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

. 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, simpli ﬁ ed 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.


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 plantsmulti-effect evaporation (MED) and multi-stage flash (MSF), mostly in the Middle East.
Seawater desalination is an energy-intensive process. 1  According to [2], the lowest energy consumptionand the closest to the minimum set by thermodynamics (1.06 kWh m À3 ) [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 À3 [4].Some plants, producing large amount of desalinated water, claim even lower energy consumption; 3.5 kW e h m À3 for Ashkelon, Israel [4]; and 2.7-3.1 kW e h m À3 (depending on temperature and membrane ageing) for Perth, Australia [5].
Thermal desalination processes consume heat, 2 in addition to electricity.Heat consumption varies between 40 and 65 kWh th m À3 for MED, and 55-80 kWh th m À3 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 e m À3 [7].MED manufacturers claim specific electricity consumptions lower than 2.5 kWh e m À3 .

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 productionleading, systematically, to higher energy costs for the thermal desalination processes compared to RO. MED's steam supply costs between 4 and 7 kWh e m À3 of "missed electricity production" according to [2].If we add 1.2-2.5 kWh e m À3 of pumping energy, we end up with an equivalent electric power consumption in the range [5.2-9.5]kWh e m À3 .
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 energetic cost of process heat is equal to the number of MW th 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 processesprecisely MEDcan be potentially more cost-effective than SWRO.The authors provided two-calculation examples -MED processes fueled by coalfired power plants in Indiafor which the cost of desalinated water is 50% lower than SWRO's.

"Nuclear steam" cost
The cost of process heat depends on the contribution, to the total cost of electricity, of fuel-related expensesa contribution widely considered to be lower for nuclearpowered 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 theenergetic and economiccost of process heat, supplied by pressurized water reactor (PWR) to a thermal desalination process.The objective is to provide a basis for comparing thermal (MED3 ) 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.

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

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 1 and HPT 2 ).The fluid is then dried-up and superheated before supplying the low-pressure stages (LPT 1 , LPT 2 and LPT 3 ).Liquid water extracted from the condenser (Condenser 2 ) is finally preheated and readmitted back to SG.
A steam extraction point was positioned between the outlet of LPT 2 and the inlet of LPT 3 .This location allows for a variable quantity (y = 0-100%) of steam (the steam normally flowing through LPT 3 ) to be diverted to an external process.The pressure at the steam extraction point (P SteamEx ) may vary between 0.05 bar (pressure at the condenser) and 2.685 bar (pressure at LPT 2 outlet), and the temperature (T SteamEx ) 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 1 , the latent heat of condensation is transferred to the external process.Condenser 2 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.

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.The model calculates the characteristics of the 23 points of the flowsheettemperature, pressure, steam quality, 5enthalpy, exergy and flowratethe 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) 6 ) to different control volumes.
_ m, mass flowrate (kg/s); _ Q, thermal power (W); _ W , mechanical power (W); h, specific enthalpy (J/kg); v 2 /2, specific kinetic energy (J/kg); g Â z, specific potential energy (J/kg); g Â z, specific potential energy (J/kg).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): e, isentropic efficiency; h in , specific enthalpy at inlet (J/kg); s in , specific entropy at inlet (J/kg/K); h out , specific enthalpy at outlet (J/kg); s out , specific entropy at outlet (J/kg/K); h out ðs out ¼ s in Þ, 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 7 are not taken into account (the calculated "net" power and heat outputs are actually "gross" power and heat outputs).

Energetic performance of the PCS
Tables 2 and 3 show the characteristics of a 2748 MW th singlepurpose plant (SPP) generating 1000 MW e of electricity.
The contribution of steam turbines to SPP's electricity output is shown in Figure 3. LPT 3 delivers 213 MW e of mechanical power, which represents 21% of the total electricity output.If all the steam normally flowing towards this turbine is redirected to the external process (T SteamEx = 80 °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 steamexactly 57.8%is diverted, the plant would produce 877 MW e of electricity and 1000 MW th of heat.The characteristics of configurationwe 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 bypassedthe 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: 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 eand 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 configurationwe 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 steamwe will call it the "Q-cost of heat" or QCH: QCH is simply obtained dividing WCH by SPP's power conversion efficiency.The increase in core power considered in this study is purely conceptual. 8Adopting 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 SteamEx .At 75 °C, each MW th h of thermal power supplied to the process costs 111 kW e h of electricity.At 100 °C, the cost increases to 169 kW e h MW th h À1 (Â1.5), and at 125 °C it reaches 223 kW e h MW th h À1 (Â2).
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)).
3 Economic cost of heat

Single-purpose plant
To evaluate the cost of electricity relative a single-purpose plant, we first calculate the minimal annual cash ingenerated from the sale of electricityrequired 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.The minimal annual cash in (ci) is related to cash outflows by equation ( 8): co cst , 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 opr , 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 dcm , 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 periods9 : 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).Annual expenses10 (construction, fuel, O&M and decommissioning) are evaluated on the basis of specific costs: -The specific cost of construction 11 : in $ per (installed) kW e .
-The specific cost of fuel: in $ per (produced) MW e h.
-The specific cost of O&M: in $ per (produced) MW e h.
-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 12 (P Elec,1Y ): 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 OECD' 2010 Projected Costs of Generating Electricity [9].

