The following assumptions and expectations for an ideal cycle simplify the analysis of complex gas cycles, such as closed Brayton cycles, to allow for an optimum operating condition with analysis to be evaluated [11] .

This study is largely focused on providing improved data on the estimation of sCO₂ components, to create a shift in the attention from cycle efficiency to techno-economics as the main driver toward commercialization of sCO₂ power generation. Hence the cycle was modelled based on the thermodynamic functionality of the individual components. The equations below defined how to determine the performance of the cycle, where the work of the compressor (W), re-compressor, turbine, and heater was defined by their thermodynamic correlations.is the mass flow rate.

Turbine $W_T={\stackrel{\dot{}}{m}}_{C{O}_2}\left(h_7-h_8\right)$ (1)

Main Compressor $W{M}_C={\stackrel{\dot{}}{m}}_{C{O}_2}X\left(h_2-h_1\right)$ (2)

Re-compressor $W{R}_C={\stackrel{\dot{}}{m}}_{C{O}_2}\left(1-X\right)\left(h_4-h_{12}\right)$ (3)

Heater $Q_H={\stackrel{\dot{}}{m}}_{C{O}_2}\left(h_7-h_6\right)$ (4)

The thermal efficiency ( ${\eta }_{th}$) of the cycle, in other words, the performance indicator of the cycle, is given by Equation (5). The cycle thermal efficiency was calculated from the turbine power, subtracting the compression power and dividing by the total heat duty input, multiplied by a hundred percent.

${\eta }_{th}=\left(\frac{W_{T-W{M}_C-W{R}_C}}{Q_H}\right)\times 100\%$ (5)

Imagine that the flow merges equation for mass conservation is ideal, which is mixed with the main flow fraction of recompression at constant pressure conditions. Then, the split fraction going into the main compressor can be expressed by Equation (6) [12] .

$X=\frac{m{\stackrel{\dot{}}{M}}_c}{m{\stackrel{\dot{}}{M}}_c+m{\stackrel{\dot{}}{R}}_c}$ (6)

But the constant flow fraction at constant pressure conditions for the main compression and recompression is given by Equation (7) where (1 − X) is the recompression fraction due to mass conservation.

$\left(1-X\right)h_4+X{h}_3=h_5$ (7)

Equations (8) and (9), on the other hand, can be expressed as the energy conservation equations in LTR and HTR, with the assumption that both recuperators have been insulated and can exchange heat only between the hot and cold streams [12] .

$\left(h_9-h_{10}\right)=X\left(h_3-h_2\right)$ (8)

$\left(h_8-h_9\right)=X\left(h_6-h_5\right)$ (9)

From Frank Incropera, in designing or estimating the performance of a heat exchanger, it is important to correlate the overall heat transfer coefficient with the heat transfer across the total surface area and the temperature variation. This can be expressed as in Equation (10), for HTR and LTR, respectively, where $\stackrel{\dot{}}{Q}$ is the heat transfer, UA is the overall heat transfer and $\Delta T{l}_m$ is the log mean temperature difference [13] .

$\stackrel{\dot{}}{Q}=\text{UA}\Delta T{l}_m$ (10)

$\Delta T{l}_m=\frac{\Delta T_1-\Delta T_2}{\ln \left(\frac{\Delta T_1}{\Delta T_2}\right)}$ (11)

where $\Delta T_1$ and $\Delta T_2$ refers to the endpoint temperature variation for counter flow heat exchangers which can be defined as in Equations (12) and (13).

$\Delta T_1=T_{hi}-T_{co}$ (12)

$\Delta T_2=T_{ho}-T_{ci}$ (13)

where $T_{hi}$ refers to the inlet temperature of the hot fluid, $T_{co}$ Refers to the inlet temperature of the cold fluid, $T_{ho}$ Refers to the outlet temperature of the hot fluid, $T_{ci}$ Refers to the outlet temperature of the cold fluid [13] .

In computing the cost of the various components used in the current study, cost functions proposed by [3] and [14] were adapted for the studied model.

