# Energetic and economic cost of nuclear heat-impact on the cost of desalination

Saied Dardour, Henri Safa

► To cite this version:

Saied Dardour, Henri Safa. Energetic and economic cost of nuclear heat-impact on the cost of desalination. EPJ N - Nuclear Sciences & Technologies, 2017, 3, pp.1. 10.1051/epjn/2016037. cea-02304980

HAL Id: cea-02304980

https://cea.hal.science/cea-02304980

Submitted on 3 Oct 2019

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
REGULAR ARTICLE

OPEN ACCESS

# Energetic and economic cost of nuclear heat – impact on the cost of desalination

Saied Dardour<fnref n="1" />,<fnref n="2" />,* and Henri Safa<fnref n="1" />,<fnref n="3" />

Received: 5 April 2016 / Received in final form: 8 November 2016 / Accepted: 8 November 2016

**Abstract.** An exploratory study has been carried out to evaluate the cost of heat supplied by a pressurized water reactor type of nuclear reactors to thermal desalination processes. In the context of this work, simplified models have been developed to describe the thermodynamics of power conversion, the energetics of multi-effect evaporation (MED), and the costs of electricity and heat cogenerated by the dual-purpose power plant. Application of these models show that, contrary to widespread belief, (nuclear-powered) MED and seawater reverse osmosis are comparable in terms of energy effectiveness. Process heat can be produced, in fact, by a relatively small increase in the core power. As fuel represents just a fraction of the cost of nuclear electricity, the increase in fuel-related expenses is expected to have limited impact on power generation economics.

## 1 Introduction

With almost 75 million cubic meter per day of worldwide installed capacity [1], desalination is the main technology used to meet water scarcity. About two third of this capacity is produced by reverse osmosis (RO) (Fig. 1). The remaining one third is produced mainly by thermal desalination plants–multi-effect evaporation (MED) and multi-stage flash (MSF), mostly in the Middle East.

Seawater desalination is an energy-intensive process.<fnref n="1" /> According to [2], the lowest energy consumption – and the closest to the minimum set by thermodynamics (1.06 kWh m⁻<fnref n="3" />) [3] – is achieved by RO processes equipped with energy recovery devices. Seawater RO (SWRO) electricity utilization ranges, in fact, between 4 and 7 kW_e h m⁻<fnref n="3" /> [4]. Some plants, producing large amount of desalinated water, claim even lower energy consumption; 3.5 kW_e h m⁻<fnref n="3" /> for Ashkelon, Israel [4]; and 2.7–3.1 kW_e h m⁻<fnref n="3" /> (depending on temperature and membrane ageing) for Perth, Australia [5].

Thermal desalination processes consume heat,<fnref n="2" /> in addition to electricity. Heat consumption varies between 40 and 65 kWh<sub>th</sub> m⁻<fnref n="3" /> for MED, and 55–80 kWh<sub>th</sub> m⁻<fnref n="3" /> for MSF [2]. MSF's electric power consumption is

higher than MED's because of pressure drops in flashing chambers and the possible presence of brine recirculation loops [6]. MSF's pumping power varies between 2.5 and 5 kWh<sub>e</sub> m⁻<fnref n="3" /> [7]. MED manufacturers claim specific electricity consumptions lower than 2.5 kWh<sub>e</sub> m⁻<fnref n="3" />.

### 1.1 Power consumption: thermal desalination systems vs. membrane-based processes

Thermal desalination systems are often coupled to power generation units to form “integrated water and power plants” (IWPPs) in which steam is supplied to the desalination unit by the power plant.

The cost of process heat provided by such plants is traditionally evaluated based on the “missed electricity production”–steam diverted to the process is no longer used for electricity production – leading, systematically, to higher energy costs for the thermal desalination processes compared to RO. MED's steam supply costs between 4 and 7 kWh<sub>e</sub> m⁻<fnref n="3" /> of “missed electricity production” according to [2]. If we add 1.2–2.5 kWh<sub>e</sub> m⁻<fnref n="3" /> of pumping energy, we end up with an equivalent electric power consumption in the range [5.2–9.5] kWh<sub>e</sub> m⁻<fnref n="3" />.

Rognoni et al. [8] suggested an alternative way to evaluating the cost of heat “duly considering the benefits of cogeneration”. The approach no longer views process heat as a “missed electricity production”, but, rather, as “a result of a (limited) raise in the primary power”–the power released from combustion. According to this approach, the

<footnote n="1">¹ Commissariat à l'Énergie Atomique et aux Énergies Alternatives, 13108 Saint-Paul-lez-Durance Cedex, France</footnote>

<footnote n="2">² DEN/DER/SESI, CEA Cadarache, Bât.1222, 13108 Saint-Paul-lez-Durance Cedex, France</footnote>

<footnote n="3">³ International Institute of Nuclear Energy, 91191 Gif-sur-Yvette Cedex, France</footnote>

<footnote>* e-mail: saied.dardour@cea.fr</footnote>

<footnote n="1">¹ Energy is, in many cases, the largest contributor to the desalted water cost, varying from one-third to more than one-half of the cost of produced water.</footnote>

<footnote n="2">² MED's top brine temperature (TBT) generally varies between 60 and 75 °C. MSF's TBT is higher, 90–110 °C.</footnote>
<figure><img src="image_2.png" /><figcaption>Fig. 1. Total worldwide installed capacity by technology.</figcaption></figure>

energetic cost of process heat is equal to the number of MW<sub>th</sub> added to the boiler thermal power output. Since fuel represents just a fraction of the cost of electricity, process heat is expected to be cheaper than predictions based on the traditional cost evaluation method. As a result, thermal desalination processes – precisely MED – can be potentially more cost-effective than SWRO. The authors provided two-calculation examples – MED processes fueled by coal-fired power plants in India – for which the cost of desalinated water is 50% lower than SWRO's.

## 1.2 "Nuclear steam" cost

The cost of process heat depends on the contribution, to the total cost of electricity, of fuel-related expenses – a contribution widely considered to be lower for nuclear-powered electricity generators compared to fossil power plants [9]. Past studies show, in fact, that heat recovery from light water reactors is economically competitive for a number of low temperature applications, including district heating [10] and seawater desalination [11].

The study described in this paper aims at evaluating the – energetic and economic – cost of process heat, supplied by pressurized water reactor (PWR) to a thermal desalination process. The objective is to provide a basis for comparing thermal (MED<sup>3</sup>) and membrane-based (SWRO) desalination processes in terms of energy costs. Simplified models, describing the thermodynamics of a generic PWR power conversion system, the energetics the MED process, and the costs of electricity and process heat produced by the dual-purpose plant (DPP), support this study. These models, and the results of their application, are presented and discussed in the next sections.

# 2 Energetic cost of heat

The energetic cost of heat was evaluated based on the power conversion system (PCS) architecture described in the next paragraph.

## 2.1 Power conversion system architecture

Figure 2 illustrates the workflow of the PCS being modeled.

The system is basically a Rankine cycle representative of the technologies commonly applied is PWRs. Steam leaving steam generators (SG) undergoes two expansions in the high-pressure body of the turbine (HPT<sub>1</sub> and HPT<sub>2</sub>). The fluid is then dried-up and superheated before supplying the low-pressure stages (LPT<sub>1</sub>, LPT<sub>2</sub> and LPT<sub>3</sub>). Liquid water extracted from the condenser (Condenser<sub>2</sub>) is finally preheated and readmitted back to SG.

A steam extraction point was positioned between the outlet of LPT<sub>2</sub> and the inlet of LPT<sub>3</sub>. This location allows for a variable quantity (y=0–100%) of steam (the steam normally flowing through LPT<sub>3</sub>) to be diverted to an external process. The pressure at the steam extraction point (P<sub>SteamEx</sub>) may vary between 0.05 bar (pressure at the condenser) and 2.685 bar (pressure at LPT<sub>2</sub> outlet), and the temperature (T<sub>SteamEx</sub>) between 33 and 129 °C. The range of temperatures generally required by thermal desalination systems generally falls within these limits.

The power plant condenser was (virtually) split in two. In Condenser<sub>1</sub>, the latent heat of condensation is transferred to the external process. Condenser<sub>2</sub> cools the condensates down to 33 °C. The heat duty of each of the two condensers strongly depends on the quantity of steam diverted to the process.

