Pierre Neveu
email: pierre.neveu@univ-perp.fr

Baptiste Rebouillat
email: baptiste.rebouillat@univ-perp.fr

Quentin Falcoz
email: quentin.falcoz@promes.cnrs.fr

Finite Dimension Thermodynamics for optimizing power plants including heat storage device

Keywords: Finite dimension thermodynamics, optimal integration, thermal storage, solar power plant. 1. Linear driving force . 3. Availability of powerful tools for industrial process optimization such as Aspen, Dymola or TRNSYS

▪ This paper deals with the optimal integration of power plants including a storage device such as concentrated solar power plants. For such systems, numerous structures are possible, involving different number of heat exchangers, and for each of them, optimal operating temperatures to be found. Moreover, the heat storage system can be located at different temperature levels offering another degree of freedom when optimizing the whole system. If process simulators are nowadays very powerful tools for optimizing complex processes, they require to propose a primary design before any optimization steps. Finite-Dimension Thermodynamics (FDT) could help engineers to propose this primary design, close to the optimal one. To this aim, FDT method have been generalized for power generation systems including a storage device and any number of heat exchangers. An model of thermal storage system is also proposed which can be included in the FDT modelling. The optimization step consists in maximizing the power generation submitted to the thermodynamics constraints (first and second Laws) related to each heat exchangers, power block and thermal storage system. Remarkable results have been found: i) all the studied structures lead to the Curzon-Ahlborn efficiency when optimized, ii) for the same driving source (same temperature and same power), the output power production varies with N -2 , N being the number of the heat exchangers, iii) Charge and discharged times scenarios have a big impact on the optimal operating temperatures and on the resulting daily energy production.

Introduction

Finite-Time Thermodynamics aims to overpass the assumption of reversible, and consequently infinitely slow, transformations inherent to the Carnot cycle definition [START_REF] Carnot | Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance[END_REF]. Chambadal [START_REF]Les centrales nucléaires[END_REF] and Novikov [START_REF] Novikov | The efficiency of atomic power stations (a review)[END_REF] were the pioneers in 1957, proposing a power plant model that associates a reversible Carnot cycle driven by a heat source at high temperature TH through an irreversible thermal resistance. These works were rediscover and completed by Curzon and Ahlborn [START_REF] Curzon | Efficiency of a Carnot engine at maximum power output[END_REF] in 1975, who added a second thermal resistance coupling the reversible power block with the heat sink at low temperature TL. All these previous studies were seeking the optimal temperatures between which the Carnot cycle must operate to ensure maximum power production. They demonstrated that at this maximum power point (MPP), the efficiency is equal to the so called 'nice radical efficiency' 𝜂 𝑀𝑃𝑃 = 1 -√𝑇 𝐿 /𝑇 𝐻 . De Vos [START_REF] Vos | Reflections on the power delivered by endoreversible engines[END_REF] and Bejan [START_REF] Bejan | Theory of heat transfer-irreversible power plants[END_REF] extended the Finite-Time Thermodynamics (FTT) to the Finite-Size or Finite-Dimension Thermodynamics (FDT) [START_REF] Bejan | Entropy generation minimization: The new thermodynamics of finite-size devices and finitetime processes[END_REF], [START_REF] Feidt | Finite Physical Dimensions Optimal Thermodynamics 2 : Complex Systems[END_REF]. The main difference is that the energy and entropy balances are here applied to the power plant itself, operating in a steady state, and not to the working fluid evolving in time over a cycle as done in FTT studies. Consequently, the method is no longer limited to Carnot engines, but can be applied for all types of engines. This leads to define endoreversible engine [START_REF] Vos | Reflections on the power delivered by endoreversible engines[END_REF] for those internal entropy production is null or negligible. In that case, FTT and FDT lead to similar results, in particular to the nice radical efficiency at the MPP. From these first studies, FTD/FDT methods have been successfully applied to a wide variety of systems such as refrigerators and heat pumps [START_REF] Blanchard | Coefficient of performance for finite speed heat pump[END_REF], distillation systems [START_REF] Schaller | The Influence of Heat Transfer Irreversibilities on the Optimal Performance of Diabatic Distillation Columns[END_REF], chemical reactions [START_REF] Ondrechen | Thermodynamics in finite time: Processes with temperature-dependent chemical reactions[END_REF] , wind power [START_REF] Gordon | Wind energy as a solar-driven heat engine: A thermodynamic approach[END_REF] or solar power [START_REF] Vos | Endoreversible thermodynamics of solar energy conversion[END_REF] However, the reliability of the results is often subjected to much criticism due to the assumptions underlying the FTT/FDT:

We obviously agree with these limitations. However, process simulators/optimisers require a mandatory first step: to propose a guessed design for initiating the optimisation process. This is the main objective of the FTT/FDT methodology: to define an initial design, deduced from thermodynamics, which could be used as a basis for further optimisation. Following this objective, the two first items seems reasonable: linear laws are commonly used in engineering pre-design, and endoreversible cycles are quite close to actual cycles as main irreversibility sources appear in the heat exchangers linking the process with the external heat sources . Moreover, Meunier et al. [START_REF] Meunier | Equivalent Carnot cycles for sorption refrigeration: Cycles de Carnot équivalents pour la production de froid par sorption[END_REF] and Castaing et al. [START_REF] Castaing-Lasvignottes | Equivalent Carnot cycle concept applied to a thermochemical solid/gas resorption system[END_REF] showed that an equivalent endoreversible cycle could be substituted to any sorption refrigeration irreversible cycles. This result also applies to any thermodynamics cycles. Conversely, passing from an endoreversible cycle to an actual cycle is also possible: in Reference [START_REF] Neveu | Optimal integration of rankine cycles in concentrated sol power plant[END_REF], FDT method was used in order to find the optimal operating conditions of a endoreversible cycle, from which the optimal operating conditions of an actual Hirn cycle were deduced (i.e condensing and evaporating pressures and inlet turbine temperature). Nevertheless, FDT models are restricted to basic architectures of power plant: a single power cycle exchanging heat with two reservoirs through two heat exchangers (HX) whereas actual power plants often include more heat exchangers. PWR nuclear power plants use a primary heat transfer fluid (HTF), that induces two heat exchangers in the hot side: the reactor and the steam generator. In contrast, BWR use a single heat exchanger in the hot side: the reactor itself. The cooling loop can also involve one (condenser for direct through cooling) or two HXs (condenser and cooling tower for indirect cooling). A question then rises: does the HX number affect the optimal operating temperatures, the output power, and the efficiency of the cycle at the MPP? Similar issue appears for Concentrated Solar Power (CSP) plants, which can also involve one (Direct Steam Generator, DSG) or two (indirect heating) hot HXs. In addition, most of CSP plants include a thermal energy storage (TES) system, which adds complexity and diversity in possible architecture: the storage system can be direct (same fluid acts as HTF and storage medium) or indirect (two different HTFs flow in the solar loop and in the power block hot loop). Consequently, thermal storage device should be integrated in the FDT analysis. Therefore, the objectives of this work are twofold:

