharge depth. It is noticeable that the optimum choice of the discharge depth is close to the baseline value (initial neutral buoyancy). This suggests that all other things being equal, initial neutral-buoyancy of the discharged seawater corresponds to a minimal disturbance of the water column with a relatively more benign impact on the thermal structure. A discharge into the mixed layer only results in 2.2 TW, or about 80% of the baseline value. The variation of global OTEC net power as a function of overall OTEC deep cold seawater flow rate is also shown in this particular case in Figure 3 (black curve); peak net power production corresponds to an OTEC cold seawater flow rate w_cw (Q_cw) of 3.5 m yr⁻¹ (11.1 Sv).

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).

3.3. Condenser-Effluent Discharge at Variable Depth (with Evaporator-Effluent Discharge within the Mixed Layer)
Here, the effect of condenser-effluent discharge depth was examined in separate discharge scenarios. Figure 6 shows the dependence of maximum OTEC net power as a function of condenser-effluent discharge depth, when the evaporator-effluent discharge is assumed to take place within the mixed layer. Maximum OTEC power production occurs at a condenser-effluent discharge depth of 705 m, i.e., once again, in the general neighborhood of the baseline value. Also, the peak power production, at 4.3 TW, is similar to the value obtained in the baseline scenario discussed in Section 3.1. This indicates that the condenser-effluent discharge depth has a much greater impact than the evaporator-effluent discharge depth.

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).

Parametric calculations where both separate discharge depths were variable yielded an overall OTEC net power maximum of 4.4 TW, i.e., hardly greater than the baseline scenario value, for evaporator and condenser effluent discharge depths of 130 m and 700 m, respectively.

Acknowledgments
The costs to publish this work in open access are waived under the sponsorship of Luis Vega, Guest Editor for the special issue of JMSE on OTEC.

Conflicts of Interest
The author declares no conflict of interest.

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PARTE I  PARTE II

Gérard Nihous
Department of Ocean and Resources Engineering, University of Hawaii, Honolulu, HI 96822, USA; Tel.: +1-808-956-2338

References

© 2018 by the author. 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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