## 2.2 Thermodynamic model

A thermodynamic model, evaluating the energetic performance of the PCS described in the previous paragraph, was developed using CEA's in-house tool ICV.<sup>4</sup>

<footnote n="3"><sup>3</sup> MSF is out of scope in this paper, as it consumes higher amounts of energy compared to MED.</footnote>

<footnote n="4"><sup>4</sup> ICV simulates the steady-state behavior of components such as boilers, heat exchangers, pumps, compressors and turbines, as well as workflows – typically heat transfer loops and power conversion cycles – based on these components. ICV has a build-in library providing the properties of steam and water [12], including saline-water [13].</footnote>
<figure><img src="image_3.png" /><figcaption>Fig. 2. Power conversion system architecture.</figcaption></figure>

<table><caption>Table 1. Assumed pressure distribution.</caption><tr><td>Steam generator outlet</td><td>70 bar</td></tr><tr><td>High pressure turbine 1 inlet</td><td></td></tr><tr><td>High pressure turbine 2 inlet</td><td>36.5091 bar</td></tr><tr><td>High pressure turbine 2 outlet</td><td>11 bar</td></tr><tr><td>Separator inlet, outlets</td><td></td></tr><tr><td>Low pressure turbine 1 inlet</td><td>10 bar</td></tr><tr><td>Low pressure turbine 2 inlet</td><td>2.685 bar</td></tr><tr><td>Low pressure turbine 3 inlet</td><td>(variable)</td></tr><tr><td>Condenser<sub>1</sub></td><td></td></tr><tr><td>Condenser<sub>2</sub></td><td>0.05 bar</td></tr><tr><td>Low pressure pump outlet</td><td>15 bar</td></tr><tr><td>Preheater outlet</td><td>12.5 bar</td></tr><tr><td>Mixer outlet</td><td>11 bar</td></tr><tr><td>High pressure pump outlet</td><td>87.5 bar</td></tr><tr><td>Reheater outlet</td><td>85 bar</td></tr></table>

The model calculates the characteristics of the 23 points of the flowsheet-temperature, pressure, steam quality,<fnref n="5" /> enthalpy, exergy and flowrate-the power of the major components of the PCS, as well as the amounts of electricity ($W_{Elec}$) and process heat ($Q_{Pro}$) cogenerated by the system.

Model inputs include:

- an assumed pressure distribution within the PCS (Tab. 1);

- SG outlet temperature (290 °C) and thermal power output ($Q_{SG}$);

- the temperature at the steam extraction point ($T_{SteamEx}$);

- the fraction of steam (normally expending through $LPT_3$) diverted to the external process (y).

The calculation of the Rankine cycle is performed sequentially, component by component, applying the mass and energy balance equations (Eqs. (1) and (2))<fnref n="6" /> to different control volumes.

$$ \sum_{\text{in}} \dot{m} = \sum_{\text{out}} \dot{m}, \quad (1) $$

$$ \begin{aligned} & \sum_{\text{in}} \dot{Q} + \dot{W} + \dot{m} \times \left( \bar{h} + \frac{v^2}{2} + g \times z \right) \\ & = \sum_{\text{out}} \dot{Q} + \dot{W} + \dot{m} \times \left( \bar{h} + \frac{v^2}{2} + g \times z \right), \end{aligned} \quad (2) $$

$\dot{m}$, mass flowrate (kg/s); $\dot{Q}$, thermal power (W); $\dot{W}$, mechanical power (W); $\bar{h}$, specific enthalpy (J/kg); $v^2/2$, specific kinetic energy (J/kg); $g \times z$, specific potential energy (J/kg); $g \times z$, specific potential energy (J/kg).

<footnote n="5">⁵ Mass of vapor to total mass in a saturated liquid-vapor mixture. Values lower than 0 or higher than 1 indicate that the fluid is either subcooled (–100) or superheated (200).</footnote>

<footnote n="6">⁶ In practice, the “kinetic + potential energies” term of equation (2) is neglected, leading to a simpler formulation of the energy conservation principle.</footnote>
<table><caption>**Table 2.** SPP (PWR 2748 MW<sub>th</sub> → 1000 MW<sub>e</sub>): thermodynamic points.</caption><thead><tr><th>Point</th><th><i>T</i> (°C)</th><th><i>P</i> (bar)</th><th><i>X</i> (%)</th><th><i>H</i> (kJ kg<sup>-1</sup>)</th><th><i>E</i> (kJ kg<sup>-1</sup> K<sup>-1</sup>)</th><th><i>F</i> (kg s<sup>-1</sup>)</th></tr></thead><tbody><tr><td>1</td><td>290</td><td>70</td><td>200</td><td>2793.98</td><td>1048.97</td><td>139.405</td></tr><tr><td>2</td><td>290</td><td>70</td><td>200</td><td>2793.98</td><td>1048.97</td><td>1385.06</td></tr><tr><td>3</td><td>245</td><td>36.5091</td><td>93.2741</td><td>2685.23</td><td>931.684</td><td>245.05</td></tr><tr><td>4</td><td>245</td><td>36.5091</td><td>93.2741</td><td>2685.23</td><td>931.684</td><td>1140.01</td></tr><tr><td>5</td><td>184.07</td><td>11</td><td>85.9332</td><td>2499.41</td><td>729.341</td><td>1140.01</td></tr><tr><td>6</td><td>184.07</td><td>11</td><td>100</td><td>2780.67</td><td>827.192</td><td>979.649</td></tr><tr><td>7</td><td>275</td><td>10</td><td>200</td><td>2997.9</td><td>902.283</td><td>979.649</td></tr><tr><td>8</td><td>145.081</td><td>2.685</td><td>200</td><td>2753.21</td><td>633.359</td><td>180.999</td></tr><tr><td>9</td><td>145.081</td><td>2.685</td><td>200</td><td>2753.21</td><td>633.359</td><td>798.65</td></tr><tr><td>10</td><td>80</td><td>0.474147</td><td>93.8269</td><td>2500.54</td><td>351.599</td><td>0</td></tr><tr><td>11</td><td>80</td><td>0.474147</td><td>93.8269</td><td>2500.54</td><td>351.599</td><td>798.65</td></tr><tr><td>12</td><td>32.8755</td><td>0.05</td><td>86.5162</td><td>2234.05</td><td>49.7124</td><td>798.65</td></tr><tr><td>13</td><td>32.8755</td><td>0.05</td><td>0</td><td>137.765</td><td>-4.22995</td><td>979.649</td></tr><tr><td>14</td><td>32.9654</td><td>15</td><td>-100</td><td>139.492</td><td>-2.72166</td><td>979.649</td></tr><tr><td>15</td><td>130.081</td><td>12.5</td><td>-100</td><td>547.394</td><td>60.0595</td><td>979.649</td></tr><tr><td>16</td><td>170.264</td><td>10</td><td>-100</td><td>720.471</td><td>110.977</td><td>1524.47</td></tr><tr><td>17</td><td>171.56</td><td>87.5</td><td>-100</td><td>730.378</td><td>120.02</td><td>1524.47</td></tr><tr><td>18</td><td>230</td><td>85</td><td>-100</td><td>991.385</td><td>216.545</td><td>1524.47</td></tr><tr><td>1s</td><td>285.83</td><td>70</td><td>0</td><td>1267.44</td><td>336.615</td><td>139.405</td></tr><tr><td>3s</td><td>245</td><td>36.5091</td><td>0</td><td>1061.49</td><td>242.266</td><td>245.05</td></tr><tr><td>5s</td><td>184.07</td><td>11</td><td>0</td><td>781.198</td><td>131.569</td><td>160.363</td></tr><tr><td>8s</td><td>129.782</td><td>2.685</td><td>0</td><td>545.456</td><td>58.7786</td><td>180.999</td></tr><tr><td>10s</td><td>80</td><td>0.474147</td><td>0</td><td>334.949</td><td>14.3207</td><td>0</td></tr></tbody></table>

The state of the fluid at the outlet of steam turbines and water pumps is determined applying an isentropic efficiency (88% for turbines and 87% for pumps):

$$ \epsilon_{\text{turbine}} = \frac{\bar{h}_{\text{in}} - \bar{h}_{\text{out}}}{\bar{h}_{\text{in}} - \bar{h}_{\text{out}}(\bar{s}_{\text{out}} = \bar{s}_{\text{in}})}, \quad (3) $$

$$ \epsilon_{\text{pump}} = \frac{\bar{h}_{\text{out}}(\bar{s}_{\text{out}} = \bar{s}_{\text{in}}) - \bar{h}_{\text{in}}}{\bar{h}_{\text{out}} - \bar{h}_{\text{in}}}, \quad (4) $$

$\epsilon$, isentropic efficiency; $\bar{h}_{\text{in}}$, specific enthalpy at inlet (J/kg); $\bar{s}_{\text{in}}$, specific entropy at inlet (J/kg/K); $\bar{h}_{\text{out}}$, specific enthalpy at outlet (J/kg); $\bar{s}_{\text{out}}$, specific entropy at outlet (J/kg/K); $\bar{h}_{\text{out}}(\bar{s}_{\text{out}} = \bar{s}_{\text{in}})$, specific enthalpy at outlet for a constant-entropy transformation.

The following assumptions were also made:

- Steam admitted to different heat exchangers is assumed to leave all its latent heat to the fluid flowing on the secondary side of the exchanger.

- A fixed pinch point temperature difference of 15 °C was systematically applied to determine the outlet fluid temperature on the secondary side.

- Energy losses<fnref n="7" /> are not taken into account (the calculated “net” power and heat outputs are actually “gross” power and heat outputs).

