Analysis of Supercritical CO₂ Cycle Using Zigzag Channel Pre-Cooler: A Design Optimization Study Based on Deep Neural Network

Abstract

The role of a pre-cooler is critical to the sCO₂-BC as it not only acts as a sink but also controls the conditions at the main compressor’s inlet that are vital to the cycle’s overall performance. Despite their prime importance, studies on the pre-cooler’s design are hard to find in the literature. This is partly due to the unavailability of data around the complex thermohydraulic characteristics linked with their operation close to the critical point. Henceforth, the current work deals with designing and optimizing pre-cooler by utilizing machine learning (ML), an in-house recuperator and pre-cooler design, an analysis code (RPDAC), and a cycle design point code (CDPC). Initially, data computed using 3D Reynolds averaged Navier-Stokes (RANS) equation is used to train the machine learning (ML) model based on the deep neural network (DNN) to predict Nusselt number ($Nu$) and friction factor ($f$). The trained ML model is then used in the pre-cooler design and optimization code (RPDAC) to generate various designs of the pre-cooler. Later, RPDAC was linked with the cycle design point code (CDPC) to understand the impact of various designs of the pre-cooler on the cycle’s performance. Finally, a multi-objective genetic algorithm was used to optimize the pre-cooler geometry in the environment of the power cycle. Results suggest that the trained ML model can approximate 99% of the data with 90% certainty in the pre-cooler’s operating regime. Cycle simulation results suggest that the cycle’s performance calculation can be misleading without considering the pre-cooler’s pumping power. Moreover, the optimization study indicates that the compressor’s inlet temperature ranging from 307.5 to 308.5 and pre-cooler channel’s Reynolds number ranging from 28,000 to 30,000 would be a good compromise between the cycle’s efficiency and the pre-cooler’s size.

1. Introduction

As part of the nationally determined contributions (NDCs) defined within the United Nations Framework Convention on Climate Change (UNFCCC), an ambitious approach from each country is required to decrease the world’s greenhouse gas emissions owing to their obligation towards the Paris Agreement. The major segment of these greenhouse gas emissions comes from the power generation and industrial sector, which comprises 69.8% of the total gas emissions worldwide. In view of this, a novel supercritical power cycle, namely the carbon dioxide Brayton cycle $\left({\mathrm{sCO}}_2-\mathrm{B}.\mathrm{C}.\right)$ coupled with phase-IV nuclear reactors could be of great value. The cycle with supercritical carbon dioxide $\left({\mathrm{sCO}}_2\right)$ as a working fluid combines the best features of both Rankine and air Brayton cycles. This cycle operates near the critical point of ${\mathrm{CO}}_2$ resulting in significantly higher thermal efficiency compared to aforementioned cycles [1] even at moderate values of turbine inlet temperature. Apart from this, the size of the components linked with the sCO₂-BC is up to 10 times smaller in comparison with the steam Rankin cycle [2]. Additionally, ${\mathrm{sCO}}_2-\mathrm{B}.\mathrm{C}.$ was accounted for as an ideal power block [3] for concentrated solar power plants ($CSP$) and is highly pertinent and cost effective for the next-generation CSP plants operating at temperatures above 600 °C [4] and the industrial waste heat recovery system [1]. Visibly, it is the future of the power cycle.

Recuperators and pre-coolers are the largest components of the${\mathrm{sCO}}_2-\mathrm{BC}$ and both can make or break the cycle’s performance. For ${\mathrm{sCO}}_2-\mathrm{BC}$, the printed circuit heat exchangers (PCHEs) are an obvious choice for recuperators as well as pre-coolers due to their compactness and ability to withstand high-pressure gradients [5,6,7,8,9]. Numerous investigations can be found in the literature to assess the heat transfer and flow characteristics linked with the zigzag channeled PCHEs functioning under the recuperator’s conditions [9,10]. These studies suggest that the thermal characteristics of zigzag channel geometries are far better than the straight channeled geometries of PCHEs [6]. On the other hand, limited information is available on zigzag channel PCHEs operating under the pre-cooler conditions. Baik et al. [11] performed both numerical and experimental evaluations to understand the performance of zigzag channels under the pre-cooler’s conditions and established heat transfer and pressure drop correlations. Cheng et al. [12] evaluated Nusselt number ($Nu$) and friction factor ($f$) correlations for the zigzag channeled pre-cooler, but the data reduction technique lacks in accommodating the sharp variation in the propterties of CO₂ in the pre-cooler’s regime. Munendra et al. [13,14,15] established a reduced-order model [16,17,18] to analyze the stability of supercritical CO₂. Saeed et al. [19,20] quantified local and overall profiles of Nusselt number ($Nu$) and friction factor ($f$) for the zigzag channeled pre-cooler. Based on their proposed new segmental averaged post processing method, they disclosed that the formerly implemented data reduction methods cannot be applied for the pre-cooler. Li et al. [18] reported that friction factor and Nusselt number correlations developed on the format of Dittus-Boelter and Jackson’s equations, for this operating regime poorly represents the actual data due to the limited degree of freedom linked with these equations. The issue can be resolved by adopting a procedure involving a higher number of degrees of freedom, such as machine learning models.