1. Extend the FDT methodology to processes that include a heat storage device. 2. Investigate the impact of TES and number of HX on the power plant optimal temperatures, power output and efficiency at the MMP. In addition to these two original contributions, the optimization problem is solved without any assumption concerning the endoreversibility of the TES and the power block: the second law is treated here as an inequality while endoreversibility is usually assumed in FDT studies.

Problem definition

The FDT formalism requires to substitute a thermodynamic equivalent system to the real components involved in the power plant. The method is detailed in [START_REF] Neveu | Optimal integration of rankine cycles in concentrated sol power plant[END_REF] and is briefly outlined here for heat exchangers and power blocks. An equivalent system is then proposed for TES systems. Finally, the optimisation problem is defined.

Equivalent heat exchanger and power block

Heat exchangers can be modelled by a thermal conductance K, transferring the heat flux 𝑞̇ between two thermostats whose temperatures are equal to the mean entropic temperatures (𝑇 ̃= ∆ℎ ∆𝑠 ⁄ ) of the cold and hot fluids. The heat flux can then be simply expressed by a Newton law:

𝑞̇= K (𝑇 ̃𝐻 -𝑇 ̃𝐿) (1)
For power blocks, the same concept applies: any power cycle can be assessed through an equivalent cycle operating between two reservoirs à 𝑇 ̃𝐻 and 𝑇 ̃𝐿. For Rankine or Hirn cycles, these two temperatures are the entropic mean temperatures related to the working fluid when crossing respectively the steam generator and the condenser. First and Second Laws then write:

𝑞̇𝐻 -𝑞̇𝐿 = 𝑤̇ (2) 𝑞̇𝐻 𝑇 ̃𝐻 - 𝑞̇𝐿 𝑇 ̃𝐿 = -σ̇𝑖 ≤ 0 (3)
with σ̇𝑖 : internal entropy production (W/K) 𝑤̇ : output power (W)

Notice that if σ̇𝑖 = 0, Eqs. ( 2) and (3) define the endoreversible cycle.

Equivalent Thermal Energy Storage system

A two-tank heat storage device is considered (Figure 1a). Upstream, 𝑚̇𝐻 flow enters the high temperature loop at temperature 𝑇 𝑜𝑢𝑡 𝐻 , and returns to the TES system at higher temperature 𝑇 𝑖𝑛 𝐻 . Downstream, 𝑚̇𝑀 flow enters the medium temperature loop at 𝑇 𝑜𝑢𝑡 𝑀 , and returns to the TES system at lower temperature 𝑇 𝑖𝑛 𝑀 . The difference between the two mass flows is stored in or taken from either of the two tanks, hot at 𝑇 𝐻 . and cold 𝑇 𝐶 . Figure 1b presents the operation scenario. Constant thermal powers are assumed here, but the method applies also for varying power. Hot loop operates during the duration 𝑡 𝐻 and the user demand is active during the duration 𝑡 𝑀 .

Figure 1. (a) 2-Tank TES system, (b) operation scenario

Assuming a periodic stationary regime and no heat losses, energy and entropy balances related to the TES system write, on a period t:

∆𝑈 = ∫ 𝑚̇𝐻 𝑡 𝐻 (ℎ 𝑖𝑛 𝐻 -ℎ 𝑜𝑢𝑡 𝐻 ) 𝑑𝑡 + ∫ 𝑚̇𝑀 𝑡 𝑀 (ℎ 𝑖𝑛 𝑀 -ℎ 𝑜𝑢𝑡 𝑀 ) 𝑑𝑡 = 0 (4) ∆𝑆 = ∫ 𝑚̇𝐻 𝑡 𝐻 (𝑠 𝑖𝑛 𝐻 -𝑠 𝑜𝑢𝑡 𝐻 ) 𝑑𝑡 + ∫ 𝑚̇𝑀 𝑡 𝑀 (𝑠 𝑖𝑛 𝑀 -𝑠 𝑜𝑢𝑡 𝑀 ) 𝑑𝑡 + ∫ 𝜎̇𝑖 𝑟𝑟 𝑑𝑡 𝑡 = 0 (5)
Eqs. ( 3) and (4) simplify in:

𝑞̇𝐻𝑡 𝐻 -𝑞̇𝑀𝑡 𝑀 = 0 (8)
𝑞̇𝐻𝑡 𝐻 𝑇 ̃𝐻 - 𝑞̇𝑀𝑡 𝑀 𝑇 ̃𝑀 = -∫ 𝜎̇𝑖 𝑟𝑟 𝑑𝑡 𝑡 ≤ 0 (9)
Energy conservation (Eq. ( 8)) permits to express the 2 nd law inequality according to the equivalent temperatures 𝑇 ̃𝐻 and 𝑇 ̃𝑀:

𝑇 ̃𝑀 -𝑇 ̃𝐻 ≤ 0 (10)

Equivalent power plant

The equivalence models allow any heat transfer fluid that undergoes a temperature change to be replaced by a thermostat whose temperature corresponds to the equivalence temperatures defined above. As an example, Figure 2 presents the flowsheet of a CSP plant with indirect storage and its related equivalent model. All heat exchangers are replaced by a conductance, and every heat transfer fluid inlet/outlet by a thermostat. The process can be divided in three loops:

In the high temperature (HT) loop, the solar thermal flux 𝑞̇𝐻 is collected and transferred to the TES system. The medium temperature (MT) loop picks up the thermal flux 𝑞̇𝑀 from the TES system and transfers it to the power block. The power block consumes 𝑞̇𝑀 , and converts it in power 𝑤̇ and thermal flux 𝑞̇𝐿 . The low temperature (LT) loop cools the power block and transfers the thermal flux 𝑞̇𝐿 to the air through the cooling tower.

