Performance Test of the Air-Cooled Finned-Tube Supercritical CO₂ Sink Heat Exchanger

In the nuclear power plant design, the consideration of multiple component failure scenarios is a motivator for the development of failure safe backup systems. One approach for a failure safe backup system currently under development is called supercritical CO₂ heat removal (sCO₂-HeRo) [1]. It is designed for boiling water reactors and pressurized water reactors (PWRs) to prevent Fukushima-like accidents, where a combined station blackout, loss of ultimate heat sink, and loss of emergency cooling occurred. The sCO₂-HeRo is such an emergency cooling system. It transports the decay heat from the reactor core through a self-propellant, self-sustaining Brayton cycle, including compressor, heat exchanger (HX) (steam-sCO₂), turbine, and sink heat exchanger to the ambient air.

The main objective of this work was to provide evidence for the concept of the air-cooled finned-tube sink HX at laboratory conditions (technical readiness levels 3–4), develop and validate a new numerical Modelica-based model for the code ClaRa suitable for modeling steady/transient scenarios in sCO₂ environment, and finally deliver valuable operational experience from the unique sCO₂ facility at Research Centre Rez (CVR).

The measurement covered both the supercritical and subcritical pressures (7–10) MPa including transition of pseudocritical region (27–36) °C in the last stages of the sink HX. The nominal parameters of the sink HX were reached: 95 kW, 7.8 MPa, 166 °C/33 °C, 0.325 kg/s for the sCO₂ side cooled by 25 °C forced air flow with ambient pressure.

A number of investigators have carried out experimental tests and analyses of the heat transfer performance of finned-tube sCO₂ gas coolers. Majority of this work was focused only on steady-state analyses [1–4]. All of these authors use ε-NTU or LMTD (i.e., lumped method and distributed method) which has limitations, especially when it comes to modeling of rapidly varying thermophysical properties in the critical region. Therefore, e.g., LMTD has to be modified using an integral approach for LMTD [5] or finite methods need to be deployed, i.e., finite volume method utilized in this paper or finite element approach found in the work by Yin et al. [6] who performed stationary calculations and optimization.

Apart from an experimental research, there are numerous studies dedicated purely to simulation tools development for sCO₂ energy systems. In the dynamic simulation software, there can be found a few in-house system codes analyzing nuclear reactors and experimental loops behavior with sCO₂ [7,8] or system codes primary developed for light water reactors safety analyses like ATHLET, RELAP, and TRACE which has been upgraded for handling sCO₂ simulations [9–12]. However, the validation of these codes in sCO₂ environment has been lacking. Therefore, this study was conducted to present a new validated Modelica code as well as to submit a new set of sCO₂ data for future benchmark.

To the best of our knowledge there has been no previous investigations reported in the literature on the sCO₂ gas coolers performing experimental work together with both, the steady-state and transient analyses.

The results in this paper will benefit to researchers, designers, software engineers, thermal hydraulic specialists, and operators of sCO₂ energy systems through the shared measured data and described operational procedures in a unique sCO₂ facility.

Figure 1 depicts the scheme of the sCO₂-HeRo retrofitted into the PWR. In case of a station black-out and the loss of ultimate heat sink accident, the reactor automatically shuts down, the turbine fast-driven valves close, and the safety valves open. However, the residual heat is produced. By nature, without the utilization of main circulation pump (MCP), natural circulation is established in the primary circuit, which transfers the decay heat to the steam generators (SG) and evaporates its water content. The steam flows into a heat exchanger (CHX), which must be very compact to fit into the limited space available in existing reactor building. The steam condenses and the liquid water, driven by gravity, flows back into the SG. Thus, the water content in the SG is preserved. Inside the compact HX the sCO₂ heats up. It flows through a turbine, which is located on the same shaft as the compressor and the generator. Downstream of the turbine, the sCO₂ gets cooled by the air in the sink HX and is delivered to the compressor and back to the compact heat exchanger. Over a large operating range, the turbine of the Brayton cycle shall produce more power than the compressor needs to operate. The excess power is transferred into electricity, which is used to power additional fans of the sink HX for better heat removal.