In the novel ${\mathrm{sCO}}_2-\mathrm{BC}$, the role of the pre-cooler is critical [1,19]. It serves as a sink and regulates the temperature and pressure settings at the compressor’s inlet. The compressor’s inlet temperature is envisioned to be kept close to the critical temperature of ${\mathrm{CO}}_2$ to accomplish superior cycle efficiencies [20]. At the same time, remarkably high values of the specific heat capacity of ${\mathrm{CO}}_2$ near its critical point (nearly forty times higher compared with water) require exceptionally high mass flow rates of the coolant (water) [21] to realize the anticipated exit temperatures of ${\mathrm{CO}}_2$. Subsequently, the power required to pump the coolant becomes high enough to disparage the cycle’s efficiency. This problem can only be mitigated by exploring new channel geometries with enhanced thermohydraulic characteristics. Moreover, ${\mathrm{sCO}}_2-\mathrm{BC}$ currently lacks investigations on the enhancement of the thermohydraulic performance of pre-coolers using efficient channel geometries. Currently installed ${\mathrm{sCO}}_2-\mathrm{BC}$ facilities, and available numerical work on the topic, focus on printed circuit heat exchangers (PCHEs) with straight channel geometries. However, it was reported [6,10] frequently in the literature that PCHEs with straight channels exhibit significantly low thermal and hydraulic characteristics compared with zigzag and airfoil channel-geometries. One of the reasons why pre-cooler’s designed with the aforementioned channel geometries were never explored was the unavailability of their thermal and hydraulic characteristics in the pre-cooler’s operating regime due to limited lab facilities and intricacies linked with swift variations in the thermophysical properties near the critical point. Given this, the proposed study focuses on the design and optimization of the pre-cooler geometry employing PCHEs with zigzag channels. To achieve this, an in-house recuperator and pre-cooler design and analysis code (RPDAC) is developed in the environment of the MATLAB©. The complex thermal and hydraulic characteristics linked to the zigzag channel geometries operating very close to the critical point are incorporated through a trained machine learning (ML) model. The ML model is based on the deep neural network (DNN) trained using the CFD data published by the same authors [5]. The in-house code is linked with the in-house cycle design point code (CDPC) to analyze various designs of the zigzag channeled pre-cooler on the performance of the recompression supercritical carbon dioxide Brayton cycle $\left(sC{O}_2-BC\right)$. Finally, the multi-objective genetic algorithm (MOGA) is used to optimize the pre-cooler’s geometry in the environment of the power cycle. Hence, the objective of the current study is to design and optimize the zigzag channeled pre-cooler by utilizing a deep neural network (DNN) and in-house codes, i.e., the recuperator and pre-cooler design and analysis code (RPDAC) in the environment of the sCO₂-BC by coupling the RPDAC with the cycle design point code (CDPC) and multi-objective genetic algorithm (MOGA).

2. Methodology

As mentioned above, the current work emphasizes the design finalization and optimization of the zigzag channeled pre-cooler in the complete power generation cycle. Various designs of pre-cooler are generated using an in-house recuperator and pre-cooler design and analysis code (RPDAC), as explained in Section 4 and Supplementary Material. Later, the impact on the performance of the supercritical carbon dioxide cycle (sCO₂-BC) is evaluated using cycle design point analysis code (CDPAC).

The flow and heat transfer characteristics linked with the zigzag channeled pre-cooler are extremely complex on account of its operation very close to the critical point of CO_2, where thermophysical properties of the hot-side working fluid (CO₂) change swiftly. Moreover, the periodic and sudden changes in the flow direction of the working fluid in the zigzag channel initiate boundary layer separation, reverse flow, and a dead flow region [5,9,22], further adding complexity and making the accurate prediction of pressure drop and heat transfer characteristics difficult. Furthermore, heat transfer and pressure correlation developed in this regime based on Dittus-Boelter and Jackson’s equations deviate considerably from the actual data [18] due to the limited degree of freedom linked with both formulations. Hence, for realistic designs of the pre-cooler, a machine learning (ML) model based on the deep neural network (DNN) is trained on the published CFD data [5] of the same authors and utilized in the above mentioned in-house recuperator and pre-cooler design and analysis code (RPDAC).

The design of the zigzag channeled pre-cooler is varied by changing design values of the effectiveness of the pre-cooler (${ϵ}_{pre}$), channel flow rate (${\dot{m}}_{ch}$), inlet temperatures of the pre-cooler and exit temperature of the pre-cooler. It is to be noted here that all design variables referred to above are varied only in the hot-side (CO₂-side) of the pre-cooler while corresponding quantities on the cold-side (waterside) were kept constant. A complete flow of the work adopted in the study is shown in Figure 1.