Optimisation problem

To be as general as possible, the problem is defined for any number of heat exchangers in each loop. Thence, the power plant to be optimised includes (Figure 3):

▪ h heat exchangers in the HT loop, ▪ m heat exchangers in the MT loop, ▪ l heat exchangers in the LT loop. According to the operation scenario displayed Figure 1b, the mechanical energy produced per cycle is:

𝑤 = 𝑤̇ 𝑡 𝑀 ( 11
)
This quantity is maximised subjected to the following equality constraints:

1. conservation of energy for the TES, Eq. ( 8), 2. Newton's Law, Eq. ( 1), for each heat exchanger, in each loop, 3. conservation of energy for the power block, Eq. ( 2), 4. Equality of 𝑇 1 𝐻 and 𝑇 l 𝐿 with the temperature of the hot and cold sources,

𝑇 1 𝐻 = T H , 𝑇 𝑙+1 𝐿 = T L (12)
Three inequality constraints also apply:

5. 2 nd law related to the power block, Eq. (3), 6. 2 nd law related to the TES system, Eq. (10), 7. Finite dimensions constraint: as each conductance must be finite, their sum is necessary also finite. Then, there exists a finite positive number 𝐾 Σ such as:

∑ 𝐾 𝑖 𝐻 ℎ 𝑖=1 + ∑ 𝐾 𝑖 𝑀 𝑚 𝑖=1 + ∑ 𝐾 𝑖 𝐿 𝑙 𝑖=1 -𝐾 Σ ≤ 0 (13)
The resolution of this optimisation problem is detailed in Appendix A. It involves 7 + 2 • (ℎ + 𝑚 + 𝑙) optimisation variables (output power 𝑤̇, heat fluxes 𝑞̇𝐻, 𝑞̇𝑀 and 𝑞̇𝐿, ℎ + 𝑚 + 𝑙 conductances 𝐾 𝑖 𝐽 , ℎ + 𝑚 + 𝑙 + 3 temperatures 𝑇 i 𝐽 ), and four optimisation parameters (durations t H and t M related to the operating scenario, and temperatures of the heat source TH and heat sink TL). An optimal solution exists whatever the number of heat exchanger is.

The main results are presented and discussed in the next section.

Results and discussion

The optimal solution is obtained for saturated inequality constraints (Eq. (A 31)-(A 33)).This shows logically that the TES system and the power block must be endoreversible, and that a higher 𝐾 Σ implies a higher output power. This last item appears clearly through the expression of the optimal power output (Eq. (A 66)):

𝑤̇ * = 1 (ℎ√𝑡 𝑀 𝑡 𝐻 ⁄ + 𝑚 + 𝑙) 2 𝐾 Σ (√𝑇 𝐻 -√𝑇 𝐿 ) 2 ( 14
)
which depends linearly on 𝐾 Σ . Equation (A 14) also shows how the number of heat exchangers affects the output power 𝑤̇ * . As 𝑡 𝐻 and 𝑡 𝑀 are of same order of magnitude, the denominator represents approximatively the total number of heat exchangers 𝑁 𝐻𝑋 = ℎ + 𝑚 + 𝑙 . Thence, output power produced by the plant is approximately inversely proportional to 𝑁 𝐻𝑋 . Impact of the TES can also be analysed. Increasing the ratio 𝑡 𝑀 𝑡 𝐻 ⁄ , (i.e., increasing the storage capacity), reduces the capacity of the power block, but increases the mechanical energy provided during a cycle given by Eq. [START_REF] Ondrechen | Thermodynamics in finite time: Processes with temperature-dependent chemical reactions[END_REF]. Influence of the number of HX is clearly displayed. For a constant 𝐾 Σ , which reflects the cost of the HXs, the output power or energy production evolves approximately with 𝑁 𝐻𝑋 -2 , as mentioned above. Concerning the production duration, it decreases the power block capacity, but increases the daily production due to a longer production duration. Another interesting result concerns the optimal conductances and driving forces. In each loop, all conductances (and consequently all driving forces because of Newton's Law) are equal. From Eqs. (A 43),(A 46) and (A 55) we have:

𝐾 𝐿 * = 𝐾 𝑀 * ∆𝑇 𝐿 * = √𝑇 𝐿 𝑇 𝐻 ⁄ ∆𝑇 𝑀 𝑞̇𝐿 * = √𝑇 𝐿 𝑇 𝐻 ⁄ 𝑞̇𝑀 (15)
𝐾 𝐻 * = √𝑡 𝑀 𝑡 𝐻 ⁄ 𝐾 * ∆𝑇 𝐻 * = √𝑡 𝑀 𝑡 𝐻 ⁄ ∆𝑇 𝑀 𝑞̇𝐻 * = (𝑡 𝑀 𝑡 𝐻 ⁄ ) 𝑞̇𝑀

For the MT and LT loop downstream the TES, conductances are equal. Heat flux 𝑞̇𝐿 * is lower 𝑞̇𝑀 * because part of 𝑞̇𝑀 * has been converted in power 𝑤̇ * . Consequently, the driving force ∆𝑇 𝐿 * is also lower than ∆𝑇 𝑀 * .