The sCO₂-HeRo system can be attached to both existing pressurized water reactors and boiling water reactors, since the thermodynamic parameters of steam are similar. Without having the sCO₂-HeRo system deployed, the water content in the SG would steadily decrease (by releasing the steam through pressure safety valve or pressure relief valve) causing overheating of the primary circuit which could eventually lead to fuel damage [13,14].

Within the European project “sCO₂-HeRo,” six partners from three European countries are working on the assessment of this cycle. The goal is to numerically and experimentally show evidence for the concept on a small-scale demonstrator of the sCO₂-HeRo system which shall be incorporated in the PWR demonstrator (a reproduction of a two-loop pressurized water reactor Siemens/Kraftwerk Union design at a scale of 1:10) at the Simulator Centre of KGS and GfS in Essen, Germany. Before assembling the small sCO₂-HeRo system in the Simulator Centre, each major component was tested in different institutions. The performance of the compact HX (microchannel type) was verified in the sCO₂ test loop (SCARLETT) in University of Stuttgart, while the air-cooled sink HX, compressor, and turbine were measured in the CVR sCO₂ experimental facility.

The design of the sink HX strongly influences the behavior of the whole sCO₂-HeRo system, as it is operated near the critical point region of CO₂ (7.8 MPa, 33 °C). Underestimated size of the HX can lead to a not self-propellant sCO₂-HeRo design. This is due to the high outlet temperature of the HX (inlet to the compressor) resulting in excessive compression work.

According to the optimized cycle calculations of the sCO₂-HeRo system, the sink HX model for the small scale sCO₂-HeRo has been specified [13].

Table 1 shows the main thermodynamic parameters for the selected two identical sinks HX's working in parallel. Each designed as finned tube HX type cooled by forced air (fan with EC motor with speed control). One of them was selected for testing and implemented into the sCO₂ loop in CVR.

The conceptual drawing with overall dimensions is shown in Fig. 2.

The internals of sink HX includes stainless steel AISI 304 tubes in staggered arrangement with rectangular aluminum fins (metal sheet). The arrangement is such that the flow on the sCO₂ side is purely horizontal (except the inclined bends placed outside the air flow), while on the air side the flow is completely vertical. An illustrative scheme is shown in Fig. 3.

The overall heat transfer area for one sink HX is 361 m². The detail geometry of sink HX is included in Table 3.

The heat transfer investigations in the sink HX test configuration took place at CVR, using sCO₂ experimental loop which was constructed within Sustainable Energy (SUSEN) project. This unique facility enables to study key aspects of the cycle (heat transfer, erosion, corrosion, etc.) with wide range of parameters: temperature up to 550 °C, pressure up to 30 MPa, and mass flow rate up to 0.35 kg/s.

Figure 4 shows the piping and instrument diagram (P&ID) of the loop. A part of the primary circuit used for the sink HX measurement is represented by thick line, and it consists of a low temperature regenerative heat exchanger (LTR) and high temperature regenerative heat exchanger (HTR), a main piston pump, and four electric heaters of the total maximum power of 110 kW. Heat exchangers HTR and LTR are designed as a counter-flow shell and tube-type from stainless steel (SS).

The electrical heater H3 with nominal power 20 kW is positioned at the bypass of the LTR in order to simulate the behavior of a recompression cycle.

For cooling purposes, two shell and tube type coolers CH₁ and CH₂ are connected to the loop. The cooler CH₁ from SS is cooled by water (temperature 20 °C, 1.4 kg/s flow rate of water), and the cooler CH₂ also from SS is used as the main cooling medium oil (Malotherm SH, Sasol, Sandton, South Africa), because of the high temperatures of the exhaust heat. Next part of the primary loop consists of two parallel electric heaters H_1/1 and H_1/2 from SS with 30 kW each, followed by one Inconel electrical heater H₂ with 30 kW. Behind the heaters, a test section TS (pressure tube which enables to insert samples) and reduction valve RV is positioned. It is used to represent a turbine expansion. The main operating parameters of the primary circuit are shown in Table 2.