3. Machine Learning Model

The particulars of the ML model employed for the in-house recuperator and pre-cooler design and analysis code (RPDAC) are provided in the current section.

3.1. Training Data Details

As explained in the above section, the data used for the training of DNN is computed using 3D RANS simulations [5]. The training data consists of 2308 data points with five input variables and two output quantities, as displayed in

Table 1. Details of the input and output variables.

Input Parameters					Output Parameter
$\mathit{P}\mathit{r}$	$\mathit{R}\mathit{e}$	$\mathit{\rho }$	$\mathit{\mu }$	$\mathit{k}$	$\mathit{N}\mathit{u}$	$\mathit{f}$
1.917557	24,272.02	232.1863	2.11 × 10−5	0.035389	168.2138	0.053858
1.984166	24,263.8	235.3596	2.12 × 10−5	0.035989	172.7054	0.054062
2.049763	24,248.63	238.5383	2.13 × 10−5	0.036591	177.4821	0.054281
2.114218	24,226.66	241.7216	2.14 × 10−5	0.037194	182.5326	0.054516
2.177501	24,198.03	244.9103	2.15 × 10−5	0.037798	187.8461	0.054769
2.239603	24,162.91	248.1036	2.16 × 10−5	0.038401	193.4112	0.055041
2.300601	24,121.43	251.3023	2.18 × 10−5	0.039004	199.217	0.055333
2.360585	24,073.77	254.5055	2.19 × 10−5	0.039606	205.2523	0.055646

.

3.2. The Deep Neural Network

A deep neural network (DNN) was utilized for training the ML model as the construction and hyperparameters of DNN are found suitable for problems similar to those studied in the current work [23].

The construction of the DNN is such that it moves the information iteratively in forward and backward directions, as shown in Figure 2a. The Figure elucidates the construction of a deep neural network with two hidden layers and a layer each for five inputs and two outputs. All the hyperparameters linked with the DNN (number of the hidden layers, number of the neurons in each layer, activation function, and optimization function) were optimized to obtain minimum values of the root mean squared (RSM) error. The deep neural network (DNN) with two hidden layers consisting of four neurons in the first and second layers provided the minimum values of RSM error without overfitting the data where Levenberg–Marquardt (L.M.) is identified as the optimal training function. Several activation functions were tested where the rectifier linear unit (ReLU) is found appropriate for the input and the two hidden layers. At the same time, the sigmoid is identified as a suitable activation function for the output layer, as shown in Figure 2.

The input layer represents the neuron with the normalized data. While the data from $i^{th}$ layer to $i+1^{th}$ move as described by Equation (1).

$${\mathrm{a}}_q^{i\to i+1}={\mathrm{f}}_q^i\left({\displaystyle \sum }_{p=1}^{N^i}{\mathrm{w}}_{qp}^i{\mathrm{L}}_p^i+{\mathrm{b}}_q^i\right)$$

(1)

where the term q varies from 1 to $N^{i+1}$.

The above Equation (1) can be expressed in the vector format as given by Equation (2).

$${\mathrm{a}}^{\mathrm{i}+1}={\mathrm{f}}^i\left({\mathrm{w}}^i{\mathrm{a}}^i-1+{\mathrm{b}}^{i+1}\right)$$

(2)

3.3. DNN Optimization Methodology

Implementing a deep neural network (DNN) learning process is known as an optimizer or optimization algorithm. Among various optimizers available for DNN, the Levenberg–Marquardt algorithm is utilized for the current work.

The learning process of the Levenberg–Marquardt algorithm is described by Equation (3), and Figure 3.

$$w^{\left(i+1\right)}=w^{\left(i\right)}-{\left({\mathrm{J}}^{\left(i\right)T}·{\mathrm{J}}^{\left(i\right)T}+{\mathsf{\lambda }}^{\mathrm{i}}\mathrm{I}\right)}^{-1}·2{\mathrm{J}}^{\left(i\right)T}·{\mathrm{e}}^{\left(i\right)})$$

(3)

In the above expression, the term $\mathsf{\lambda }$ is a damping factor while J represents the Jacobian matrix.

The Jacobian matrix is defined by Equation (4).

$${\mathrm{J}}_{i,j}=\frac{\partial e_i}{\partial {\mathrm{w}}_{\mathrm{j}}}\mathrm{for}i=1,\dots ,m;\mathrm{and}j=1,\dots ,n$$

(4)

4. Recuperator and Pre-Cooler Design and Analysis Code (RPDAC)

An in-house recuperator and pre-cooler design and analysis code (RPDAC) is developed and utilized to design and simulate the recuperators and pre-cooler of the sCO₂-BC.