For the HT loop, heat flux 𝑞̇𝐻 * is higher then 𝑞̇𝑀 * , but here, the optimal solution shares equally this increase between the driving force ∆𝑇 𝐾 * and the conductances 𝐾 𝐻 * Finally, the energy efficiency of the plant can be deduced from (A 65) and (A 66):

𝜂 * = 𝑤̇ * 𝑡 𝑀 𝑞̇𝐻 * 𝑡 𝐻 = 𝑤̇ * 𝑞̇𝑀 * = 1 -√ 𝑇 𝐿 𝑇 𝐻 (17)
Thus, the optimal efficiency neither depends on the number of HXs, nor on the operating scenario, and is equal to the Curzon-Ahlborn efficiency. Figure 5 compares the efficiency given by Eq.( 17) with experimental efficiencies evaluated from available data sets covering many power plant architectures. The ORCs [START_REF] Landelle | Organic Rankine cycle design and performance comparison based on experimental database[END_REF] correspond to the simpler structure: no TES ( 𝑡 𝑀 = 𝑡 𝐻 ), no MT loop ( 𝑚 = 0 ), direct or indirect cooling (𝑙 = 1 𝑜𝑟 2). Most of those displayed are laboratory prototypes. This could explain efficiencies sometimes much lower than 𝜂 * . For nuclear power plants [START_REF] Agency | Operating Experience with Nuclear Power Stations in Member States[END_REF], the hot source temperature (cladding maximum temperature) has been supposed to be 50°C higher than the reactor outlet temperature. Two architectures are presented, BWR (ℎ = 1) and PWR (ℎ = 2), the other parameters being similar (𝑡 𝑀 = 𝑡 𝐻 , 𝑚 = 0, 𝑙 = 1 𝑜𝑟 2).

Most of them reach or even overcome the optimal efficiency. Solar power plant [START_REF] Thonig | CSP.guru 2022-07-01[END_REF] are also commercial plants, all including a TES, with various storage capacity (2 ℎ < 𝑡 𝑠𝑡𝑜𝑟𝑎𝑔𝑒 < 13 ℎ), storage technology (m = 1 or 2), and involving a cooling tower (l = 2). Similarly to nuclear, the only hot temperature available in database was the solar field outlet temperature. Hot source temperatures (maximum temperature of the receiver wall) have been supposed to be 20°C (parabolic trough) or 50°C (tower) higher than the solar field outlet temperature. This figure clearly displays the fact that commercial plants, which have benefited for more than 40 years (solar) or 70 years (nuclear) of R&D fit quite well the 𝜂 * curve. That proves that Eq. ( 17) gives a good approximation of the efficiency of heat engines that have benefited from several decades of research and development. Therefore, we can reasonably assume that it also gives a good assessment of what the efficiency of emerging technologies will be in their future commercial form.

Figure 5 : Energy efficiency of power plants from [START_REF] Landelle | Organic Rankine cycle design and performance comparison based on experimental database[END_REF], [START_REF] Agency | Operating Experience with Nuclear Power Stations in Member States[END_REF], [START_REF] Thonig | CSP.guru 2022-07-01[END_REF] data.

Conclusion

This study aimed to extend FDT results to many power plant structures that may involve different numbers of heat exchangers and may operate with or without a TES. A general model was developed that determined the operating temperatures maximizing the power plant's output energy. A remarkable result is that the optimal efficiency is equal to the Curzon-Ahlborn efficiency, regardless the architecture of the plant and the operation scenario. However, the architecture of the plant impacts a lot the output power which varies approximatively with 𝑁 𝐻𝑋 -2 . Concerning the operation scenario, increasing the storage capacity decreases the installed capacity of the power block but increases the daily energy output. Finally, the comparison of the actual efficiency of power plants shows that technologies that have benefited from several years of R&D achieve the Curzon Ahlborn efficiency. The latter could then be used to estimate the future performance of emerging technologies, such as combined cycles, or supercritical cycles. These technologies could take advantage of a higher temperature provided by generation IV nuclear reactors or future solar towers. Integrating these new technologies in the present analysis is one of the perspectives of this work.

Figure 2 .

 2 Figure 2. CSP plant with indirect storage. Schematic flowsheet and related equivalent system.

Figure 3 .

 3 Figure 3. Equivalent system to be optimised.

Figure 4 Figure 4 .

 44 Figure 4. Output power 𝑤̇ * (a) and mechanical energy per day (b) according to the production duration.

Appendix A

The optimization problem is: min(-𝑤̇ 𝑡 𝑀 ) s.t.

𝑞̇𝐻 𝑡 𝐻 -𝑞̇𝑀 𝑡 𝑀 = 0 (A 1)

𝐾 𝑖 𝐽 (𝑇 𝑖 𝐽 -𝑇 𝑖+1 𝐽 ) -𝑞̇𝑖 𝐽 = 0 , ∀ 𝐽 ∈ {𝐻, 𝑀, 𝐿}, ∀ 𝑖 ∈ [1 ⋯ 𝑛 𝐽 ] (A 2)

𝑞̇𝐻 -𝑞̇𝐿 -𝑤̇= 0 (A 3) The durations (𝑡 𝐻 , 𝑡 𝑀 ), the heat source temperatures (T H ,T L ) and the numbers of heat exchangers 𝑛 𝐻 = ℎ, 𝑛 𝑀 = 𝑚, 𝑛 𝐿 = 𝑙) being taken as parameters, the Lagrangian function of this optimization problem writes:

We first demonstrate that 𝜇 1 , 𝜇 2 and 𝜇 3 cannot be null and then solve this system.