For testing of the sink HX, the low pressure side (behind the reduction valve) of the LTR and the HTR as well as the oil cooler CH₂ were by-passed in order to achieve desired inlet temperatures (max. 170 °C) to the sink HX. The by-pass is marked in thick red line with squares. The omitted piping is marked in thin gray line. Pressure in the system is controlled either by the electric heaters, i.e., by the temperature in the circuit, or by the filling compressor/release valves (to the outside atmosphere) by which it is possible to control the amount of CO₂ in the loop, thus the pressure.

Figure 5 shows the sCO₂ loop and the installed sink HX configuration, which is outside of the experimental hall.

Component geometry of the sCO₂ loop is summarized in Table 3.

This section contains the measurement procedure of the performed tests on sink HX within sCO₂ experimental facility in CVR.

Operational limits of the test facility (Table 4) must be taken into account and they should not be exceeded during the performance test.

For carrying out the experiments, the primary circuit was first evacuated and then filled by CO₂ (99.995%).

Figure 6 shows the sink HX outside of the experimental hall with in-coming and out-going pipelines together with all measurement devices.

The measurement campaigns covered both supercritical and subcritical regions including transition through the pseudocritical region in the last stages of the sink HX. The critical point of the CO₂ is 7.39 MPa and 31.1 °C. The controlled (independent) and resulted (dependent) parameters are summarized in Table 5.

Measurement campaigns were carried out with different inlet conditions on both sides of the sink HX. The measurement time took about 15 min at each measurement point in order to reach stable conditions. The operational procedure was as follows:

1. (1)

   hold p_{_sCO2_in} = 7.8 MPa at nominal

2. (2)

   hold ṁ_{_sCO2} and T_{_sCO2_in} at certain value (0.1, 0.2, or 0.32) kg/s and (50, 100, 166) °C, respectively

3. (3)

   vary V̇_{_air_out}, i.e., frequency of the fan (50, 75, 100) % of nominal 50 Hz, while for each frequency a measurement was recorded

4. (4)

   increase/decrease m_{_sCO2} while keeping the T_{_sCO2_in} and repeat step 3 and repeat this procedure for all variants of ṁ_{_sCO2} (0.1, 0.2 or 0.32) kg/s

5. (5)

   increase/decrease T_{_sCO2_in} to new value and repeat steps 3 and 4 to record all variants of T_{_sCO2_in} (50, 100, 166) °C

   With this procedure the influence of m_{_sCO2}, T_{_sCO2_in}, and V̇_{_air_out} was studied. In order to see impact of p_{_sCO2_in} following steps were taken:

6. (6)

   hold T_{_sCO2_in} at certain value (100 °C)

7. (7)

   hold m_{_sCO2} and p_{_sCO2_in} at certain value (0.1, 0.2, or 0.3) kg/s and (7, 7.4, 8.5, 9.4, 10) MPa, respectively, and vary V̇_{_air_out}

8. (8)

   increase/decrease m_{_sCO2} while keeping the p_{_sCO2_in} and repeat step 3 repeat this procedure for all variants of ṁ_{_sCO2} (0.1, 0.2 or 0.32) kg/s

9. (9)

   increase/decrease p_{_sCO2_in} to new value and repeat step 3 and 7 to record all variants of p_{_sCO2_in} (7*, 7.4, 8.5, 9.4*, 10*) MPa.

*Not all ṁ_{_sCO2} (0.1, 0.2 or 0.32) kg/s were possible to implement due to the limited power of filling pump.

Figure 4 shows the piping and installation diagram (P&ID diagram) of the modified sCO₂ loop with the main components together with all installed measurement devices, such as a mass flow meter, volume flow meter, Pt-100 sensors, thermocouples, and pressure sensors. The nomenclature of the measurement devices respects the KKS identification system for power plants.