<table><caption>**Table 3.** SPP (PWR 2748 MW<sub>th</sub> → 1000 MW<sub>e</sub>): mechanical and thermal powers.</caption><thead><tr><th>Component</th><th>Power (MW)</th></tr></thead><tbody><tr><td>Steam generators</td><td>2747.99</td></tr><tr><td>High pressure turbine 1</td><td>-150.624</td></tr><tr><td>High pressure turbine 2</td><td>-211.837</td></tr><tr><td>Low pressure turbine 1</td><td>-239.709</td></tr><tr><td>Low pressure turbine 2</td><td>-201.796</td></tr><tr><td>Low pressure turbine 3</td><td>-212.827</td></tr><tr><td>Condenser<sub>1</sub> (Process)</td><td>0</td></tr><tr><td>Condenser<sub>2</sub> (Tertiary circuit)</td><td>-1747.99</td></tr><tr><td>Low pressure pump</td><td>1.69183</td></tr><tr><td>High pressure pump</td><td>15.102</td></tr><tr><td>Sum</td><td>-1.14 × 10<sup>-13</sup></td></tr><tr><td>Net power output</td><td>-1000</td></tr><tr><td>Power conversion efficiency (%)</td><td>36.3902</td></tr></tbody></table>

## 2.3 Energetic performance of the PCS

Tables 2 and 3 show the characteristics of a 2748 MW<sub>th</sub> single-purpose plant (SPP) generating 1000 MW<sub>e</sub> of electricity.

The contribution of steam turbines to SPP's electricity output is shown in Figure 3. LPT<sub>3</sub> delivers 213 MW<sub>e</sub> of mechanical power, which represents 21% of the total electricity output.

<footnote n="7">⁷ Thermal losses at heat exchangers. Mechanical losses at pumps, turbines and generators. Electrical power consumption, internal to the power plant and the external process.</footnote>
<figure><img src="image_4.png" /><figcaption>Fig. 3. Contribution of steam turbines to SPP's electricity output.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 4. Available heat for the external process vs. temperature at the steam extraction point. Blue bar: PWR 1000 MWe (2748 MWth); orange bar: PWR 1650 MWe (4534 MWth).</figcaption></figure>

If all the steam normally flowing towards this turbine is redirected to the external process ($T_{SteamEx} = 80^{\circ}C$), the plant would generate 787 MW$_e$ of electricity and 1730 MW$_{th}$ of process heat. The reactor's process heat generation capacity depends, in fact, on the core power, and on the temperature at the steam extraction point, as shown in Figure 4.

Now, if only a portion of this steam – exactly 57.8% – is diverted, the plant would produce 877 MW$_e$ of electricity and 1000 MW$_{th}$ of heat. The characteristics of configuration – we will call it DPP$_1$ (dual-purpose plant) – are listed in Tables 4 and 5.

The differences between SPP and DPP$_1$ are highlighted (underlined) in Tables 2–5. The two Rankine cycles have identical characteristics except for points 10–12. In DPP$_1$, turbine LPT$_3$ is partly bypassed – the exergy of the rerouted steam is later “destructed” in Condenser$_1$ – resulting in a 123 MW$_e$ decrease in power generation compared to SPP.

The number of MW$_e$ of electricity production lost for each MW$_{th}$ supplied to the external process (123 kW$_e$ per MW$_{th}$ in the case of DPP$_1$) is a traditional measure of the energetic cost of process heat. This measure will be referred

to as the “W-cost of heat” or WCH:

$$WCH = \left[ \frac{\Delta \dot{W}_{Elec}}{\dot{Q}_{Pro}} \right] \dot{Q}_{SG=Constant} \quad (5)$$

This “loss” in electricity production can be avoided by increasing the thermal power of the core. To keep the electricity generation capacity at 1000 MW$_e$ – and the heat production level at 1000 MW$_{th}$ – SG have to deliver an additional 338 MW$_{th}$. The portion of diverted steam has also to be adjusted (51.5%). This configuration – we will call it DPP$_2$ (Tabs. 6 and 7) – not only offers higher power conversion efficiency (32.40%) compared to DPP$_1$ (31.91%), but also results in lower heat cost, as we will see in Section 2.

The number of MW$_{th}$ added to core power, per MW$_{th}$ supplied to the external process (338 kW$_{th}$ per MW$_{th}$ in the case of DPP$_2$) provides an alternative measure of the energetic cost of steam – we will call it the “Q-cost of heat” or QCH:

$$QCH = \left[ \frac{\Delta \dot{Q}_{SG}}{\dot{Q}_{Pro}} \right] \dot{W}_{Elec=Constant} \quad (6)$$
<table><caption>**Table 4.** DPP₁ (PWR 2748 MWth → 877 MWe + 1000 MWth at 80 °C): thermodynamic points.</caption><thead><tr><th>Point</th><th><i>T</i> (°C)</th><th><i>P</i> (bar)</th><th><i>X</i> (%)</th><th><i>H</i> (kJ kg<sup>-1</sup>)</th><th><i>E</i> (kJ kg<sup>-1</sup> K<sup>-1</sup>)</th><th><i>F</i> (kg s<sup>-1</sup>)</th></tr></thead><tbody><tr><td>1</td><td>290</td><td>70</td><td>200</td><td>2793.98</td><td>1048.97</td><td>139.405</td></tr><tr><td>2</td><td>290</td><td>70</td><td>200</td><td>2793.98</td><td>1048.97</td><td>1385.06</td></tr><tr><td>3</td><td>245</td><td>36.5091</td><td>93.2741</td><td>2685.23</td><td>931.684</td><td>245.05</td></tr><tr><td>4</td><td>245</td><td>36.5091</td><td>93.2741</td><td>2685.23</td><td>931.684</td><td>1140.01</td></tr><tr><td>5</td><td>184.07</td><td>11</td><td>85.9332</td><td>2499.41</td><td>729.341</td><td>1140.01</td></tr><tr><td>6</td><td>184.07</td><td>11</td><td>100</td><td>2780.67</td><td>827.192</td><td>979.649</td></tr><tr><td>7</td><td>275</td><td>10</td><td>200</td><td>2997.9</td><td>902.283</td><td>979.649</td></tr><tr><td>8</td><td>145.081</td><td>2.685</td><td>200</td><td>2753.21</td><td>633.359</td><td>180.999</td></tr><tr><td>9</td><td>145.081</td><td>2.685</td><td>200</td><td>2753.21</td><td>633.359</td><td>798.65</td></tr><tr><td>10</td><td>80</td><td>0.474147</td><td>93.8269</td><td>2500.54</td><td>351.599</td><td><u>461.769</u></td></tr><tr><td>11</td><td>80</td><td>0.474147</td><td>93.8269</td><td>2500.54</td><td>351.599</td><td><u>336.881</u></td></tr><tr><td>12</td><td>32.8755</td><td>0.05</td><td>86.5162</td><td>2234.05</td><td>49.7124</td><td><u>336.881</u></td></tr><tr><td>13</td><td>32.8755</td><td>0.05</td><td>0</td><td>137.765</td><td>-4.22995</td><td>979.649</td></tr><tr><td>14</td><td>32.9654</td><td>15</td><td>-100</td><td>139.492</td><td>-2.72166</td><td>979.649</td></tr><tr><td>15</td><td>130.081</td><td>12.5</td><td>-100</td><td>547.394</td><td>60.0595</td><td>979.649</td></tr><tr><td>16</td><td>170.264</td><td>10</td><td>-100</td><td>720.471</td><td>110.977</td><td>1524.47</td></tr><tr><td>17</td><td>171.56</td><td>87.5</td><td>-100</td><td>730.378</td><td>120.02</td><td>1524.47</td></tr><tr><td>18</td><td>230</td><td>85</td><td>-100</td><td>991.385</td><td>216.545</td><td>1524.47</td></tr><tr><td>1s</td><td>285.83</td><td>70</td><td>0</td><td>1267.44</td><td>336.615</td><td>139.405</td></tr><tr><td>3s</td><td>245</td><td>36.5091</td><td>0</td><td>1061.49</td><td>242.266</td><td>245.05</td></tr><tr><td>5s</td><td>184.07</td><td>11</td><td>0</td><td>781.198</td><td>131.569</td><td>160.363</td></tr><tr><td>8s</td><td>129.782</td><td>2.685</td><td>0</td><td>545.456</td><td>58.7786</td><td>180.999</td></tr><tr><td>10s</td><td>80</td><td>0.474147</td><td>0</td><td>334.949</td><td>14.3207</td><td><u>461.769</u></td></tr></tbody></table>

QCH is simply obtained dividing WCH by SPP's power conversion efficiency.

The increase in core power considered in this study is purely conceptual.<fnref n="8" /> Adopting QCH as a measure of the energetic cost of steam makes it possible, in fact, to take into account the advantages cogeneration offers.

Figure 5 shows how WCS and QCH vary with $T_{\text{SteamEx}}$. At 75 °C, each MWth,h of thermal power supplied to the process costs 111 kWc,h of electricity. At 100 °C, the cost increases to 169 kWc,h MWth,h⁻¹ (×1.5), and at 125 °C it reaches 223 kWc,h MWth,h⁻¹ (×2).