𝜇 3 = 0

Equation (A 14) shows that all the 𝜆 𝑖 𝐽 should be zero, as 𝑇 𝑖 𝐽 -𝑇 𝑖+1 𝐽 = 0 would imply, from (A 23), 𝑞̇𝐽 = 0 which is not acceptable. Therefore, 𝜇 2 = 0 from (A 19) or (A 20), and (A 13) implies that 𝜆 2 = 0,which is forbidden from (A 10), the duration 𝑡 𝑀 being strictly positive: [START_REF] Agency | Operating Experience with Nuclear Power Stations in Member States[END_REF] shows that 𝜆 1 𝑀 should be zero as 𝐾 1 𝑀 = 0 would imply, , from (A 23), 𝑞̇𝑀 = 0, which is not acceptable. Therefore, Eq. (A 15) implies that all 𝜆 𝑖 𝑀 = 0 because 𝐾 j 𝑀 = 0 would also implies 𝑞̇𝑀 = 0. Thence, 𝜇 2 = 0 from Eq. (A 19), which is not acceptable as seen above. Hence, 𝜇 1 ≠ 0

𝜇 2 = 0

Equations (A 19) shows that 𝜆 𝑚 𝑀 = 0 as 𝐾 𝑚 𝑀 = 0 is not acceptable, because it would imply, from (A 23), 𝑞̇𝑀 = 0. Therefore, 𝜇 2 = 0 from (A 14), which is forbidden as seen above. Hence, 𝜇 2 ≠ 0 4. 𝜇 1 ≠ 0 and 𝜇 2 ≠ 0 and 𝜇 3 ≠ 0 Equations (A 28) and (A 29) then write:

Equations (A 31) and (A 32) demonstrates that, quite logically:

▪ the optimal TES system operates reversibly, ▪ the optimal power block is an endoreversible engine.

Equations (A 14) and (A 23) imply:

Replacing 𝜆 𝑖 𝐽 in Eq. (A 15) gives:

showing that, in each loop (i.e., HT, MT, or LT loops), all the conductances are equal.

In a similar way, we have, using (A 26)

Thence, Eq. (A 51) simplifies in:

showing that:

Combining (A 38), (A 44) and (A 55) gives:

Temperatures

The optimal temperature bounding the power block write, using (A 56) and (A 57):


Amandine Asselin

Jean-Philippe Charron
email: jean-philippe.charron@polymtl.ca

Clélia Desmettre

Farid Benboudjema

Cécile Oliver-Leblond

Numerical simulations for the determination of chloride diffusivity in reinforced concrete under tensile load

Keywords: Simulations, Chloride migration, reinforced concrete, tensile load, cracking

In service conditions, reinforced concrete structures are multi-cracked due to the loads they are submitted to. It enables aggressive agents, such as chlorides, to penetrate the concrete cover and could initiate steel rebar corrosion leading to more structural damage. Some experimental programs focusing on chloride penetration were conducted on plain or cracked concrete, but mainly unloaded and unreinforced concrete specimens were used for the measurement. These tests highlight the linear dependence of chloride diffusivity to crack opening. However, they do not take into account the presence of rebar, and cracks are partially or totally closed up during the test, which is different from service conditions. This research project aims to understand the influence of micro-and macro-cracks on chloride diffusivity in conditions close to the service ones. To achieve this objective, three steady-state accelerated migration tests under electrical field are to be carried out on a same reinforced concrete specimen kept under a tensile load representative of a structural one. This non-standard chloride penetration test requires the adaptation of the experimental protocol. This paper presents preliminary adaptation work using numerical simulations developed with Comsol Multiphysics®. The impact of testing configuration and parameters, as well as cracking was investigated. Comparison of simulations and preliminary experimental results are also given.

Introduction

Reinforced concrete structures are often multi-cracked because they are submitted to mechanical and environmental loads during their service life. These cracks are a preferential path for aggressive agents carried by water, such as chlorides, to enter the concrete cover. When chlorides reach steel rebars in sufficient quantity, they initiate rebar corrosion, which progressively damages the structure. Chloride diffusivity measurements allow to quantify the chloride penetration capability and give an indicator of concrete durability. However, chloride diffusivity measurements are often conducted on plain concrete without reinforcement [START_REF] Sanchez | Étude comparative de la diffusion d'espèces anioniques et cationiques dans les matériaux cimentaires : étude expérimentale et numérique[END_REF] and, in some cases, on cracked concrete without reinforcement. In the latter case, specimens are damaged with a splitting test and then unloaded before the diffusivity measure [START_REF] Ismail | Effect of crack opening on the local diffusion of chloride in cracked mortar samples[END_REF][START_REF] Kessler | Effect of freeze-thaw damage on chloride ingress into concrete[END_REF][START_REF] Fu | Chloride penetration into concrete damaged by uniaxial tensile fatigue loading[END_REF]. Through these studies, it has been shown that chloride diffusivity linearly increases with the crack opening until it reaches the chloride diffusivity value in water when the crack opening is over about 80 µm [START_REF] Djerbi | Influence of traversing crack on chloride diffusion into concrete[END_REF]. Nevertheless, these studies do not reflect the reality of the concrete structures in service conditions, which are reinforced, cracked and loaded continuously.

This research project aims to understand the impact of the micro-and macro-cracks found in service conditions in reinforced concrete structures on chloride diffusivity. Accelerated migration tests are simultaneously conducted on three different parts of a reinforced concrete tie specimen, which is kept under a service-conditions tension load. The chloride diffusivity measurements are performed under an electrical field in steadystate conditions. Because the thickness of the tie-specimen, representative of concrete cover in real structures, is larger than specimens used in standard migration tests, and also due to the presence of three migration cells on the tie-specimen, the standard migration test had to be adapted for this project. This paper presents preliminary numerical works conducted to evaluate the required adaptations to the NT Build 335 procedure [START_REF]Nordtest method: Chloride diffusion coefficient from migration cell experiments[END_REF]. It investigates variations in the electrical field and the upstream chloride concentration with Comsol Multiphysics® simulations. The effect of proximity between migration cells and the impact of a straight crack is also analysed. Comparisons of simulations and preliminary experimental results are given.