The uncertainties provided by the measurement devices, transducer, input card, and control system are summarized in Table 6. The errors correspond to calibration certificates and manufacturer's instructions.

The error propagations are described in Annex A.

The results for the design (nominal) conditions of the sink HX have shown 15% error propagation of the heat transfer on the sCO₂ side Q_sCO2 and 8% for the air side Q_{_air.}

This section contains experimental results for steady-state and transient operation.

Figure 7 shows the experimental results of $Q_air=m˙_air·cp_air·(T_air−out−T_air_in)$ and $Q_sCO2=m˙_sCO2·(h_sCO2−in−h_sCO2−out)$⁠. For all of the 34 measurements, the heat transfer ratio R = Q__air/Q__sCO2 stayed within the limits (115%/85%). The base source of the errors propagation for the Q__sCO2 is the uncertainty of the thermocouple measurement of the outlet sCO₂ (far less than at the inlet). This is due to the fact that the pseudocritical region (around 34 °C) is crossed here and each small error of the temperature determination leads to high errors in evaluation of enthalpies (up to 60 kJ/kg), i.e., heat power (15 kW). Figure 6 shows the sink HX standing outside of the experimental hall with pipelines and measurement devices.

The honey combs are utilized to stabilize the flow at the outlet of the air pipe and more importantly, in front of the Wilson grid which is used to measure volumetric flowrate throughout the pitot arrays. These consist of a row of vertical tubes, with alternate rows of holes facing up and down stream, measuring the total and substatic pressures from which dynamic pressures are calculated. As shown in Fig. 7, the air side heat flow rate Q__air exceeds the CO₂ heat flow rate Q__sCO2. by max. 15%.

The potential of the sCO₂-HeRo system to deal with a range of different accident scenarios and beyond-design accidents will need to be proven with the help of thermal hydraulic codes. Therefore, heat transfer models were compared with the experimental data.

The heat transfer at the tube side where sCO₂ flows is geometrically characterized by the inner diameter and shape of the tubes and has been thoroughly studied. Numbers of correlations are discussed in the literature [16–18].

The air, which is pulled through the cooler by a fan mounted at the top of the unit, flows around the tube bundle with fins. This is geometrically much more complex. It includes definition of transverse and longitudinal tube spacing, tube outer diameter, number of tube rows, fin spacing, fin thickness, and fin type. Besides this complexity, the air local heat transfer coefficient is by one order of magnitude smaller than of the sCO₂ side. Thus, the air side determines the size of the whole HX.

For the calculation of efficiency of the rectangular fins η_fin, a formula stated in Ref. [24] was used. For the given geometry it resulted in η_fin = 0.95.

Equations (4) and (5) are taken from Refs. [24] and[25].

The graph in Fig. 8 shows a comparison of resulted averaged overall heat transfer coefficients k_{_calc_avg} calculated (using Gnielinski [18] for sCO₂ and IPPE [23] for the air) and experimentally determined k_{_exp_avg} for all the 34 measurement points. The overall k_{_exp_avg} was calculated from the measured temperatures, pressures, mass flow rates on both the sCO₂ and air sides using the following formula $Q=kexp_avg·Aouter·ΔT′(W)$ describing the heat transferred in each control volume of the sink HX. The positive errors suggest that the calculated values, using correlations, overestimate the experimental values for the negative errors and vice versa. It can be seen that the discrepancy is reasonable low + 25% and −10%.

From the graph Fig. 9, it can be concluded that both correlations according to IPPE and VDI are in perfect match.

The effect of the mass flux on the local heat transfer coefficient of sCO₂ is illustrated in Fig. 10. At the same pressure, the local heat transfer coefficient of sCO₂ increases with mass flux due to higher Reynolds number.