The model works on the following assumptions:

• The recuperators and pre-coolers are designed as zigzag channeled PCHEs, as indicated in Figure 4. Figure 1 lists the details and specifications of the zigzag channel used for the current work;
• Pressure losses in both headers are ignored;
• Uniform flow distribution is assumed among all channels;
• External walls of the PCHEs are considered adiabatic;
• It can be deducted from the assumptions made above; there are no temperature gradients normal to the flow directions.

4.1. Properties of CO₂

The sharp variation in the properties of CO₂ (Figure 5a) demonstrates that the precise implementation of the properties is mandatory for the realistic designs of the recuperator and pre-cooler in RPDAC. In this reference, NIST’s REFPROP [24,25] is coupled with the RPDAC through a subroutine.

4.2. Mathematical Model

The in-house recuperator and pre-cooler design and analysis code (RPDAC) is developed on a discretized model utilizing the log mean temperature difference (LMTD) method shown in Figure 5. Further details of the discretization can be found in the author’s previous work [6,26].

The calculation in the RPDAC is carried out in an interactive method consisting of several steps. The procedure adopted for these calculations is shown in Figure 6.

Step 1: The RPDAC is initialized with the given input parameters, i.e., $h_{\mathrm{in}}^{\mathrm{h}},P_{\mathrm{in}}^{\mathrm{h}},h_{\mathrm{in}}^{\mathrm{c}},P_{\mathrm{in}}^{\mathrm{c}},{\dot{m}}^{\mathrm{c}},{\dot{m}}^{\mathrm{h}},ϵ,{\mathrm{N}}_{\mathrm{nodes}},d\mathrm{x}$.

Step 2: The iterative process begins with the estimation of $\mathsf{\Delta }{P}^c$and $\mathsf{\Delta }{P}^h$. Now with known values of pressure at both outlets, temperature values at the exit of both are computed using the definition of the effectiveness ($ϵ)$, as demonstrated in Figure 6.

Step 3: As indicated in the Figure, the whole discretized domain can be initialized, presuming a linear variation of the specific enthalpy and pressure drop along the pre-cooler’s length.

Step 4: The cell-centered quantities are used to compute the flow and heat transfer characteristics. The cell-centered values are computed using the arithmetic mean of the cell face node values.

Step 5: The cell-centered values for enthalpy and pressure are used to compute the thermophysical properties linked with the former.

Step 6: This step utilized the trained deep neural network to compute update values of pressure drop $(d{P}_{\mathrm{new}}$), and heat flux $\left(d{q}_{\mathrm{new}}\right)$, as displayed in Figure 6.

$${\left(\mathrm{U}.\mathrm{A}.\right)}^i=\frac{1}{\frac{1}{{\left({\mathrm{htc}}_{\mathrm{local}}^{\mathrm{h}}\right)}^i}+\frac{\mathrm{k}}{\mathrm{t}}+\frac{1}{{\left({\mathrm{htc}}_{\mathrm{local}}^{\mathrm{c}}\right)}^i}}$$

(5)

$${\left(d{q}_{new}\right)}^i=A_{ht}^c{\left(UA\right)}^i{\left(\mathrm{LMTD}\right)}^i$$

(6)

where the definition of ${\left(\mathrm{LMTD}\right)}^i$ is given by Equation (3).

$$\left\{\begin{array}{c}{\left({\theta }_2\right)}^{\mathrm{i}}={\left(T^{\mathrm{h}}\right)}^i-{\left(T^{\mathrm{c}}\right)}^i\\ {\left({\theta }_1\right)}^{\mathrm{i}}={\left(T^{\mathrm{h}}\right)}^{i+1}-{\left(T^{\mathrm{c}}\right)}^{i+1}\\ {\left(\mathrm{LMTD}\right)}^{\mathrm{i}}=\frac{\max \left({\left({\theta }_1\right)}^i,{\left({\theta }_2\right)}^{\mathrm{i}}\right)-\min \left({\left({\theta }_1\right)}^i,{\left({\theta }_2\right)}^i\right)}{\log \left(\frac{\max \left({\left({\theta }_1\right)}^i,{\left({\theta }_2\right)}^i\right)}{\min \left({\left({\theta }_1\right)}^i,{\left({\theta }_2\right)}^i\right)}\right)}\end{array}\right.$$

(7)

Step 7: The updated values of pressure drop ($d{P}_{\mathrm{new}}$) are determined in step 6, in conjunction with the pressure drop values calculated in the previous iterations.

Step 8: This step moves the calculation from ith node to $i^{th}$ to ${\left(i+1\right)}^{th}$ node, if the convergence criteria displayed in Figure 6 are fulfilled.

Step 9: This step controls the intermediate loop by putting a check on the exit conditions.

Step 10: This step monitors the outermost loop that corrects the required exit conditions.