Methodology

Accelerated migration test

Accelerated migration tests are generally carried out on cylindrical specimens with a thickness of 0.5 to 5 cm [START_REF] Sanchez | Étude comparative de la diffusion d'espèces anioniques et cationiques dans les matériaux cimentaires : étude expérimentale et numérique[END_REF][START_REF] Ismail | Effect of crack opening on the local diffusion of chloride in cracked mortar samples[END_REF][START_REF] Kessler | Effect of freeze-thaw damage on chloride ingress into concrete[END_REF][START_REF] Fu | Chloride penetration into concrete damaged by uniaxial tensile fatigue loading[END_REF][START_REF] Djerbi | Influence of traversing crack on chloride diffusion into concrete[END_REF][START_REF]Nordtest method: Chloride diffusion coefficient from migration cell experiments[END_REF][START_REF] Andrade | Experimental procedure for the calculation of chloride diffusion coefficients in concrete from migration tests[END_REF]. In this project, they are conducted on tie-specimens with thickness of 9 cm, offering a classical cover thickness (40 mm around a rebar of 10 mm in diameter). Migration test implies two cells attached to two opposite sides of the specimen. The first cell (upstream) contains a chloride solution, whereas the other one (downstream) contains a solution without chlorides. A constant electrical potential is then applied between these cells to accelerate the migration of chlorides from the upstream to the downstream cells through the concrete. The electrical potential is applied via an electrode placed in each cell: it is called anode in upstream cell and cathode in downstream cell. Fig. 1 illustrates the set-up for a preliminary test carried out on a small 9 cm thick concrete prism. At the beginning of a migration test, downstream chloride concentration remains null until chlorides have begun to pass through concrete. Then concentration increases in a non-linear way during the non-steady state. Finally, downstream concentration in chlorides 𝒄 𝒅𝒐𝒘𝒏 [mmol/L] linearly increases with time 𝒕 [s] at the steady state, during which the slope of the chloride concentration evolution is proportional to diffusivity 𝑫 𝒆 [m2/s] according to Eq. [START_REF] Sanchez | Étude comparative de la diffusion d'espèces anioniques et cationiques dans les matériaux cimentaires : étude expérimentale et numérique[END_REF].

𝑐 𝑑𝑜𝑤𝑛 = 𝑐 𝑢𝑝 × Δ𝑉 × 𝑆 × 𝐹 𝑅 × 𝑇 × 𝑉 × 𝑒 𝐷 𝑒 𝑡 (1)
Where 𝑐 𝑢𝑝 [mmol/L] is the upstream concentration, Δ𝑉 [V] is the potential difference, 𝑆 [m 2 ] is the surface of the specimen exposed to the diffusion tests, 𝐹 [C/mol] is the Faraday constant, 𝑅 [J/mol/K] is the molar gas constant, 𝑇 [K] is the temperature, 𝑉 [m 3 ] is the downstream cell volume and 𝑒 [m] is the thickness of the specimen.

Eq. ( 1) is derived from the Nernst-Planck equation and is applicable under the following assumptions:

-Diffusion and electrical fields are both assumed to be unidirectional; -The concrete is saturated, which means convection is negligible and is not considered in the modelling; -Interactions between chlorides and cementitious matrix are neglected; -Migration of the chlorides under the electrical field is predominant compared to natural diffusion; -The upstream chloride concentration is considered constant.

In view of the numerous parameters involved in Eq. ( 1) and the various assumptions made, several numerical simulations were carried out to study the impact of electrical field and upstream concentration variations. The minimal distance between migration cells and the effect of a crack were also investigated.

Numerical simulation

The following numerical simulations have been carried out using modules "Transport of Diluted Species in Porous Media" and "Electrostatic" in Comsol Multiphysics®.

The aim of this project is to make simultaneous migration tests on three different parts of a tie-specimen with a 9 cm x 9 cm cross-section and 61 cm length. Dimensions were chosen according to a previous project concerning water permeability measures [START_REF] Desmettre | Contribution à l'étude de la perméabilité du béton armé sous sollicitations statiques et cycliques[END_REF]. A 2D reference simulation has thus been conducted with a concrete specimen with these dimensions (Fig. 2). This reference simulation considers only one migration test carried out at mid-height of the specimen (Fig. 2). The upstream concentration cup (1060 mmol/L) and the electrical ground are imposed on a height of 15 cm corresponding to the diameter of the upstream cell (Fig. 1). It is generally recommended to impose an electrical potential difference of 12 V for 5 cm thick concrete specimens to keep natural diffusion negligible compared to migration transport [START_REF] Djerbi | Influence of traversing crack on chloride diffusion into concrete[END_REF][START_REF]Nordtest method: Chloride diffusion coefficient from migration cell experiments[END_REF]. That is why the electrical potential, imposed facing on the opposite side, has been chosen equal to 12 V. At the same location, the downstream cell, with dimensions of 15 cm by 15 cm, is modelled and filled with water, of which diffusivity is equal to Dw = 2x10 -9 m²/s. The surface average of chloride concentration is evaluated in the latter cell to determine diffusivity. Finally, the chloride flux and the electrical load are set to null on the rest of the specimen boundary. Concrete porosity is not physically considered in the modelling, it is included numerically by setting the chloride diffusivity Dc = 1x10 -11 m²/s to the concrete specimen. The electrical field is constant between both cells, except on the borders where irregularities can be observed (Fig. 3a). This is linked to the end of the electrode and is expected also present in the experimental test. Indeed, field lines are parallel and regularly distributed in the middle of electrodes. But, on the electrodes rim, field lines are incurved and join the electrodes at the same place, leading to field lines concentration and a higher value of the electrical field. The same phenomenon can be observed with the chloride concentration: a higher value is noticeable close to the electrode ends (Fig. 3b). These perturbations at the extremities lead to a slight decrease in diffusivity. Thus, by evaluating the diffusivity from the evolution of the chloride concentration in the downstream cell, a value of De equal to 0.91x10 -11 m²/s is determined (Fig. 3c), which is 9% less than the one imposed for the concrete specimen in the simulation (Dc in Fig. 2).

Results of the sensitivity studies

This section aims to evaluate the impact of some testing parameters by modifying the reference simulation. Each different value of parameters (one point on following figures) corresponds to one simulation.