Figure 11 presents the local heat transfer coefficient of sCO₂ for different cooling pressures ranging from 7.1 MPa to 9.4 MPa at a given mass flux. For the supercritical pressures (higher than 7.4 MPa), the peak values in the local heat transfer coefficient are shown at the same pseudo-critical temperatures. Higher pressure has lower local heat transfer coefficient because the specific heat is lower. At the subcritical pressure (7.1 MPa), the local heat transfer coefficient increases toward colder temperatures and even exceeds the values of supercritical pressure due to the higher specific heat at this region. There has been considerable prior research done in the area of sCO₂ coolers with similar findings [20,21].

During the performance measurement of the sink HX a transient test was performed. The volumetric flow rate of the air was stepwise changed from the value 12,250 m³/h through 9400 m³/h (75% fan speed) to 6400 m³/h (50% fan speed) while keeping the nominal sCO₂ mass flow rate at 0.32 kg/s. Before each change a steady-state was reached such that p_{_sCO2_in} = 7.8 MPa T_{_sCO2_in} = 166 °C. Each drop of V̇_{_air_out} resulted in a rise of pressure (2–4 bars) in the primary circuit due to a higher mean temperature in the system, particularly in the sink HX. This was compensated with the pressure control system feeding additional sCO₂ by a booster compressor. At time 1450 s (6400 m³/h, 0.32 kg/s), frequency of the main circulation pump started to stepwise decrease the ṁ_{_sCO2}. As consequence of the ṁ_{_sCO2} reduction, the inlet temperature to the sink HX T_{_sCO2_in} abruptly increased, until it reached its maximum limit 170 °C at 1820s, even though the air fan was switched back to its nominal 100%. The automatic control system switched off all heaters which were at this time almost at their maximum, i.e., H_1/1—28 kW, H_1/2—30 kW, H₂—26 kW, and H₂—20 kW. Switching off the electric heaters resulted in sudden drops of the temperatures and pressures in the system. However, there was some reaction time of the control system, and the inlet temperature to the sink HX was slightly exceeded. The controlled parameters are summarized in Table 7.

The experimentally measured data of the sCO₂ loop from the transient scenario described in the Transient Operation section was used for code benchmark to test and validate thermal hydraulic Modelica-based code ClaRa [26,27].

The pipe model includes equations derived from the general form of the conservation equations by the finite volume approach. The finite volume approach was used to derive a set of ordinary differential equations from partial differential equations, such that they can be implemented in a computer and numerically solved. In many situations (e.g., pipe model which is our case), it is reasonable to simplify models by restricting to one-dimensional mass flows which can be then spatially discretized and modeled by number of control volumes. For each control volume, we can write mass, momentum, and energy balance equations which are implemented in ClaRa.

The dynamic sCO₂ loop model includes all major components of the CVR test facility according to the P&ID. The main circulation pump MP is speed-controlled with preset input parameters. Heaters with PID controllers provide desired temperatures at the sink HX. The outlet temperature of cooler CH₁ is handled with PID-operated water flow rate. The pressure in the system is controlled by feeding additional sCO₂ (by a booster compressor) or releasing sCO₂ through orifices, modeled in the computational model in a simple manner by the sCO₂ source, and the PID controller. The air flow rate through the sink HX is handled with defined input.

The obtained results of the computational model (Fig. 12) for the nominal parameters can be found in Fig. 13, where temperatures of the sCO₂ and air along the length of the sink HX tubes are displayed.

The main resulted parameters from both, the measurement and transient simulation, are shown in Fig. 14. They show fair agreement, demonstrating reasonable accuracy of the simulation tool. There is an evident deviation at the peak inlet temperature of sCO₂ to the sink HX (by 13 K) leading to 3 bar pressure difference and 2 K discrepancy at the sink HX outlet. Apparently, this results from a smaller heat capacity of the numerical model than in reality. A faster temperature change (sCO₂) at the sink HX inlet justifies that. The model neglects all pipe supports, flanges, and bolts.

This paper reports the performance tests of the supercritical air-cooled finned-type sink HX (tube Ø 12 mm x 0.7 mm) and presents a high quality numerical model. Altogether 34 measurement points were collected which were used for system code validation. Additionally, transients were logged, aiming to understand the energy and mass storage effects in the component.