The energetic cost of heat depends, actually, on the enthalpy at the steam extraction point, which is a function of the level of temperature required by the external process (Eq. (7)).

$$ \text{WCH}(T_{\text{SteamEx}}) = \frac{\bar{h}_{\text{SteamEx}}(T_{\text{SteamEx}}) - \bar{h}_{\text{LPT}_{3,\text{outlet}}}}{\bar{h}_{\text{SteamEx}}(T_{\text{SteamEx}}) - \bar{h}_{\text{Consenser}_{1,\text{outlet}}}} \quad (7) $$

# 3 Economic cost of heat

## 3.1 Single-purpose plant

To evaluate the cost of electricity relative a single-purpose plant, we first calculate the minimal annual cash in-generated from

<table><caption>**Table 5.** DPP₁ (PWR 2748 MWth → 877 MWc + 1000 MWth at 80 °C): mechanical and thermal powers.</caption><thead><tr><th>Component</th><th>Power (MW)</th></tr></thead><tbody><tr><td>Steam generators</td><td>2747.99</td></tr><tr><td>High pressure turbine 1</td><td>-150.624</td></tr><tr><td>High pressure turbine 2</td><td>-211.837</td></tr><tr><td>Low pressure turbine 1</td><td>-239.709</td></tr><tr><td>Low pressure turbine 2</td><td>-201.796</td></tr><tr><td>Low pressure turbine 3</td><td>-89.7732</td></tr><tr><td>Condenser₁ (Process)</td><td>-1000</td></tr><tr><td>Condenser₂ (Tertiary circuit)</td><td>-871.044</td></tr><tr><td>Low pressure pump</td><td>1.69183</td></tr><tr><td>High pressure pump</td><td>15.102</td></tr><tr><td>Sum</td><td>1.25 × 10<sup>-13</sup></td></tr><tr><td>Net power output</td><td>-876.946</td></tr><tr><td>Power conversion efficiency (%)</td><td>31.9123</td></tr></tbody></table>

the sale of electricity-required to have a positive NPV.NPV refers here to the net present value of future free cash flows:

- annual expenses related to, construction, purchase of nuclear fuel, operation and maintenance (O&M), and decommissioning, on one hand;

- annual revenue generated from the sale of electricity, on the other hand.

<footnote n="8">⁸ Increasing the fission power of the core is not always technologically feasible, especially for plants that are already “big”.</footnote>
<table><caption>**Table 6.** DPP₂ (PWR 3086 MW<sub>th</sub> → 1000 MW<sub>e</sub> + 1000 MW<sub>th</sub> at 80°C): thermodynamic points.</caption><thead><tr><th>Point</th><th><i>T</i> (°C)</th><th><i>P</i> (bar)</th><th><i>X</i> (%)</th><th><i>H</i> (kJ kg<sup>-1</sup>)</th><th><i>E</i> (kJ kg<sup>-1</sup> K<sup>-1</sup>)</th><th><i>F</i> (kg s<sup>-1</sup>)</th></tr></thead><tbody><tr><td>1</td><td>290</td><td>70</td><td>200</td><td>2793.98</td><td>1048.97</td><td>156.56</td></tr><tr><td>2</td><td>290</td><td>70</td><td>200</td><td>2793.98</td><td>1048.97</td><td>1555.5</td></tr><tr><td>3</td><td>245</td><td>36.5091</td><td>93.2741</td><td>2685.23</td><td>931.684</td><td>275.204</td></tr><tr><td>4</td><td>245</td><td>36.5091</td><td>93.2741</td><td>2685.23</td><td>931.684</td><td>1280.29</td></tr><tr><td>5</td><td>184.07</td><td>11</td><td>85.9332</td><td>2499.41</td><td>729.341</td><td>1280.29</td></tr><tr><td>6</td><td>184.07</td><td>11</td><td>100</td><td>2780.67</td><td>827.192</td><td>1100.2</td></tr><tr><td>7</td><td>275</td><td>10</td><td>200</td><td>2997.9</td><td>902.283</td><td>1100.2</td></tr><tr><td>8</td><td>145.081</td><td>2.685</td><td>200</td><td>2753.21</td><td>633.359</td><td>203.272</td></tr><tr><td>9</td><td>145.081</td><td>2.685</td><td>200</td><td>2753.21</td><td>633.359</td><td>896.926</td></tr><tr><td>10</td><td>80</td><td>0.474147</td><td>93.8269</td><td>2500.54</td><td>351.599</td><td>461.769</td></tr><tr><td>11</td><td>80</td><td>0.474147</td><td>93.8269</td><td>2500.54</td><td>351.599</td><td>435.158</td></tr><tr><td>12</td><td>32.8755</td><td>0.05</td><td>86.5162</td><td>2234.05</td><td>49.7124</td><td>435.158</td></tr><tr><td>13</td><td>32.8755</td><td>0.05</td><td>0</td><td>137.765</td><td>-4.22995</td><td>1100.2</td></tr><tr><td>14</td><td>32.9654</td><td>15</td><td>-100</td><td>139.492</td><td>-2.72166</td><td>1100.2</td></tr><tr><td>15</td><td>130.081</td><td>12.5</td><td>-100</td><td>547.394</td><td>60.0595</td><td>1100.2</td></tr><tr><td>16</td><td>170.264</td><td>10</td><td>-100</td><td>720.471</td><td>110.977</td><td>1712.06</td></tr><tr><td>17</td><td>171.56</td><td>87.5</td><td>-100</td><td>730.378</td><td>120.02</td><td>1712.06</td></tr><tr><td>18</td><td>230</td><td>85</td><td>-100</td><td>991.385</td><td>216.545</td><td>1712.06</td></tr><tr><td>1s</td><td>285.83</td><td>70</td><td>0</td><td>1267.44</td><td>336.615</td><td>156.56</td></tr><tr><td>3s</td><td>245</td><td>36.5091</td><td>0</td><td>1061.49</td><td>242.266</td><td>275.204</td></tr><tr><td>5s</td><td>184.07</td><td>11</td><td>0</td><td>781.198</td><td>131.569</td><td>180.097</td></tr><tr><td>8s</td><td>129.782</td><td>2.685</td><td>0</td><td>545.456</td><td>58.7786</td><td>203.272</td></tr><tr><td>10s</td><td>80</td><td>0.474147</td><td>0</td><td>334.949</td><td>14.3207</td><td>461.769</td></tr></tbody></table>

<table><caption>**Table 7.** DPP₂ (PWR 3086 MW<sub>th</sub> → 1000 MW<sub>e</sub> + 1000 MW<sub>th</sub> at 80°C): mechanical and thermal powers.</caption><thead><tr><th>Component</th><th>Power (MW)</th></tr></thead><tbody><tr><td>Steam generators</td><td>3086.14</td></tr><tr><td>High pressure turbine 1</td><td>-169.159</td></tr><tr><td>High pressure turbine 2</td><td>-237.905</td></tr><tr><td>Low pressure turbine 1</td><td>-269.206</td></tr><tr><td>Low pressure turbine 2</td><td>-226.627</td></tr><tr><td>Low pressure turbine 3</td><td>-115.962</td></tr><tr><td>Condenser<sub>1</sub> (Process)</td><td>-1000</td></tr><tr><td>Condenser<sub>2</sub> (Tertiary circuit)</td><td>-1086.14</td></tr><tr><td>Low pressure pump</td><td>1.90001</td></tr><tr><td>High pressure pump</td><td>16.9604</td></tr><tr><td>Sum</td><td>1.31 × 10<sup>-12</sup></td></tr><tr><td>Net power output</td><td>-1000</td></tr><tr><td>Power conversion efficiency (%)</td><td>32.4029</td></tr></tbody></table>

The minimal annual cash in (ci) is related to cash outflows by equation (8):

$$ \begin{aligned} & - co_{\text{cst}} \times \text{npv}(1\$, \text{cst}) + (ci - co_{\text{opr}}) \times \text{npv}(1\$, \text{opr}) \\ & - co_{\text{dcm}} \times \text{npv}(1\$, \text{dcm}) = 0, \end{aligned} \quad (8) $$

co<sub>cst</sub>, annual cash out, construction period, ($); npv (1$, cst), NPV of a fixed expense of 1$ per year, spent during the construction period, ($); ci, annual revenue generated from the sale of electricity, ($); co<sub>opr</sub>, annual expenses related to fuel and O&M, economic lifetime of the plant, ($); npv (1$, opr), NPV of a fixed expense of 1$ per year, spent over the economic lifetime of the plant; co<sub>dcm</sub>, annual cash out, decommissioning period, ($); npv (1$, dcm), NPV of a fixed expense of 1$ per year, spent during the decommissioning period.