Electrical field

In this section, the voltage was changed while keeping the same configuration as the reference simulation to study the impact of the potential difference on the diffusivity. The diffusivity value increases with the potential difference to reach a 27% higher value than the reference when increasing the electrical potential difference to 110V. It means that the results obtained during migration tests are very sensitive to voltage variations and can be overestimated for high potential differences. Therefore, even if a high potential is a way to achieve test results more quickly, it will be essential to use reasonable value as well as same potential difference to compare results in experimental tests.

Considering the previous conclusions as well as the risk of formation of undetectable chlorinated compounds such as dichlore or hypochlorite when the voltage is too high, a potential difference of 58.5V was chosen for all the experimental tests [START_REF] Asselin | Évaluation du coefficient de diffusion des chlorures dans le béton armé sous chargement mécanique[END_REF]. This electrical potential value was numerically tested with Comsol Multiphysics® and involved a diffusivity equal to 1.09x10 -11 m²/s and a difference of 19% with the reference value, as shown in Fig. 4.

Chloride upstream concentration

The influence of upstream chloride concentration has been studied and shown in Fig. 5. The upstream cell chloride concentration does not influence chloride diffusivity in the concrete. However, it is assumed that upstream concentration is constant during the simulation, which is not the case experimentally when the upstream solution is not regularly renewed. Another simulation with an upstream cell filled initially with a chloride solution and free to equilibrate with the system has shown a decrease in concentration with time, as shown in Fig. 6. This decrease reaches 9% of the initial upstream chloride content after 100h of migration test. Afterwards, this linear decrease was set as a chloride content boundary evolution instead of a constant one. The diffusivity obtained was 0.88x10 - 11 m²/s, which is only 3% lower than the reference value. This observation means that variation of chloride concentration in the upstream cell should have a negligible effect in experiments.

Distance between 2 cells

In this section, two sets of chloride migration cells are considered in the simulations and the impact of the distance between cells is studied. Cells are located in such a way that geometry is symmetrical to the middle horizontal axis, which means migration phenomena are the same in both migration cells by symmetry. Results of diffusivity are given according to the distance between cells in Fig. 7. When cells are closer than 10 cm, diffusivity is higher than the reference simulation including only one set of migration cells. Indeed, electrical fields of both sets of cells are superposed and locally higher, which drives more easily chlorides and lead to an overestimation of diffusivity. For larger distance between the sets of cells, the diffusivity decreases until it reaches the reference value for a distance of 15 cm. Then, it starts to increase again when the sets of cells become close to the top and bottom specimen border. Indeed, a reflection against the top and bottom surfaces reduces the chloride transit time from the upstream to the downstream cells, artificially raising the diffusivity.

The same simulations were thus conducted on a 2m-long specimen to eliminate the specimen length effect (Fig. 8).

Fig. 8. Impact of the distance between 2 cells on the chloride diffusivity in a 2m-long specimen.

Increasing the specimen length from 0.6 to 2 m brings a very little change in the value of De when in presence of one centred cell (0.92x10 -11 m²/s in comparison to 0.91x10 -11 m²/s for the 0.61m-long specimen). The new value of 0.92x10 -11 m²/s was then used as the reference in Fig. 8 for proper comparison of the effect of two cells versus one cell on the 2m-long specimen. As observed in Fig. 7, diffusivity still decreases in Fig. 8 when the distance between cells increases up to 10 cm. However, it then remains constant. This confirms that the increase of diffusivity observed in Fig. 7 when the sets of cells were moved away from each other (for distances superior to 15 cm) is related to the specimen length effect. Consequently, recommendation for the minimal distance between migration sets of cells is 10 cm to avoid spurious effects, whereas the minimal distance to specimen border is 5 cm. These values have been experimentally verified. A migration test has been carried out on ordinary concrete, reinforced with a 10M epoxy-coated steel rebar. The water/binder ratio was 0.60 and the cement was a standard North American GU type Portland cement. At the end of the test, the specimen was split. A solution of silver nitrate was then sprayed to determine which part of the concrete had been reached by chlorides. The surface of concrete contaminated with chlorides becomes white due to the formation of silver chloride, whereas it stays brown in the absence of chloride [START_REF] Collepardi | Penetration of chloride ions into cement pastes and concretes[END_REF]. Fig. 9 shows that chloride diffused in concrete in a zone wide of 17 cm, corresponding to a distance of 4.5 cm outside the cell. Therefore, sets of cells must be spaced at least by more than 9 cm and located at more than 4.5 cm from the specimen top and bottom surfaces to avoid interferences. This is consistent with numerical results.

Influence of a straight crack

A straight and constant opening crack filled with water was added at mid-height of the specimen with a unique set of migration cells (Fig. 10a). Several numerical simulations were performed for crack openings ranging from 0.05 mm to 1 mm. A comparison of those simulations with the theory is made. For this, a parallel theoretical model is used: it considers two plain concrete surfaces surrounding a water surface corresponding to the crack. The diffusivity of the cracked concrete is then deduced from the diffusivities of the plain concrete and the water [START_REF] Djerbi | Influence of traversing crack on chloride diffusion into concrete[END_REF] (Fig. 10b). The variability in diffusivity appears to be between 12.4 and 19.5% [START_REF] Aît-Mokhtar | Experimental investigation of the variability of concrete durability properties[END_REF]. To be sure to quantify the effect of a crack and not the variability, a variability threshold equal to 20% of the reference diffusivity value was considered (Fig. 11). It means the crack opening has to reach 0.1 mm to cause a significant diffusivity variation compared to inherent concrete variability. Experimental loading will be adapted to reach the crack opening.

Conclusion

This research project aims to understand the impact of micro-and macro-crack on chloride diffusion in reinforced concrete structures in service conditions. To this end, three simultaneous accelerated migration tests under an electrical field are carried out on the same reinforced concrete tie-specimen under tensile load. These non-standard migration tests have required some numerical work to establish the optimal testing configuration. Comsol Multiphysics® was used to model chloride migration in the concrete specimen in different conditions. A set of electrical and transport parameters were imposed to the specimen, whereas the evolution of downstream chloride concentration provides an estimation of chloride diffusivity based on the Nernst-Planck equation.