The following conclusions can be drawn from the experimental results:

• The pressure, mass flux, and temperature of sCO₂ have significant effects on the local heat transfer coefficient, especially near pseudo-critical region. The local heat transfer coefficient is decreased when cooling pressure is increased (for p_sCO2 > 7.4 MPa) otherwise increased when mass flux is increased. The local heat transfer coefficient along the sink HX changes rapidly with the temperature of the fluid. It reaches a peak near the pseudo-critical temperature due to the highest heat capacity.

• The experimentally determined heat balances from the measured parameters on both sides (sCO₂ and air) Q__air and Q__sCO2 are in good agreement (±15%) with each other.

• The results of calculated averaged overall heat transfer coefficients k_{_calc_avg} using correlations (Gnielinski [18] for sCO₂ and IPPE [23] or VDI [24] for the air) and experimentally determined values k_{_exp_avg} show for the performed tests reasonably low error of + 25% and −10%. Therefore, using the correlations for the estimation of the heat transfer in the sink HX with a similar design and similar conditions gives a fair error and thus is recommended. It is straightforward. Utilizing the measured data for look up tables for the HT of the sink HX is rather complicated to program.

• The analyzed correlations for heat transfer on the air side according to IPPE and VDI are in perfect match with each other.

• The sink HX heat exchanger configuration is able to remove planned 95 kW under design conditions, 7.8 MPa, 166 °C/33 °C, 0.325 kg/s (for the sCO₂ side) and 24 °C (design is 25 °C), 3.65 kg/s for the forced air flow with ambient pressure.

• Air-cooled finned-tube sink HX is suitable for the sCO₂-HeRo system.

• For a transient scenario—step-wise drop of ṁ_sCO₂ followed by loss of electric heating power, a Modelica code with newly implemented sink HX model was used. Simulation matches the measurement results well with mean deviations (ṁ_{_sCO2} 5%, V̇_{_air_out} 5%, T_{_sCO2_in} 2%, T_{_sCO2_out} 3%, p_{_sCO2_in} 3%, T_{_air_out} 3%).

Authors thank Johannes Brunnemann and Timm Hoppe from XRG Simulation who provided insight and expertise of Modelica/ClaRa and wish to acknowledge the help of Martina Fruhbauerova with the final editing and proof read.

• European Union's Horizon 2020 research and training/research and innovation programme (662116/No 764690).

• Ministry of Education, Youth and Sport Czech Republic – project LQ1603 Research for SUSEN.

A =

area, m²

c_p =

specific heat capacity, J·kg⁻¹·K⁻¹

d =

diameter, m

h =

enthalpy, J·kg⁻¹

h′ =

height of fin, m

H_flow =

enthalpy flow, W

K =

overall heat transfer coefficient, W·m⁻²·K⁻¹

L =

length, m

ṁ =

mass flow rate, kg/s

n =

number of fins of 1 tube

Nu =

Nusselt number

p =

pressure, Pa

P =

electric power, W

Pr =

Prandtl number

Q =

heat power, W

Re =

Reynolds number

s₁ =

pitch of tubes perpendicular to the air flow direction, m

s₂ =

pitch of tubes of HX above each other from the air flow sense, m

s₃ =

pitch of tubes behind each other (diagonal) from the air flow sense, m

T =

temperature, K

u =

gap between fins of 1 tube, m

V̇ =

volumetric flow rate, m³·s⁻¹

w =

velocity, m/s

Δp =

pressure drop, Pa

ΔT′ =

difference in temperatures of the mediums (air/sCO₂) within one segment of a heat exchanger, K