5. In-House Cycle Design Point Code (CDPC)

Power cycle analysis falls into three groups, i.e., (1) cycle design point analysis (CDPA), (2) cycle off-design analysis (CODA), and (3) transient cycle analysis (TCA) [1]. CDPA is conducted to evaluate the conditions across each cycle’s components and its loads in the current study. The data ise later used for the design of corresponding components by the manufacturers. For the cycle design point analysis, the in-house cycle design point code (CDPC) is utilized to study various designs of the pre-cooler for a $10M{W}_e$ recompression supercritical carbon dioxide Brayton cycle ($sC{O}_2-BC$) as discussed above. The schematic of the recompression $sC{O}_2-BC$ is shown in Figure 7a along with its T.S. diagram and variation of the Prandtl number (Pr) in various cycle components (Figure 7b). The data presented in Figure 7b is computed using CODC using the boundary conditions listed in

Table 2. Boundary conditions for the cycle design point code (CDPC).

Inlet temperature, main compressor	$T_1^{cyc}$	Pre-cooler’s exit temperature governed by RPDAC
Inlet pressure, main compressor	$P_1^{cyc}$	Pre-cooler’s exit pressure governed by RPDAC
Upper pressure	$P_2^{cyc}$	25 MPa
Inlet temperature, turbine	$T_7^{cyc}$	873 K
Effectiveness, HTR	${\epsilon }_{HTR}^{cyc}$	0.90
Effectiveness, LTR	${\epsilon }_{LTR}^{cyc}$	0.90
Effectiveness, Pre	${\epsilon }_{pre}^{cyc}$	governed by RPDAC based on $T_1^{cyc}$
Efficiency, turbine	${\eta }_T$	0.90
Efficiency, compressor	${\eta }_C$	0.85
Split mass fraction	$\mathrm{x}$	Corresponding to maximum efficiency

. The variation of the Pr in various cycle components is shown to compare the variation in the thermophysical properties of working in those components. It can be observed by looking at the data of Figure 7b that variation in the properties of CO₂ (Prandtl number) in the pre-cooler is substantially higher in comparison with corresponding changes in the HTR and LTR.

The numerical model for the $sC{O}_2-BC$mainly comprises the sub-models for turbomachinery (compressor and turbine) and heat exchanger models. The recompression $sC{O}_2-BC$ consists of four heat exchanger intermediate heat exchangers, a high-temperature recuperator (HTR), a low-temperature recuperator (LTR), and a pre-cooler. Turbomachinery of the cycle is modeled using turbomachinery models (TMs) based on the fixed isentropic efficiencies Equations (8)–(10) provide. On the other hand, detailed models for the heat exchanger are required for the current for accurate calculations [6,27]. Therefore, to model HTR, LTR, and the pre-cooler, an in-house recuperator and pre-cooler design and analysis code (RPDAC) is developed and used in the current work. The details of the RPDAC are provided in Section 4.

The flow diagram of the CDPC is displayed in Figure 8, while the cycle boundary conditions are listed in Table 1. The governing equations for each component of the cycles are described in this section, and the detailed models of heat exchangers are in the following section. Turbomachinery of the cycle is modeled using turbomachinery models (TMs) based on the isentropic efficiencies that Equations (8)–(10) provide.

$${\eta }_{MC}=\frac{h_{2s}^{cyc}-h_1^{cyc}}{h_2^c-h_1^c}$$

(8)

$${\eta }_{RC}=\frac{h_{5s}^{cyc}-h_{10}^{cyc}}{h_5-h_{10}}$$

(9)

$${\eta }_T=\frac{h_7^{cyc}-h_8^{cyc}}{h_7^{cyc}-h_{8s}^{cyc}}$$

(10)

Split fraction $“\mathrm{x}\text{”}$ is calculated employing Equation (11).

$$\mathrm{x}=\frac{{\dot{m}}_{MC}}{{\dot{m}}_{MC}+{\dot{m}}_{RC}}$$

(11)

Since ideal mixing is assumed

$$h_4^{cyc}=\mathrm{x}{h}_3^{cyc}+\left(1-\mathrm{x}\right)h_5^{cyc}$$

(12)

6. Optimization of Pre-Cooler Geometry

The geometry of the pre-cooler is optimized in the atmosphere of the power cycle employing cycle efficiency and the volume of the pre-cooler as two objective functions for the existing problem where the former is maximized and the latter is minimized. The optimization algorithm employed for the optimization study is a multiobjective genetic algorithm [28,29,30,31,32] in the environment of Matlab©. The CDPC embedded with RPDAC is utilized as a fitness function. In contrast, design variables studied for the present optimization study are exit temperature of the pre-cooler, channel’s Reynolds number on the hot side, and split mass fraction (x). It is to be noted here that the value of the split mass fraction is not a design parameter for the pre-cooler; however, the optimal value of x would change with the changing design of the pre-cooler. Therefore, the inclusion of x as a member of the design variables would result in the realistic optimization of the pre-cooler, as discussed in the author’s previous studies [20,27,33,34,35].

The schematic of the methodology opted for the employed optimization problem is displayed in Figure 9. The list of design variables with their upper and lower bounds are listed in

Table 3. Upper and lower bounds for design variables.