A reference simulation was used to evaluate the impact of the different sets of parameters. The electrical field map in the specimen allows reporting the boundary effect of electrodes with a field lines concentration, leading to a locally higher concentration. Diffusivity found in the simulation is slightly lower than the one imposed, but other diffusivity estimations will be compared to this reference value.

Several electrical potential differences in migration test, as well as upstream cell chloride concentrations, were tested. While upstream cell chloride concentration has no impact on the diffusivity, this latter increases with the potential difference. Therefore, keeping the same voltage during the experimental campaign will be mandatory to compare results with each other. The assumption of constant upstream concentration was also verified. Even if the chloride content has decreased by 9% 100h after the beginning of the accelerated migration test, diffusivity is almost not impacted. It means that the change of upstream cell solution is not necessary during the experimental test.

In the final experimental configuration, at least two sets of migration cells will be installed simultaneously on the same specimen. Therefore, the impact of one on the other was verified by simulations. Interferences between the cells are observed when they are separated by less than 10 cm. Furthermore, spurious effects are noticed when sets of cells are too close to specimen extremities. The addition of silver nitrate on a concrete specimen after a migration test showed chloride diffusion up to 4.5 cm around each cell. It corroborates the simulation results.

Fig. 1 .

 1 Fig. 1. Accelerated chloride migration test device.

Fig. 2 .

 2 Fig. 2. Geometry and parameters of the reference simulation.

Fig. 3

 3 Fig.3gives a 2D representation of the electrical field, and concentration 100h after the beginning of the simulation. Time-evolution of the chloride concentration in the downstream cell is also given.

Fig. 3 .

 3 Fig. 3. Result of a simulation, a) electrical field (V/m), b) chloride concentration (mol/m 3 ) and flow lines for the reference test at t=100h and, c) chloride concentration evolution in the downstream cell.

Fig. 4 Fig. 4 .

 44 Fig. 4 illustrates the modification of the diffusivity when increasing the potential difference.

Fig. 5 .

 5 Fig. 5. Evolution of diffusivity according to upstream concentration.

Fig. 6 .

 6 Fig. 6. Decrease of the chloride concentration in the upstream cell.

Fig. 7 .

 7 Fig. 7. Impact of the distance between 2 cells on the chloride diffusivity.

Fig. 9 .

 9 Fig. 9. Concrete contaminated with chloride (white surface) and without chlorides (brown) revealed by the addition of silver nitrate.

Fig. 10 .

 10 Fig. 10. Impact of crack, a) cracked specimen simulated, b) decomposition of cracked concrete in plain concrete and crack.

Fig. 11 .

 11 Fig. 11. Comparison between theoretical and simulated diffusivity values with a crack opening in the specimen. A variability threshold corresponding to 20% of the reference diffusivity has been added.

A concrete specimen with a straight crack was also simulated and compared to a theoretical model. Numerical and theoretical results are similar. The conclusion is that the crack opening must reach 0.1 mm to modify significantly the diffusivity in view of the variability of the diffusivity of plain concrete.


Marine Dumarest

Camille Victoire Migné

Manon Chevrier

Naomie Ramage

Teheipuaura Helle

Scott C Weaver

Nolwenn M Dheilly

Sylvie Lecollinet

Stéphan Zientara

Cécile Beck

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Improvement of molecular and serological tools for the detection of Venezuelan Equine Encephalitis Virus (VEEV)

Improvement of molecular and serological tools for the detection of Venezuelan Equine Encephalitis Virus (VEEV) VEEV belongs to Alphavirus genus within the Togaviridae family.

Correspondance: LRUE maladies équines: adm.eurl-equinediseases@anses.fr marine.dumarest@anses.fr gaelle.gonzalez@anses.fr

CONCLUSIONS & PERSPECTIVES

  of the sensitivity of VEEV RT-qPCR method  Detection of 100% of VEEV tested strains  Specificity : 100%  Sensitivity: estimated at 50 genome copies/µl of serological tools A. Use of Chimeric virus Sindbis/VEEV to perform VNT in BSL2 B. VNT in BSL2 using chimeric SINV/VEEV  Natural neutralizing antibodies recognize SINV/VEEV chimeric viruses  Equal and Higher sensitivity using chimeric viruses  No cross-reactivity with other EEEV and WEEV indirect « in-house » ELISA based on total inactivated particles camille.migne@anses.fr  Improvement of molecular detection of VEEV by RT-qPCR  Work in progress on « easy to use » serological tools Europe needs to be prepared in case of emergence CONTEXT OBJECTIVES RESULTS

  Marine Dumarest 1 , Camille V. Migné 1 , Manon Chevrier 1 , Naomie Ramage 1 , Teheipuaura Helle 1 , Scott C. Weaver 2 , Nolwenn M. Dheilly 1 , Sylvie Lecollinet 1 , Stephan Zientara 1 , Cécile Beck 1 , Gaëlle Gonzalez 1* Institute for Human Infection and Immunity, Department of Microbiology and Immunology, Galveston, TX, USA.

1 Anses, INRAE, Ecole Nationale Vétérinaire d'Alfort, UMR Virologie, Laboratoire de Santé Animale, 94700 Maisons-Alfort. 2 University of Texas Medical Branch, REFERENCES Vina-Rodriguez A, et al. A Quantitative Real-Time RT-PCR Assay for the Detection of Venezuelan equine encephalitis virus Utilizing a Universal Alphavirus Control RNA. Biomed Res Int. 2016;2016:8543204. Toussaint JF, et al. Bluetongue virus detection by two real-time RT-qPCRs targeting two different genomic segments. J Virol Methods. 2007;140(1-2):115-23. Ni H, et al. Recombinant alphaviruses are safe and useful serological diagnostic tools. Am J Trop Med Hyg. 2007; 76(4):774-781. WOAH 2019, chapter 3.5.5, Equine Encephalomyelitis.