NPV terms of equation (8) are estimated based on a fixed discount rate (r) applicable for the three periods<fnref n="9" />:

$$ \text{npv}(1\$, \text{period}) = \sum_{Y=Y_{\text{beginning,period}}}^{Y_{\text{end,period}}} (1+r)^{-Y}. \quad (9) $$

Equation (8) assumes fixed values of future inflows and outflows over the three key phases of the lifetime of the plant: construction (cst), operation (opr) and decommissioning (dcm).

<footnote n="9">⁹ Traditionally, the rate used in discounted cash flow analysis is adjusted for risk, period by period. This is not the case for this exercise.</footnote>
<figure><img src="image_6.png" /><figcaption>Fig. 5. WCH and QCH vs. temperature at the steam extraction point.</figcaption></figure>

Annual expenses<fnref n="10" /> (construction, fuel, O&M and decommissioning) are evaluated on the basis of specific costs:

- The specific cost of construction<fnref n="11" />: in $ per (installed) $kW_e$.

- The specific cost of fuel: in $ per (produced) $MW_{eh}$.

- The specific cost of O&M: in $ per (produced) $MW_{ch}$.

- The specific cost of decommissioning: in $ per (installed) $kW_e$.

Once minimal annual cash in (ci) is evaluated, the cost of electricity is deduced by dividing ci by the annual electricity production volume<fnref n="12" /> ($P_{Elec,1Y}$):

$$c_{kW_e h} = \frac{ci}{P_{Elec,1Y}}, \qquad (10)$$

$c_{kW_e h}$, cost of electricity; $P_{Elec,1Y}$, annual electricity production volume ($kW_e h$). ($$ per $kW_e h$$).

A numerical example of electricity cost calculation for a 1000 MW_e PWR is provided in Table 8. The results show good agreement with the evaluation reported in OECD¹ 2010 Projected Costs of Generating Electricity [9].

## 3.2 Dual-purpose plant

The traditional method (Method 1) for evaluating the cost of process heat consists in multiplying the cost of electricity, as calculated for SPP, by the expected decrease in electricity production.

Consider the 1000 MW_e PWR example of Table 8. According to the thermodynamic model described in the previous section, the reactor can produce up to 1730 MW_th of process heat at 80°C. Each MW_th supplied to the

<table><caption>Table 8. SPP (PWR 2748 MW_th → 1000 MW_e): electricity cost.</caption><tr><td>Reference core thermal power (MW<sub>th</sub>)</td><td>2748</td></tr><tr><td>Reference electric power generation capacity (MW<sub>e</sub>)</td><td>1000</td></tr><tr><td>Specific construction cost ($ per installed kW<sub>e</sub> (electric power))</td><td>4101.51<sup>a</sup></td></tr><tr><td>Specific fuel cost ($ per produced MW<sub>eh</sub> (electric power))</td><td>9.33<sup>a</sup></td></tr><tr><td>Specific O&M cost ($ per produced MW<sub>eh</sub> (electric power))</td><td>14.74<sup>a</sup></td></tr><tr><td>Specific decommissioning cost ($ per installed kW<sub>e</sub> (electric power))</td><td>820.30<sup>b</sup></td></tr><tr><td>Length of the construction period (years)</td><td>7<sup>a</sup></td></tr><tr><td>Economic lifetime of the plant (years)</td><td>60<sup>a</sup></td></tr><tr><td>Average availability of the plant (%)</td><td>85<sup>a</sup></td></tr><tr><td>Length of the decommissioning period (years)</td><td>5</td></tr><tr><td>Discount rate (%)</td><td>5</td></tr><tr><td>Cost of electricity (10<sup>-2</sup> $ per kW<sub>eh</sub>)</td><td>5.816</td></tr><tr><td>Percentage allocated to construction (%)</td><td>58.15%</td></tr><tr><td>Percentage allocated to fuel (%)</td><td>16.04%</td></tr><tr><td>Percentage allocated to O&M (%)</td><td>25.34%</td></tr><tr><td>Percentage allocated to decommissioning (%)</td><td>0.46%</td></tr></table>

a Values suggested in the OECD¹ 2010 Projected Costs of Generating Electricity [9, p. 103].

b 20% of the specific construction cost.

external process at this temperature will cause the reactor's net power output to decrease by 123 kW_e (W-cost of heat). With a cost of electricity of 5.82 cents per kW_e, the cost of heat would be equal to 7.15 $ per MW_th (0.715 cents per kW_th).

An alternative method of evaluating the cost of heat (Method 2) consists of considering a modified reactor design (DPP₂, cf. Tabs. 6 and 7) offering higher core power output compared to SPP. Such plant would generate the

<footnote n="10">¹⁰ All expenses are considered “overnight”, i.e. interest free. Inflation (fuel cost escalation in particular) is not taken into account.</footnote>

<footnote n="11">¹¹ Owner's, construction and contingency costs.</footnote>

<footnote n="12">¹² The annual electricity production volume is evaluated from the reference electric power generation capacity assuming a constant average availability of the plant.</footnote>
<figure><img src="image_7.png" /><figcaption>Fig. 6. Number of entries (vertical axis) for which i equals a certain value (horizontal axis).</figcaption></figure>

same amount of electricity as SPP (1000 MW<sub>e</sub>) while meeting the demand of the external process in terms of thermal power (1000 MW<sub>th</sub> at 80 °C).

At 80 °C, the Q-cost of heat is equal to 338 kWh<sub>th</sub> per MW<sub>th</sub>. This means that, in order to produce 1000 MW<sub>th</sub> of process heat at 80 °C, without affecting the electric power generation capacity, the core power has to be raised from 2748 to 3086 MW<sub>th</sub> (+12.3%).

The effect of increasing core power on construction costs can be estimated based on the formula:

$$ \frac{\text{cost}_{\text{DPP}}}{\text{cost}_{\text{SPP}}} = 0.75 + 0.25 \times \left( \frac{\dot{Q}_{\text{Core,DPP}}}{\dot{Q}_{\text{Core,SPP}}} \right)^{0.6} . \quad (11) $$

Equation (11) assumes that:

- Nuclear Island represents roughly x=25% of the costs.

- The cost relative to Nuclear Island:

• Depends on core power exclusively.

• Can be scaled-up applying a capital scaling function<fnref n="13" /> with a scaling exponent equal to n=0.6.<fnref n="14" />

- The remaining 75% of the costs depend solely on the plant power generation capacity (which is the same for both SPP and DPP).

The Single-Purpose 1000 MW<sub>e</sub> PWR example of Table 8 costs 4.102 billion $ to construct. Adding 338 MW<sub>th</sub> to core power would increase this cost by i=1.8%. If x and n-which are rather uncertain-are uniformly distributed, in [15-35] (%) for x, and in [0.4-0.8] for n, i would have the distribution<fnref n="15" /> shown in Figure 6 (mean value for cost increase: 1.8%, standard deviation: 0.56%). A cost increase of 3.5% appears to be an upper limit.

Increasing core power has also an impact on fuel costs. A simple way to take it into account is to apply a correction factor (f) to SPP's specific fuel cost (Eq. (12)). Although SPP and DPP₂ have the same power generation capacity, the annual electricity production volume can differ between the two plants depending on the availability of DPP₂ vs. SPP. If we assume a 1% decrease in availability for DPP₂ compared to SPP (84% for DPP₂ vs. 85% for SPP), the increase in fuel costs would be equal to 12.31%.

$$ f = \frac{P_{\text{Elec,1Y,SPP}}}{P_{\text{Elec,1Y,DPP}}} \times \frac{P_{\text{Core,1Y,DPP}}}{P_{\text{Core,1Y,SPP}}}, \quad (12) $$

$P_{\text{Elec,1Y,SPP}}$, annual electricity production volume, SPP (kW<sub>e</sub>h); $P_{\text{Elec,1Y,DPP}}$, annual electricity production volume, DPP (kW<sub>e</sub>h); $P_{\text{Core,1Y,SPP}}$, annual production volume, thermal power, SG, SPP (kW<sub>th</sub>h); $P_{\text{Core,1Y,DPP}}$, annual production volume, thermal power, SG, DPP (kW<sub>th</sub>h).

The rise in O&M expenses is expected to be less sensitive to the increase in core power compared to fuel costs. The correction factor (f'), applicable to SPP's specific O&M cost, is assumed to be the following:

$$ f' = \frac{1 + f}{2}. \quad (13) $$

Table 9 provides a preliminary economic evaluation of DPP₂ vs. SPP. The cost of heat reported in this table is calculated following the steps listed below:

- The minimal annual cash in required to have a positive NPV-(ci<sub>DPP</sub>) is calculated for DPP₂.

- We assume that all electricity generated by DPP₂ is sold at 5.82 cents per kW<sub>e</sub>h-i.e. the cost of electricity as produced by SPP (c<sub>kWeh,SPP</sub>).