	${\mathit{T}}_1$	$\mathit{R}{\mathit{e}}_{\mathit{i}\mathit{n}}$	x
Lower bound	305	10,000	0.5
Upper bound	310	40,000	0.99

. The procedure begins by random initialization of a defined population size of 100 combinations of the input variables. The generated population is then examined by the fitness function (CDPC with embedded RPDAC) for the defined objective function (cycle efficiency and volume of the pre-cooler). The top 5% of the population are categorized as elite and are passed to the next generation, while the remaining 95% of the new populations are generated through crossover and mutation. The procedure persists until the stopping criteria (shown in Figure 9) are satisfied. Further aspects of the G.A. are available in the literature [36].

7. Results

7.1. Training of the ANN Neural Network

In the current section, the training results of the ML model are elaborated. The Levenberg–Marquardt (L.M.) optimizer is utilized for training the data. For the present study, two hidden layers are utilized with four neurons in each layer. The data set used for the current study is divided into three subsets, i.e., training validation and testing data sets. The training data set consists of 70% of the total data, while the remaining 30% is equally divided into validation and testing data sets. Figure 10 shows the mean squared error (MSE) values during the training process. The Figure shows that training of the data required only 58 epoch to achieve minimum mean squared error value. Further, it is obvious from the Figure 10 that the converged mean squared error values for the training, validation, and test set data are almost identical. This shows that the trained DNN model is not overfitting the data. It is to be noted here that two DNN models were trained for the Nu and the second for f. Further 10-fold cross-validation was used while training the ML model.

Figure 11 shows regression data for the friction factor, providing a comparison between the target and output values of training, validation, and testing data sets. Figure 11 shows R-values for the training data, validation, test, and the whole data set are 0.988, 0.989, 0.987, and 0.988, respectively. This shows that the trained model can predict 98.8% of the variability in the output data for the combined data set containing all data from training, validation, and test sets. At the same time, the capacity of the trained data ML model to estimate individual percentage variability for training, validation, and test data set is 98.8%, 98.9%, and 98.7%, respectively. At the same time, the R-value for the validation training and test data sets are almost identical; therefore, it can be inferred that the trained DNN model is not overfitting the data.

Figure 12 shows regression data for the Nusselt number values, supplying a comparison among the target and output values of training, validation, and testing data sets. It can be observed from Figure 12 that R-values for all data sets, i.e., training data, validation, test, and the whole data set, is 99.9. This signifies that the trained model can predict 99.9% of the variability in the output data for the combined data set containing all data from training, validation, and test sets. Similarly to the friction factor data, the R-value for the Nusselt number’s validation training and test data sets are almost identical. The trained DNN model is not overfitting the data.

7.2. Validation of the Recuperator and Pre-Cooler Design and Analysis Code

It is shown in the above section that the trained ML model is extremely accurate in predicting the complex heat transfer and pressure drop characteristics in the zigzag channel even when operating very close to the critical. Thus, the trained ML model is utilized in the recuperator and pre-cooler design and analysis code (RPDAC) to compute the realistic and accurate design of the recuperator and pre-cooler, as discussed in Section 4. To validate the in-house RPDAC data obtained from the published CFD calculation [5] was compared with the RPDAC calculation under the same operating condition. The comparison of RPDAC and CFD Nusslet number distribution along the channel length data is displayed in Figure 13. The comparison shows that RPDAC data is quite in agreement with the computed data using the 3D-RANS model. Further, the 3D-RANS model computation for a single simulation needs up to 10 to 26 on a high-performance computational (HPC) facility [5]. At the same time, RPDAC based on the DNN model can imitate the same simulation in 2 to 10 min only.

7.3. Performance of the Pre-Cooler in the Environment of the Power Cycle Calculations

The validated recuperator and pre-cooler design and analysis code is finally linked with the cycle design point code to model recuperators and pre-cooler of the recompression Brayton cycle, as explained in Section 5 and shown in Figure 8. It is to be noted here that the code results present in this section and reported in the author’s previous work [5,26] would differ entirely as the latter was conducted using a fixed length of pre-cooler and imposed certain conditions at the inlet. On the contrary, results presented in the current section are computed by simulating the RPDAC in the environment of the power cycle where the conditions at the inlet and exit of the pre-cooler are dictated by cycle calculation.