Table 2 illustrates constant values applicable to the cost functions. $C_1$ and $C_2$ used in the cost function for the compressor and turbine are correction factors that will permit better cost estimation. These constant values are consistently used in the literature on sCO₂ components cost estimates.

Values for the intercooler, recuperator, and flue gas heat exchanger as proposed by [3] were multiplied by a thousand in this present study analysis to be able to compensate for any lost in the overall heat transfer (UA) value. It is assumed that these values will minimize the doubts in the cost estimation to about plus or minus 30%, subject to the maximum material temperature and pressure required.

The turbine was modeled using a general guideline method in ASPEN plus simulation with the isentropic efficiency and pressure ratio specification. These coefficients were assumed to increase the overall cycle efficiency. Equation (14) was used for the turbine cost estimation, and the formula was adapted from [3] .

Gas Turbine = $\frac{C_1\times A_{1}m\left(\frac{kg}{s}\right)\ln \left(PR\right)\times \left(1+{\exp }^{\left(0.036\times \left(T_{\max }-54.4\times C_2\right)\right)}\right)}{0.92-{\eta }_T}$ (14)

where $\stackrel{\dot{}}{m}$ is the mass flow rate (kg/s), $P_R$ is the pressure ratio, ${\eta }_T$is the isentropic efficiency of the turbine and $T_{\max }$ is the maximum temperature of the turbine.

Compressor: the compression unit has been designed using reheating, intercooling, and split recompression to account for high compressibility. Equation (15) was used for the cost estimation [3] .

Compressor = $\frac{{\text{C}}_1\times {\text{A}}_2\text{m}\left(\frac{\text{kg}}{\text{s}}\right)\times \text{PR}\ln \left(\text{PR}\right)}{0.9-{\text{η}}_{\text{C}}}$ (15)

Heat exchanger: The printed circuit heat exchanger was used for this model because they are effective in operation, less expensive, robust, and very compact, and can also withstand high pressure and temperature up to 60 MPa (600 bars) and 900˚C [15] .

Because the closed Brayton cycle depends on a great sum of internal heat recuperation to increase the cycle efficiency, the model used two recuperators, the LTR and HTR. However, following an investigation of the literature, the cost of the heat exchanger can contribute to about 30% or more of the entire cycle cost [16] . Hence, there was the need to compute the cost of the heat exchanger by linking that to its thermodynamic measured parameters because the performance of a heat exchanger is dependent on the overall heat transfer coefficient and the log mean temperature difference. Equation (16) was used for the cost estimation as proposed by [3] .

Recuperator $R_{HE}=A_4\times \text{UA}$ (16)

Heater: this acts as the central heating point of the working fluid, transferring the hot sCO₂ from the primary heating source to the turbine by further raising the sCO₂ pressure and temperature. The main parameter which defines its thermodynamic condition is the overall heat transfer coefficient. However, the overall heat transfer coefficient not only describes the thermal condition but also defines the size and capital cost [3] .

Heater $C_{HE}=A_5\times \text{UA}$ (17)

Intercooler: this serves as an intermediate cooling device between the LTR and the main compressor, its ultimate purpose is to cool sCO₂ exiting the LTR and entering the inlet of the main compressor. The magnitude of heat rejection will determine the thermodynamic operating point of the main compressor feed. Although a preview from the literature has indicated that the intercooler and heaters thus have the same cost function, this present study analyses the intercooler cost separately from the heaters using the cost function in Equation (18).

Intercooler $I_{HE}=A_3\times \text{UA}$ (18)

Generator: the cost estimation for the generator in this model is an inclusive cost of the following: electrical equipment and materials, piping, oil seals and bearings, gearing systems, oil lubricating system, balance of plant (BOP), instrumentation and controls. The cost was estimated using the net power produced by the constant introduced in Equation (19) as proposed by [3] .

Generator = $A_6\times \text{Net power}$ (19)

To accomplish the main goal of conducting a thermo-economics analysis, there was the need to investigate the capital expenditure of the study model. Nonetheless, taking into consideration some onsite and offsite factors, but in order to obtain the optimal output of an electricity generating facility to meet the system load at the lowest possible cost subject to transmission and operational constraints. An annual capital cost was predicted based on 20-year period assumption with a 10% discount rate annually, assuming the plant will run for 8000 hours annually. Table 3 indicates techno-economic assumptions applied to the present study as proposed by [3] .