Greek Symbols

α =

coefficient of heat transfer, W·m⁻²·K^–1

β =

auxiliary variable to calculate an efficiency of a fin

δ =

thickness, m

ζ =

pressure drop coefficient

η =

dynamic viscosity, Pa·s

η_fin =

efficiency of a fin

λ =

thermal conductivity of a medium, W·m⁻¹·K⁻¹

ρ =

density, kg·m⁻³

σ_cp =

error propagation of specific heat capacity, J·kg⁻¹·K⁻¹

σ_h =

error propagation of enthalpy, J/kg

σ_m =

error propagation of mass flow rate, kg·s⁻¹

σ_Q =

error propagation of heat power transferred, W

σ_ρ =

error propagation of density, kg·m⁻³

σ_V̇ =

error propagation of volumetric flow rate, m³·s⁻¹

Subscipts

air =

air

adv =

advection

calc_avg =

calculated + averaged

cross =

cross section

e =

equivalent

exp_avg =

experimentally determined + averaged

fin =

fin of the heat exchanger

fric =

frictional

grav =

gravitational

h =

hydraulic

H_1/1, H_1/2, H₂, and H₃ =

heaters H_1/1, H_1/2, H₂, and H₃

Ideal =

ideal (e.g., α_ideal is coefficient heat transfer for η_fin = 1)

in =

inlet

inner =

inner side (of tube/HX)

out =

outlet

outer =

outer side (of tube/HX)

outer_tube_fin =

outer side among fins

sCO₂ =

supercritical CO₂

tube =

tube of the heat exchanger

Acronyms

CAD =

computer-aided design

CH₁ =

water cooler

CH₂ =

oil cooler

CVR =

Research Centre Rez

EC =

electronically communicated

GfS =

The Simulator Centre in Essen, Germany

H_1/1, H_1/2, H₂ and H₃ =

electric heaters

HT =

heat transfer

HTR =

high temperature regenerative heat exchanger

HX =

heat exchanger

IPPE =

Institute of Physics and Power Engineering

KKS =

identification system for power plants

LMTD =

logarithmic mean temperature difference

LTR =

low temperature regenerative heat exchanger

LWR =

light water reactor

MP =

main pump

MCP =

main circulation pump

NTU =

number of transfer unit

P&ID =

piping and installation diagram

PID =

proportional–integral–derivative

PWR =

pressurized water reactor

sCO₂ =

supercritical carbon dioxide

sCO₂-HeRo =

supercritical carbon dioxide heat removal system

SG =

steam generator

SS =

stainless steel

SUSEN =

Sustainable Energy project

TG =

turbine generator

VDI =

VDI - Heat Atlas

When a function (e.g., enthalpy) is a set of nonlinear combination of the variables, an interval propagation could be performed in order to compute intervals which contain all consistent values for the variables. In a probabilistic approach, the function (e.g., enthalpy) must usually be linearized by approximation to a first-order Taylor series expansion.

For the calculation of the sCO₂ enthalpy uncertainty at the inlet of the sink HX $σhsCO2−in$ four enthalpies were used. The first one $h_sCO2−inT_sCO2_in/p_sCO2_in_max$ was calculated with the measured sCO₂ inlet temperature T_{_sCO2_in} and the maximum possible inlet pressure p_{_sCO2_in_max} = p_{_sCO2_in} + 0.11 MPa, the second one $h_sCO2−inT_sCO2_in/p_sCO2_in_min$ with the measured sCO₂ inlet temperature T_{_sCO2_in} and the minimum possible inlet pressure p_{_sCO2_in_min} = p_{_sCO2_in} – 0.11 MPa, the third one $h_sCO2−inT_sCO2_in_max/p_sCO2_in$ with the measured sCO₂ inlet pressure p_{_sCO2_in} and the maximum possible inlet temperature T_{_sCO2_in_max} = T_{_sCO2_in} + 1.75 K and the fourth one $h_sCO2−inT_sCO2_in_min/p_sCO2_in$ with the measured sCO₂ inlet pressure p_{_sCO2_in} and the minimum possible inlet temperature T_{_sCO2_min} = T_{_sCO2_in} − 1.75 K. The propagated sCO₂ enthalpy uncertainty at the outlet of the sink HX $σhsCO2−out$ was calculated in the similar manner as for $σhsCO2−in$⁠.