Figure 14 shows the pressure drop in the hot side of the pre-cooler at various imposed temperature conditions at the inlet and outlet of the pre-cooler by the cycle design point code. In Figure 14, the compressor’s inlet temperature is plotted on the graph’s horizontal axis while the inlet temperature on the CO₂ sides and pressure drop in the pre-cooler are plotted on the right and left vertical axis. Figure 14 suggests that the inlet temperature of the pre-cooler increases with the increasing inlet temperature of the compressor. The inlet temperature of the pre-cooler is governed by the exit temperature of the low-temperature recuperator (LTR). The LTR exit temperature increases as it is linked with the exit temperature of the main compressor and split mass fraction. As the main compressor inlet temperature increases, its exit temperature and split mass fraction both increase as a result of an increase in the values of the inlet t temperature in the cold side of LTR; the exit temperature of the hot side of LTR (inlet temperature of the pre-cooler) increases. It can be observed that the inlet temperature of the pre-cooler increases substantially, i.e., 347 K to 396 K, by the increase in inlet temperature of the compressor from 305 K to 310 K. Because the inlet temperature of the compressor is the exit temperature of the pre-cooler, the difference along the pre-cooler would increase with the increase in the value of the compressor inlet temperature. At the same time, the pressure drop of the values corresponding to all values of the Reynolds number specified at the inlet of the pre-cooler decreases significantly with the increase in the value of the compressor’s inlet temperature. This is because of the high specific heat capacity (Cp) values near the critical point as discussed in the author’s previous work [5] and the smaller computed length of the pre-cooler are explained below. Further, it can be observed from the Figure that pressure drop increases substantially with the increase in the design value of the inlet Reynolds number, and the effect is more prominent at lower values of the compressor’s inlet temperature. On the other hand, the inlet temperature of the pre-cooler is not affected by the design value of the channel Reynolds number.

Figure 15 reflects pressure drop in the cold side of the pre-cooler and pumping power at various imposed temperature conditions at the inlet and outlet of the pre-cooler by the cycle design point code. In Figure 15, the compressor’s inlet temperature is plotted on the graph’s horizontal axis, while the length of the pre-cooler and the volume of the pre-cooler are plotted on the right and left vertical axis, respectively. It can be observed that pressure drop on the cold side, and consequently, the pumping power of the water increases significantly with the disease in the compressor’s inlet temperature. The increase in the pumping power is due to the increase in the length of the pre-cooler and decrease the compressor’s inlet temperature, as shown in Figure 15. A detailed discussion on the increase in the length of the pre-cooler by decreasing the compressor’s inlet temperature can be found in the author’s previous work [5,9]. Interestingly, despite the constant flow rate maintained on the cold side, pressure drop in the cold side increases substantially by increasing the design value of the channel’s Reynolds number (channel’s mass flow rate) on the hot side. The increase in the pressure drop on the cold side can be explained based on the fact that computed values of the length of the channel increase with the increase in the channels’ Reynolds number on the hot side, as is obvious from the data displayed in Figure 16.

On the other hand, the water’s pumping power in relation to the channel’s Reynolds number on the hot side is tricky, to some extent. It can be observed that at higher compressor’s inlet temperatures, the pumping power corresponding to a higher channel Reynolds number in the hot side is less than the pumping power linked with the pre-cooler design, with lower design values of the Reynolds number on the hot side. This is because the length of the channels at a higher compressors’ inlet temperature has differed slightly. The effect of the channel length on the pumping power (discussed above) is suppressed by the dominant effect caused by the conditions at the exit pre-cooler. This means that at a higher compressor’s inlet temperature, the change in the lengths of the pre-cooler with the Reynolds number is too small to affect the pumping power. It is to be noted here that the higher inlet temperature of the compressor is referred to as the inlet temperature values of the compressor that are higher than pseudo-critical temperature, i.e., $T_1^{cyc}>308\mathrm{K}$. On the other hand, the channels’ length changes substantially with the channel Reynolds number on the hot side at compressor’s inlet values lower than 308 K. In this situation, the influence of the channel length becomes dominant, as is obvious from the data displayed in Figure 16.

Data of the pre-cooler’s length and volumes displayed in Figure 16 suggest a similar trend as discussed for the pre-cooler’s pumping water. The length of the channels increases by decreasing the compressor’s inlet temperature and increasing the channels Reynolds number. At the same time, the channels’ volume decreases by increasing the channels Reynolds number for the compressor’s inlet temperature values smaller than 308 K. This can be explained using the data for the number of channels required displayed in Figure 17. It is clear from Figure 17 that the number of channels decreases by increasing the channels Reynolds number.

It is important to know that the pre-cooler’s design considered in the present study is built on the cycle running at its design point (D.P.). As the cycle is working at D.P., the cycle’s mass flow rate and the split mass fraction (x) (the fraction of the flow going to the main compressor and the re-compressor) [20,37] are fixed. Bearing in mind that the total mass flow rate through the heat exchanger would be fixed as the cycle is operating at its D.P., the flow rate through each channel can be altered by changing the number of flow channels (the cross-section of the precoder). Therefore, the Reynolds number referred to in the current study is the channel Reynolds number that corresponds to the flow rate of the working fluid through a single channel. In this context, the length of the channels increases with the increase in the values of the channel’s Reynolds number. Consequently, the required number of channels and cross-section of the heat exchanger will decrease. Further, it is to be noted here that the size of the heat exchanger is referred to as the volume of the heat exchange that is multiple times that of the heat exchanger’s length and crossectional area.