Other important concern parts of a cost analysis of power cycles are costs associated with procuring or maintaining the life cycle of the plant, such as fixed and variable operation and maintenance costs and levelized cost of electricity. The capital recovery factor was calculated using Equation (20). [3]

Capital Recovery Factor (CRF).

This is expressed as a ratio of a constant annuity of the current value receiving that annuity for a given period, where n is the annuity. [3]

$\text{CRF}=i\times \frac{{\left(1+i\right)}^n}{{\left(1+i\right)}^n-1}$ (20)

Cost Rate (CR) $/S

The method used to calculate the cost rate of the present study in Equation (21), was based on the evaluation of the capacity recovery factor, which corresponds to the sum of the equipment purchase cost divided by the operating hours of the plant. The total cost rate was defined from the cost rate and variable operation and maintenance (O&M) using Equation (22) as proposed by [3] . However, the fuel cost rate (FR) expressed in Equation (22) has been assumed to be zero since the cycle is assumed to integrate into a pre-existing combustion cycle.

CR = $\frac{\text{CRF}\times \sum \text{purchased equipment cost}+\text{Fixed O}\&\text{M}}{\text{OPerating Hours}\times 3600}$ (21)

Total Cost Rate (TR) $/S

The total cost rate is the sum of the cost rate, fuel cost, and variable operation and maintenance.

TR = CR + FR + Variable O & M (22)

Cost of Electricity (COE) $/kWh

The COE was determined through Equation (23), as suggested by [3] . This equation was used to estimate the revenue expected by running the generator per net kWh for the first annual operation of the plant.

COE = $\frac{\text{TR}\times 3600\times {10}^5}{\text{Net Power}}$ (23)

Capacity Factor (CF)

The capacity factor of the plant has been expressed as a ratio of the plant’s actual output over a period of time to its potential output, assuming the plant could be operated at a maximum capacity [16] . The capacity factor was calculated using Equation (24); this is to allow estimation and comparison based on the impact of capital and variable operating hours.

CF = $\frac{\text{Energy generated}}{8760\times \text{Net power}}$ (24)

Levelized Cost of Electricity (LCOE) $/kWh

This is an economic assessment that determines the cost of electricity generated over the lifespan of the plant. In other words, this also permits the comparison of diverse technologies with different scales of operation, investment, and operating periods so that policymakers can make informed decisions [17] . Equation (25) has been used to calculate the LCOE for the present study as proposed by [3] .

LCOE $=\frac{\text{CRF}\times \text{MF}\times {\displaystyle \sum \text{PEC}}+\text{FR}+\text{Variable O}\&\text{M}}{8760\times \text{Capacity factor}}$ (25)

Operation and Maintenance Expenditure (OPEX) ($/kW-yr)

Operation and maintenance costs are the expenditures related to charges connected to operating and maintaining the plant over its expected lifespan. This cost is made up of fixed operation and maintenance, which is expressed over the annual hour, while variable operation and maintenance is expressed over the operating hours, as indicated in Equation (26).

OPEX = $\frac{\text{fixed O}\&\text{M}}{8760}+\frac{\text{variable O}\&\text{M}}{\text{Oh}}$ (26)

Profit Measure Analysis

Following the cost analysis, profitability measures were conducted to assess the prospects of the sCO₂ power cycle’s investments.

Return on Investment (ROI)

Return on investment was introduced as a performance measure to evaluate the amount of return on sCO₂ power cycle cost relative to the total investment expenditure. ROI was calculated based on the annual earnings over the total capital investment, as shown in Equation (27) proposed by [18] .

ROI = $\frac{\text{Annual earnings}}{\text{Total capital investment}}$ (27)

Net Present Value (NPV)

The net present value was expressed using Equation (28), as proposed by [19] . This is defined as the sum of the CAPEX and OPEX by the discount rate to the lifetime of the cycle.