- We use the difference between, the minimal annual cash in required to have a positive NPV-and, the annual revenue generated from the sale of electricity-as a basis for evaluating the cost of heat (Eq. (14)).

$$ c_{\text{kW}_{\text{th,h,DPP}}} = \frac{\text{ci}_{\text{DPP}} - c_{\text{kWeh,SPP}} \times P_{\text{Elec,1Y,DPP}}}{P_{\text{Heat,1Y,DPP}}}. \quad (14) $$

<footnote n="13">¹³ Capital cost scaling functions are often used to account for economies of scale (as the nuclear island gets larger in size, it gets progressively cheaper to add additional capacity). Examples from the power generation industry are provided in [14].</footnote>

<footnote n="14">¹⁴ When n is unknown, a value of 0.6 is generally assumed (rule of six-tenths).</footnote>

<footnote n="15">¹⁵ Figure 6 was obtained after (Latin Hypercube) sampling of two inputs, carried out using CEA's open source software URANIE [15].</footnote>
<table><caption>**Table 9.** DPP₂ (PWR 3086 MWₜh → 1000 MWₑ + 1000 MWₜh at 80°C): electricity and heat costs.</caption><thead><tr><th></th><th>SPP</th><th>DPP<sub>2</sub></th></tr></thead><tbody><tr><td>Reference core thermal power (MW<sub>th</sub>)</td><td>2748</td><td>3086 (+12.3%)</td></tr><tr><td>Reference electric power generation capacity (MW<sub>e</sub>)</td><td>1000</td><td>1000</td></tr><tr><td>Reference process heat generation capacity (MW<sub>th</sub> at 80°C)</td><td>—</td><td>1000</td></tr><tr><td>Specific construction cost ($ per installed kW<sub>e</sub> (electric power))</td><td>4101.51</td><td>4175.451 (+1.8%)</td></tr><tr><td>Specific fuel cost ($ per produced MW<sub>e</sub>h (electric power))</td><td>9.33</td><td>10.478 (+12.31%)</td></tr><tr><td>Specific O&amp;M cost ($ per produced MW<sub>e</sub>h (electric power))</td><td>14.74</td><td>15.647 (+6.15%)</td></tr><tr><td>Specific decommissioning cost ($ per installed kW<sub>e</sub> (electric power))</td><td>820.30</td><td>835.090 (+1.8%)</td></tr><tr><td>Length of the construction period (years)</td><td>7</td><td>7</td></tr><tr><td>Economic lifetime of the plant (years)</td><td>60</td><td>60</td></tr><tr><td>Average availability of the plant (%)</td><td>85</td><td>84 (–1 point)</td></tr><tr><td>Length of the decommissioning period (years)</td><td>5</td><td>5</td></tr><tr><td>Discount rate (%)</td><td>5</td><td>5</td></tr><tr><td>Minimal annual cash in required to have a positive NPV (million $)</td><td>433.378</td><td>450.979 (+17.601)</td></tr><tr><td>Cost of electricity (10<sup>-2</sup> $ per kW<sub>e</sub>h)</td><td>5.816</td><td></td></tr><tr><td>Cost of heat (10<sup>-2</sup> $ per kW<sub>th</sub>h at 80°C)</td><td>—</td><td>0.308</td></tr><tr><td>Cost of heat (DPP) to cost of electricity (SPP)</td><td>—</td><td>5.30%</td></tr></tbody></table>

<figure><img src="image_8.png" /><figcaption>**Fig. 7.** Cost of heat vs. temperature at the steam extraction point.</figcaption></figure>

The cost of heat, as calculated by this method (Method 2), is equal to 0.308 cent per kWₜh (80°C), which represents 5.30% of the cost of electricity produced by SPP. This cost is 57% percent lower than the cost calculated by Method 1. Figure 7 shows how the cost varies with the level of temperature required by the external process.

At 75°C, each kWₜh of thermal power supplied to the process costs 0.282 c$. At 100°C, the cost rises to 0.408 c$ kWₜh⁻¹ (×1.45), and at 125°C it reaches 0.525 c$ kWₜh⁻¹ (×1.86). These costs, estimated based on Method 2, represent 4.9–9.0% of the cost of electricity, depending on the steam extraction temperature (Fig. 8).

The ratio–cost of heat to cost of electricity–will be referred to as the E-cost of heat (ECH). ECH is subject to the size effect (Fig. 9). It is also sensitive to availability of the cogeneration plant, as shown in Figure 10.

Method 2 provides an alternative approach to converting MWₜh to MWₑ, considering the benefits of cogeneration –it allocates CAPEX and OPEX to the two byproducts –but also, the constraints introduced by the integrated system–higher expenses, extended construction period, lower availability, etc.

In the next section, we will use this method to compare two nuclear-powered integrated water and power plants, based on either, multi-effect distillation, or, seawater RO.
<figure><img src="image_9.png" /><figcaption>Fig. 8. Cost of heat to cost of electricity vs. temperature at the steam extraction point.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 9. Cost of heat to cost of electricity vs. temperature at the steam extraction point (Method 2) for different values of process thermal power (MW<sub>th</sub>).</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 10. Cost of heat to cost of electricity vs. temperature at the steam extraction point (Method 2) for different values of (DPP<sub>2</sub>) availability.</figcaption></figure>
# 4 Impact on the cost of desalination

## 4.1 MED process performance model

The MED process performance model aims at evaluating its specific thermal power consumption, in $kW_{th} \cdot h \cdot m^{-3}$ for fresh water produced by the plant. Based on the simplified approach already implemented in the DEEP Code [16], the model follows the three steps described below:

- First, the number of MED stages is determined (Eq. (15)) based on:

* The temperatures at the first stage (top brine temperature) and the final condenser.

* The average temperature drop between stages.

$$N_{\text{Stages}} = \text{int}\left(\frac{T_{\text{max}} - T_{\text{min}}}{\Delta T_{\text{Stages}}}\right), \quad (15)$$

$N_{\text{Stages}}$, number of stages; int (function), round down real numbers to the nearest integer; $T_{\text{max}}$, top brine temperature, (°C); $T_{\text{min}}$, temperature at the final condenser, (°C); $\Delta T_{\text{Stages}}$, average temperature drop between stages, (°C).

<table><caption>**Table 10.** Example of MED process performance calculation (1).</caption><tr><td>Top brine temperature (°C)</td><td>70</td></tr><tr><td>Temperature at the final condenser (°C)</td><td>33</td></tr><tr><td>Average temperature drop between stages (°C)</td><td>2</td></tr><tr><td>Number of stages (–)</td><td>18</td></tr><tr><td>GOR to number of stages</td><td>0.8</td></tr><tr><td>GOR (–)</td><td>14.4</td></tr><tr><td>Pinch point temperature difference, first effect (°C)</td><td>5</td></tr><tr><td>Steam supply temperature (°C)</td><td>75</td></tr><tr><td>Specific heat consumption ($kWh_{th} \cdot m^{-3}$)</td><td>44.76</td></tr></table>

- The gain output ratio (GOR) (kilograms of fresh water produced per kilogram of steam supplied to the process) is then estimated based on an average effect efficiency of 0.8:

$$\text{GOR} = 0.8 \times \text{NS.} \quad (16)$$

- The specific power consumption ($kW_{th} \cdot h \cdot m^{-3}$) is finally deduced:

$$\text{SHC} = \frac{L}{3.6 \times \text{GOR}}, \quad (17)$$

$L$, latent heat at steam supply temperature, (kJ kg$^{-1}$).

A numerical example of MED process performance calculation is provided in Table 10.

The specific thermal power consumption evaluated by this model is sensitive to both, the temperature difference between MED effects, and, the stage average efficiency, as illustrated by Figures 11 and 12.

## 4.2 MED equivalent specific electric power consumption

The calculations, reported in this paragraph, are based on the following assumptions:

- MED model inputs are basically those listed in Table 10. Only the top brine- and steam supply-temperatures vary.

- A (pinch point temperature) difference of 5°C between MED's steam supply temperature and the temperature at the steam extraction point ($T_{\text{SteamEx}}$, power conversion system).

- Conversion of MED specific power thermal consumption to an electric equivalent is performed based on either:

* the W-cost of heat (cf. Sect. 2.3) (Method 1), or,

* the cost of heat to cost of electricity-ratio (ECH) as calculated by Method 2 (cf. Sect. 3.2).

<figure><img src="image_12.png" /><figcaption>**Fig. 11.** MED specific thermal power consumption vs. top brine temperature. 1.5, 1.75, 2.0, 2.25, 2.5: average temperature drop between stages (°C).</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>Fig. 12. MED specific thermal power consumption vs. top brine temperature. 0.7, 0.75, 0.8, 0.85, 0.9: GOR to number of stages.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 13. MED energy cost vs. top brine temperature. Based on Method 1. 1, 2, 3, 4: MED specific electric consumption equal to 1, 2, 3 and $4\,\text{kW}_e\,\text{m}^{-3}$ respectively. min, mid, and max: minimal, medium and maximal specific electric consumption of the SWRO process as reported in literature.</figcaption></figure>

Figure 13 shows how MED's equivalent electric power consumption<fnref n="16" /> varies with the top brine temperature (TBT). Conversion from $W_{th}$ to $W_e$ is based, in this case, on the W-cost of heat (WCH).