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 h supplied to the external process at this temperature will cause the reactor's net power output to decrease by 123 kW e h (W-cost of heat).With a cost of electricity of 5.82 cents per kW e h, the cost of heat would be equal to 7.15 $ per MW th h (0.715 cents per kW th h).An alternative method of evaluating the cost of heat (Method 2) consists of considering a modified reactor design (DPP 2 , cf.Tabs.6 and 7) offering higher core power output compared to SPP.Such plant would generate the  same amount of electricity as SPP (1000 MW e ) while meeting the demand of the external process in terms of thermal power (1000 MW th at 80 °C).
At 80 °C, the Q-cost of heat is equal to 338 kWh th per MW th .This means that, in order to produce 1000 MW th 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 th (+12.3%).
The effect of increasing core power on construction costs can be estimated based on the formula: 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 13with a scaling exponent equal to n = 0.6.14 -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 e PWR example of Table 8 costs 4.102 billion $ to construct.Adding 338 MW th to core power would increase this cost by i = 1.8%.If x and nwhich are rather uncertainare uniformly distributed, in  (%) for x, and in [0.4-0.8] for n, i would have the distribution 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 DPP 2 have the same power generation capacity, the annual electricity production volume can differ between the two plants depending on the availability of DPP 2 vs. SPP.If we assume a 1% decrease in availability for DPP 2 compared to SPP (84% for DPP 2 vs. 85% for SPP), the increase in fuel costs would be equal to 12.31%.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 0 ), applicable to SPP's specific O&M cost, is assumed to be the following: Table 9 provides a preliminary economic evaluation of DPP 2 vs. SPP.The cost of heat reported in this table is calculated following the steps listed below: -Theminimal annual cash in required to have a positive NPV -(ci DPP ) is calculated for DPP 2 .-We assume that all electricity generated by DPP 2 is sold at 5.82 cents per kW e hi.e. the cost of electricity as produced by SPP (c kWeh,SPP ).-We use the difference between, theminimal annual cash in required to have a positive NPVand, theannual revenue generated from the sale of electricityas a basis for evaluating the cost of heat (Eq.( 14)).
c kW th h;DPP ¼ ci DPP À c kW e h;SPP Â P Elec;1Y ;DPP P Heat;1Y ;DPP : ð14Þ The cost of heat, as calculated by this method (Method 2), is equal to 0.308 cent per kW th h (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 kW th h of thermal power supplied to the process costs 0.282 c$.At 100 °C, the cost rises to 0.408 c$ kW th h À1 (Â1.45), and at 125 °C it reaches 0.525 c$ kW th h À1 (Â1.86).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 ratiocost of heat to cost of electricitywill 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 MW th to MW e , considering the benefits of cogeneration it allocates CAPEX and OPEX to the two byproducts but also, the constraints introduced by the integrated systemhigher 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.The MED process performance model aims at evaluating its specific thermal power consumption, in kW th h m À3 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 at the first stage (top brine temperature) and the final condenser.
The average temperature drop between stages.
N Stages , number of stages; int (function), round down real numbers to the nearest integer; T max , top brine temperature, (°C); T min , temperature at the final condenser, (°C); DT Stages , average temperature drop between stages, (°C).
-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: -The specific power consumption (kW th h m À3 ) is finally deduced: L, latent heat at steam supply temperature, (kJ kg À1 ).
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 .

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 brineand steam supplytemperatures vary.-A (pinch point temperature) difference of 5 °C between MED's steam supply temperature and the temperature at the steam extraction point (T 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, thecost of heat to cost of electricityratio (ECH) as calculated by Method 2 (cf.Sect.3.2).
Table 10.Example of MED process performance calculation Figure 13 shows how MED's equivalent electric power consumption 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 with a specific electric consumption lower than 1 kW e m À3 , for which the equivalent electric power consumption is in the range 6-7 kW e m À3 .
If thecost of heat to cost of electricityratio (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][2][3][4] kW e m À3 , MED's equivalent electric power consumption varies between 3 and 6 kW e m À3 , 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, 17 decreasing the average temperature drop between MED stages, 18 or, increasing MED effects' efficiency 19 (GOR to number of stages), as illustrated by the example provided in Table 11.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 kW e m À3 respectively.min, mid, and max: minimal, medium and maximal specific electric consumption of the SWRO process as reported in literature. 16Electric equivalent of the thermal power supplied by the nuclear reactor, plus, electric power consumption internal to the process. 17Raising 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 antiscalants. 18Reducing the temperature difference requires larger heat transfer surfaces. 19Effect efficiency can be improved by reducing thermal losses.

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 powerand 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 ratiocost of heat to cost of electricitytaking into account cogeneration's benefits and constraints.
The three costs -WCH, QCH, and ECHdepend 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.-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 watera 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 CO 2 into useful and reusable products such as sodium bicarbonate [18].These processesstill under developmentcan 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, theintake, pretreatment, post-treatment and brine rejectstructures, and the variability of the demand for power and water.

Fig. 6 .
Fig. 6.Number of entries (vertical axis) for which i equals a certain value (horizontal axis).

Fig. 7 .
Fig. 7. Cost of heat vs. temperature at the steam extraction point.