NPV= CAPEX + $\underset{k=0}{\overset{n}{{\displaystyle \sum }}}\frac{OPEX}{{\left(1+r\right)}^n}$ (28)

Pay Back Period (PBP)

The duration to recover the cost of investment was calculated using Equation (29) as proposed by [19] . Where SR is the sales revenue. But sales revenue is calculated using Equation (30).

PBP = $\frac{TotalCapitalcost}{SR-\left(Operationcost\right)-\left(Tax\right)}$ (29)

The supercritical carbon dioxide (sCO₂) Brayton Cycle Power Plants offer promising advantages over traditional power cycles, including higher efficiency and reduced environmental impact. However, their widespread adoption is hindered by several operational challenges. Here are some of the main challenges:

Material Selection: Operating at high temperatures and pressures necessitates materials that can withstand these conditions without degrading. The availability of suitable materials is limited and often expensive [20] .

Corrosion and Oxidation: sCO₂ can be corrosive, especially at high temperatures, which can lead to material degradation. Developing materials or coatings resistant to these effects is a significant challenge [21] .

Compressor and Turbine Efficiency: Designing a turbomachinery that can efficiently handle the unique properties of sCO₂ is complex. Traditional designs often need significant modifications [22] .

Sealing and Leakage: Ensuring proper sealing is crucial because sCO₂ can easily leak through traditional seals, leading to efficiency losses and safety hazards.

Compact Heat Exchangers: The high density and thermal capacity of sCO₂ necessitate compact heat exchanger designs, which require advanced manufacturing techniques and materials to handle high pressure and thermal stresses [23] .

Fouling and Maintenance: Heat exchangers can suffer from fouling, reducing efficiency and increasing maintenance requirements. Cleaning and maintenance can be challenging due to the compact design [24] .

Dynamic Response: sCO₂ systems require precise control systems to manage rapid changes in pressure and temperature, ensuring safe and efficient operation [25] .

Integration with Other Systems: Integrating sCO₂ cycles with existing energy systems (like solar, nuclear, or fossil fuel) requires sophisticated control strategies to manage different energy inputs and outputs.

Thermal Stresses: The thermal stresses during startup and shutdown can be significant, leading to material fatigue and failure. Proper procedures and systems must be developed to minimize these stresses.

Transient Behavior: Understanding and controlling the system’s transient behavior during startup and shutdown is essential to prevent damage and inefficiencies.

Figure 4 shows a sensitivity study conducted to analyze the effect of engineering and supervision costs on the CAPEX. However, the results from the graph in Figure 4 reveal that when the percentage of engineering and supervision costs relative to the PEC increases, the sub-total cost for engineering and supervision also increases the CAPEX. At 10%, an amount of $4 million is added to the CAPEX as compared to $19 million at 40%.

Figure 5 shows a sensitivity evaluation conducted on the contingency cost. From the graph, it can be seen that when there is an increase in the percentage of contingency cost, the total indirect cost increases proportional to the sub-total of contingency cost. In this present study, contingency cost was estimated at 30% of the total direct cost. This corresponds to $22 million from the graph as indicated earlier in Table 5 . However, when the contingency cost is reduced to 10%, the sub-total cost for contingency becomes $7 million.

A sensitivity analysis was also carried out for land and civil works, as shown in Figure 6 . Results, however, reveal that incremental changes in land and civil works prices place marginal increments on the total direct cost. However, the CAPEX records a significant positive value due to an increase in the direct cost. In this study results, the total direct cost contributes to about 70% of the CAPEX.

Calculated results from the profitability measures were used for these sensitivity analyses. Key variables such as the discount rate in time, the CAPEX, and OPEX were the main determinants in this study evaluation. From the preliminary cost estimation, the annual cash flow is estimated to be $88 million with a projected NPV of $19 million for the estimated period and with an estimated payback period of 9 years. Figure 7 shows sensitivity results on Net Present Value (NPV). From the graph, it can be seen that the NPV reduces as the period increases; this is an indication that the investment will yield profitable income as time increases. The sensitivity result indicates that profitability with the NPV depends much on the assumptions made and the projected lifetime period.