The power required to produce a cubic meter of fresh water, as calculated by Method 1, is higher for MED than for SWRO, except for processes operating at a TBT higher than 60, with a specific electric consumption lower than $1\,\text{kW}_e\,\text{m}^{-3}$, for which the equivalent electric power consumption is in the range $6–7\,\text{kW}_e\,\text{m}^{-3}$.

If the cost of heat to cost of electricity–ratio (ECH), as calculated by Method 2, is used as a basis for converting $W_{th}$ to $W_e$, MED's efficiency, in terms of energy utilization, is globally improved, as illustrated by Figure 14.

Figure 14 shows that, for specific consumptions in the range [1–4] $\text{kW}_e\,\text{m}^{-3}$, MED's equivalent electric power consumption varies between 3 and $6\,\text{kW}_e\,\text{m}^{-3}$, matching the range of the RO specific electric consumption as reported in literature.

MED's equivalent electric power consumption can be further reduced by, raising the TBT,<fnref n="17" /> decreasing the average temperature drop between MED stages,<fnref n="18" /> or, increasing MED effects' efficiency<fnref n="19" /> (GOR to number of stages), as illustrated by the example provided in Table 11.

<footnote n="16">¹⁶ Electric equivalent of the thermal power supplied by the nuclear reactor, plus, electric power consumption internal to the process.</footnote>

<footnote n="17">¹⁷ Raising the TBT exposes the plant to severe corrosion and scaling problems. In recent years, many of these problems have been solved thanks to improvements in materials and anti-scalants.</footnote>

<footnote n="18">¹⁸ Reducing the temperature difference requires larger heat transfer surfaces.</footnote>

<footnote n="19">¹⁹ Effect efficiency can be improved by reducing thermal losses.</footnote>
<figure><img src="image_15.png" /><figcaption>**Fig. 14.** MED energy cost vs. top brine temperature. Based on Method 2. 1, 2, 3, 4: MED specific electric consumption equal to 1, 2, 3 and 4 $kW_e m^{-3}$ respectively. min, mid, and max: minimal, medium and maximal specific electric consumption of the SWRO process as reported in literature.</figcaption></figure>

<table><caption>**Table 11.** Example of MED process performance calculation (2).</caption><tr><td>Top brine temperature (°C)</td><td>75</td></tr><tr><td>Temperature at the final condenser (°C)</td><td>33</td></tr><tr><td>Average temperature drop between stages (°C)</td><td>1.85</td></tr><tr><td>Number of stages (-)</td><td>22</td></tr><tr><td>GOR to number of stages</td><td>0.85</td></tr><tr><td>GOR (-)</td><td>18.7<sup>a</sup></td></tr><tr><td>Pinch point temperature difference, first effect (°C)</td><td>5</td></tr><tr><td>Steam supply temperature (°C)</td><td>80</td></tr><tr><td>Specific heat consumption ($kWh_{th} m^{-3}$)</td><td>34.285</td></tr><tr><td>Equivalent electric power consumption ($kW_e m^{-3}$) - Basis: 1 $kW_e m^{-3}$</td><td>2.97</td></tr><tr><td>Equivalent electric power consumption ($kW_e m^{-3}$) - Basis: 2 $kW_e m^{-3}$</td><td>3.97</td></tr><tr><td>Equivalent electric power consumption ($kW_e m^{-3}$) - Basis: 3 $kW_e m^{-3}$</td><td>4.97</td></tr><tr><td>Equivalent electric power consumption ($kW_e m^{-3}$) - Basis: 4 $kW_e m^{-3}$</td><td>5.97</td></tr></table>

## 5 Conclusion

Process heat has an energetic and an economic cost that affects the cost of desalination. The exploratory study, described in this paper, attempted to evaluate these costs based on simplified models.

The power conversion system model provided a basis for assessing the “W-cost of heat” (WCH) – number of $kW_e$ of “missed electricity production” per $MW_{th}$ of process power – and the “Q-cost of heat” (QCH) – number of $kW_{th}$ of additional core power (required to keep a constant level of electricity production) per $MW_{th}$.

The economic model helped evaluate the “E-cost of heat” (ECH), defined as the ratio-cost of heat to cost of electricity-taking into account cogeneration's benefits and constraints.

The three costs-WCH, QCH, and ECH-depend primarily on the level of temperature required by the process. ECH also depends on the economic model's inputs.

This work confirms two conclusions from an earlier study by Rognoni et al. [8]:

- Evaluating the heat cost on the basis of WCH (and the cost of electricity generated by a single-purpose power plant) leads to higher energy costs for MED compared to SWRO.

<footnote>a MED manufacturers claim a GOR of 10–16 in working units and up to 30 in designed prototypes [2].</footnote>
- A rigorous techno-economic approach, duly considering the benefits of cogeneration, results in lower heat costs, and comparable equivalent electric power consumptions between MED and SWRO.

Energy is an important contributor to the cost of desalted water - a contributor among many others: construction, O&M, chemicals, insurance, labor . . . -. Evaluating the cost of desalted water should take into account all the expenses related to the project, including the investments needed to construct (or extend) water transfer and supply networks (IWPPs are generally located far from urban and industrial areas).

Water desalination plants produce huge amounts of reject brine. This brine can be turned into salt [17] or used to convert CO₂ into useful and reusable products such as sodium bicarbonate [18]. These processes - still under development - can potentially improve the economics of seawater desalination while minimizing the impact of brine discharge on the environment.

To identify the most appropriate reactor-process combination for a given site, case-specific evaluations have to be performed, considering the precise characteristics of the power generation system, the reactor to process heat transfer loop, the seawater desalination unit, and the water transport system. Other important factors have also to be considered such as the final use of the product, the quality of the feed, the intake, pretreatment, post-treatment and brine reject-structures, and the variability of the demand for power and water.

## Abbreviations

<table><tr><td>CAPEX</td><td>capital expenditures</td></tr><tr><td>DEEP</td><td>desalination economic evaluation program (Software)</td></tr><tr><td>DPP</td><td>dual-purpose plant</td></tr><tr><td>ECH</td><td>E-cost of heat</td></tr><tr><td>ED</td><td>electrodialysis</td></tr><tr><td>GOR</td><td>gain output ratio</td></tr><tr><td>HPT</td><td>high pressure turbine</td></tr><tr><td>IAEA</td><td>international atomic energy agency</td></tr><tr><td>ICV</td><td>interconnected control volumes (software)</td></tr><tr><td>IWPP</td><td>integrated water and power plant</td></tr><tr><td>LPT</td><td>low pressure turbine</td></tr><tr><td>MED</td><td>multi-effect evaporation</td></tr><tr><td>MSF</td><td>multi-stage flash</td></tr><tr><td>NPV</td><td>net present value</td></tr><tr><td>O&M</td><td>operation and maintenance</td></tr><tr><td>OPEX</td><td>operating expenditures</td></tr><tr><td>PCS</td><td>power conversion system</td></tr><tr><td>PWR</td><td>pressurized water reactor</td></tr><tr><td>QCH</td><td>Q-cost of heat</td></tr><tr><td>RO</td><td>reverse osmosis</td></tr><tr><td>SG</td><td>steam generator</td></tr><tr><td>SPP</td><td>single-purpose plant</td></tr><tr><td>SteamEx</td><td>steam extraction point</td></tr><tr><td>SWRO</td><td>seawater reverse osmosis</td></tr><tr><td>TBT</td><td>top brine temperature</td></tr><tr><td>WCH</td><td>W-cost of heat</td></tr></table>

## Appendix

<table><caption>Cost of heat vs. temperature at the extraction point.</caption><thead><tr><th>T</th><th>WCH</th><th>QCH</th><th>ECH</th><th>T</th><th>WCH</th><th>QCH</th><th>ECH</th></tr></thead><tbody><tr><td>33</td><td>0</td><td>1</td><td>7</td><td>81</td><td>125</td><td>345</td><td>54</td></tr><tr><td>34</td><td>3</td><td>9</td><td>8</td><td>82</td><td>128</td><td>351</td><td>55</td></tr><tr><td>35</td><td>6</td><td>17</td><td>9</td><td>83</td><td>130</td><td>358</td><td>56</td></tr><tr><td>36</td><td>9</td><td>25</td><td>10</td><td>84</td><td>133</td><td>364</td><td>57</td></tr><tr><td>37</td><td>12</td><td>32</td><td>11</td><td>85</td><td>135</td><td>371</td><td>57</td></tr><tr><td>38</td><td>15</td><td>40</td><td>12</td><td>86</td><td>137</td><td>377</td><td>58</td></tr><tr><td>39</td><td>17</td><td>48</td><td>14</td><td>87</td><td>140</td><td>384</td><td>59</td></tr><tr><td>40</td><td>20</td><td>56</td><td>15</td><td>88</td><td>142</td><td>390</td><td>60</td></tr><tr><td>41</td><td>23</td><td>63</td><td>16</td><td>89</td><td>144</td><td>396</td><td>61</td></tr><tr><td>42</td><td>26</td><td>71</td><td>17</td><td>90</td><td>147</td><td>403</td><td>62</td></tr><tr><td>43</td><td>28</td><td>78</td><td>18</td><td>91</td><td>149</td><td>409</td><td>63</td></tr><tr><td>44</td><td>31</td><td>86</td><td>19</td><td>92</td><td>151</td><td>415</td><td>63</td></tr><tr><td>45</td><td>34</td><td>93</td><td>20</td><td>93</td><td>153</td><td>422</td><td>64</td></tr><tr><td>46</td><td>37</td><td>101</td><td>21</td><td>94</td><td>156</td><td>428</td><td>65</td></tr><tr><td>47</td><td>39</td><td>108</td><td>22</td><td>95</td><td>158</td><td>434</td><td>66</td></tr><tr><td>48</td><td>42</td><td>116</td><td>23</td><td>96</td><td>160</td><td>440</td><td>67</td></tr><tr><td>49</td><td>45</td><td>123</td><td>24</td><td>97</td><td>162</td><td>447</td><td>68</td></tr><tr><td>50</td><td>47</td><td>131</td><td>25</td><td>98</td><td>165</td><td>453</td><td>68</td></tr><tr><td>51</td><td>50</td><td>138</td><td>26</td><td>99</td><td>167</td><td>459</td><td>69</td></tr><tr><td>52</td><td>53</td><td>145</td><td>27</td><td>100</td><td>169</td><td>465</td><td>70</td></tr><tr><td>53</td><td>55</td><td>152</td><td>28</td><td>101</td><td>171</td><td>471</td><td>71</td></tr><tr><td>54</td><td>58</td><td>160</td><td>29</td><td>102</td><td>174</td><td>478</td><td>72</td></tr><tr><td>55</td><td>61</td><td>167</td><td>30</td><td>103</td><td>176</td><td>484</td><td>73</td></tr><tr><td>56</td><td>63</td><td>174</td><td>31</td><td>104</td><td>178</td><td>490</td><td>73</td></tr><tr><td>57</td><td>66</td><td>181</td><td>32</td><td>105</td><td>180</td><td>496</td><td>74</td></tr><tr><td>58</td><td>68</td><td>188</td><td>33</td><td>106</td><td>183</td><td>502</td><td>75</td></tr><tr><td>59</td><td>71</td><td>195</td><td>34</td><td>107</td><td>185</td><td>508</td><td>76</td></tr><tr><td>60</td><td>74</td><td>202</td><td>35</td><td>108</td><td>187</td><td>514</td><td>77</td></tr><tr><td>61</td><td>76</td><td>209</td><td>36</td><td>109</td><td>189</td><td>520</td><td>78</td></tr><tr><td>62</td><td>79</td><td>216</td><td>36</td><td>110</td><td>191</td><td>526</td><td>78</td></tr><tr><td>63</td><td>81</td><td>223</td><td>37</td><td>111</td><td>193</td><td>532</td><td>79</td></tr><tr><td>64</td><td>84</td><td>230</td><td>38</td><td>112</td><td>196</td><td>538</td><td>80</td></tr><tr><td>65</td><td>86</td><td>237</td><td>39</td><td>113</td><td>198</td><td>544</td><td>81</td></tr><tr><td>66</td><td>89</td><td>244</td><td>40</td><td>114</td><td>200</td><td>550</td><td>82</td></tr><tr><td>67</td><td>91</td><td>251</td><td>41</td><td>115</td><td>202</td><td>556</td><td>82</td></tr><tr><td>68</td><td>94</td><td>258</td><td>42</td><td>116</td><td>204</td><td>562</td><td>83</td></tr><tr><td>69</td><td>96</td><td>265</td><td>43</td><td>117</td><td>206</td><td>568</td><td>84</td></tr><tr><td>70</td><td>99</td><td>271</td><td>44</td><td>118</td><td>208</td><td>574</td><td>85</td></tr><tr><td>71</td><td>101</td><td>278</td><td>45</td><td>119</td><td>211</td><td>580</td><td>85</td></tr><tr><td>72</td><td>104</td><td>285</td><td>46</td><td>120</td><td>213</td><td>586</td><td>86</td></tr><tr><td>73</td><td>106</td><td>292</td><td>47</td><td>121</td><td>215</td><td>592</td><td>87</td></tr><tr><td>74</td><td>109</td><td>298</td><td>48</td><td>122</td><td>217</td><td>598</td><td>88</td></tr><tr><td>75</td><td>111</td><td>305</td><td>49</td><td>123</td><td>219</td><td>603</td><td>89</td></tr><tr><td>76</td><td>113</td><td>312</td><td>49</td><td>124</td><td>221</td><td>609</td><td>89</td></tr><tr><td>77</td><td>116</td><td>318</td><td>50</td><td>125</td><td>223</td><td>615</td><td>90</td></tr><tr><td>78</td><td>118</td><td>325</td><td>51</td><td>126</td><td>225</td><td>621</td><td>91</td></tr><tr><td>79</td><td>121</td><td>332</td><td>52</td><td>127</td><td>227</td><td>627</td><td>92</td></tr><tr><td>80</td><td>123</td><td>338</td><td>53</td><td>128</td><td>230</td><td>633</td><td>92</td></tr></tbody></table>

T, temperature at the extraction point (°C); WCH, W-cost of heat (kW_e of "missed electricity production" per MW_th supplied to the process); QCH, Q-cost of heat (kW_th of additional core thermal power per MW_th supplied to the process); ECH, E-cost of heat (kW_e of electricity per MW_th supplied to the process).
References

1. GWI, *IDA Desalination Yearbook 2012–2013* (Global Water Intelligence, 2012)

2. R. Semiat, Energy issues in desalination processes, Environ. Sci. Technol. **42**, 8193 (2008)

3. M. Elimelech, Seawater desalination, in *NWRI Clarke Prize Conference, Newport Beach, California*, 2012, available online at: http://www.nwri-usa.org/documents/Elimelech_000.pdf

4. S. Miller, H. Shemer, R. Semiat, Energy and environmental issues in desalination, Desalination **366**, 2 (2014)

5. S. Burn, M. Hoang, D. Zarzo, F. Olewniak, E. Campos, B. Bolto, O. Barron, Desalination techniques – a review of the opportunities for desalination in agriculture, Desalination **364**, 2 (2015)

6. G.M. Zak, Thermal desalination: structural optimization and integration in clean power and water, Doctoral dissertation, Massachusetts Institute of Technology, 2012

7. A. Al-Karaghouli, L.L. Kazmerski, Energy consumption and water production cost of conventional and renewable-energy-powered desalination processes, Renew. Sustain. Energy Rev. **24**, 343 (2013)

8. M. Rognoni, M.P. Ramaswamy, J.R. Paden, Energy cost for desalination evaporation versus reverse osmosis, Int. J. Nucl. Desalin. **4**, 277 (2011)

9. IEA, OECD, NEA, *Projected Costs of Generating Electricity – 2010 Edition* (International Energy Agency and Nuclear Energy Agency, France, 2010)

10. H. Safa, Economics of district heating using light water reactors, in *NEA/IAEA Workshop on Nuclear Cogeneration, Paris, 4–5 April 2013*, 2013, available online at: http://www.oecd-nea.org/ndd/workshops/nucogen/presentations/4_Safa_Economics-District-Heating-Light.pdf

11. S. Nisan, S. Dardour, Economic evaluation of nuclear desalination systems, Desalination **205**, 231 (2007)

12. W. Wagner, J.R. Cooper, A. Dittmann, et al., The IAPWS industrial formulation 1997 for the thermodynamic properties of water and steam, J. Eng. Gas Turbines Power **122**, 150 (2000)

13. M.H. Sharqawy, J.H. Lienhard, S.M. Zubair, Thermophysical properties of seawater: a review of existing correlations and data, Desalin. Water Treat. **16**, 354 (2010)

14. J. Black, *Capital Cost Scaling Methodology* (NETL, DOE, USA, 2013)

15. F. Gaudier, URANIE: the CEA/DEN uncertainty and sensitivity platform, Proc. Soc. Behav. Sci. **2**, 7660 (2010), URANIE is available online at: http://sourceforge.net/projects/uranie/

16. K.C. Kavvadias, I. Khamis, The IAEA DEEP desalination economic model: a critical review, Desalination **257**, 150 (2010)

17. M. Ahmed, A. Arakel, D. Hoey, M. Thumarukudy, M. Goosen, M. Al-Haddabi, A. Al-Belushi, Feasibility of salt production from inland RO desalination plant reject brine: a case study, Desalination **158**, 109 (2003)

18. A. Dindi, D. Viet Quang, M. Abu-Zahra, Simultaneous carbon dioxide capture and utilization using thermal desalination reject brine, Appl. Energy **154**, 298 (2015)

Cite this article as: Saied Dardour, Henri Safa, Energetic and economic cost of nuclear heat – impact on the cost of desalination, EPJ Nuclear Sci. Technol. **3**, 1 (2017)