This is because the total flow rate through the pre-cooler is fixed; therefore, when the channel Reynolds number is increased by increasing the channel flow rate, fewer channels are required.

Figure 18 shows the variation in the computed values of the main compressor work and specific work of the cycle by CDPC corresponding to various designs of the pre-cooler. The values of specific work displayed in the Figure do not include pumping work. The data displayed in the graph shows that the main compressor work and specific work changes significantly by changing the compressor inlet temperature; however, change in the channels Reynolds number of the pre-cooler has no significant impact on the parameters mentioned above. At the same time, Figure 19 shows the cycle’s efficiency with and without considering the pumping power of the pre-cooler corresponding to its various designs for the pre-cooler. In the literature, previous studies have ignored the effect of pumping power in the simulations [20,22,27,33,38,39,40]; however, the data in Figure 19 indicates that pre-cooler’s work greatly impacts the cycle’s efficiency as there is a substantial difference in the efficacy values with and without considering the pumping power.

Figure 20 compares the cycle’s efficiency and the size of the pre-cooler corresponding to various designs of the pre-cooler. Data in the Figure reveals that, when decreasing the compressor’s inlet temperature, both cycle’s efficiency and pre-cooler’s sizes increase swiftly until $T_1^{cyc}<307 \mathrm{K}$. However, the cycle’s efficacy becomes constant or declines on further decreasing the compressor’s inlet temperature. The decline in the cycle efficiency on decreasing the compressor’s inlet temperature contradicts the previously published studies [20] that ignore the pre-coolers’ work while analyzing the cycle’s performance. On the other hand, current work suggests that pre-coolers’ pumping power becomes extremely high which, in turn, decreases the cycle’s efficiency if the compressor inlet temperature has to be maintained below 307 K. On the other hand, the values of the pre-cooler’s size keeps on increasing with the increase in the compressor’s inlet temperature.

As it is discussed above, the increase in the cycle’s efficiency comes at the cost of the increase in the size of the pre-cooler. This calls for optimization of the pre-cooler geometry to achieve maximum efficiency while keeping the size of the pre-cooler at a minimum level. In this reference, an optimization study is conducted, details of which are available in Section 6. The results of the optimization study are displayed in Figure 21 and listed in Appendix A. Figure 21 shows Pareto fronts for the cycle’s efficiency and heat exchanger’s volume. Every point of the Pareto front is an optimized solution for a pre-cooler’s design. It corresponds to the maximum value of the efficiency that can be achieved employing that design. In this case, if one wants to improve one of the two parameters, the improvement would come by sacrificing the other. Figure 21 shows that the maximum value of the cycle’s efficiency (54%) corresponds to an inlet temperature of 306.2 K and a pre-cooler’s channel Reynolds number of 11,200. The maximum cycle efficiency corresponds to maximum pre-cooler size, i.e., $0.44 {\mathrm{m}}^3$. However, the optimal size of the pre-cooler can be decreased by compromising the cycle’s efficiency. The optimal value of the minimum size of the pre-cooler, i.e., 0.24, can be achieved by designing the pre-cooler for a comparatively higher compressor’s inlet temperature (T1 = 310 K) and the design value of the inlet Reynolds number (Re = 33,000) corresponds to ${\eta }_{cyc}^{WP}=50$. A good compromise between the cycle’s efficiency and size can be found in the central part of the Pareto front encircled (${\mathrm{Hx}}_{\mathrm{vol}}$ ranging from 0.28 to 0.30 with corresponding cycle efficiency ranging from 53.2% to 53.5%) in Figure 21. Moving in left or right from this location to improve any parameter (cycle’s efficiency or pre-cooler size) requires a sufficiently large compromise of the second parameter.

8. Conclusions

An optimization study for the design of a pre-cooler working in a sCO₂-BC was conducted, and the following deductions were made based on the results discussed above:

• Cycle simulations using various pre-cooler designs suggest that pre-cooler should only be studied as an embedded component of the whole cycle system. Standalone design calculations of the pre-cooler can lead to unrealistic results;
• The cycle simulation findings show that the pre-cooler’s design significantly influences both the cycle’s performance and its layout size. If the compressor’s inlet temperature is maintained below 308 K (T₁ < 308 K), as suggested by the literature, the pumping power and the size of the pre-cooler rise considerably and, in turn, the cycle’s specific work and efficiency deteriorate noticeably;
• Previous studies that ignore the pre-cooler’s pumping power in the cycle’s calculations suggest that the closer the inlet temperature (of the main compressor) to the critical point of CO_2, the higher the cycle’s efficiency. However, the current optimization study suggests that an inlet temperature T₁ ranging from 307.5 K to 308.5 K and channel Reynolds number ranging from 2800 to 30,000 would be a good compromise between the cycle’s efficiency and the pre-cooler’s size;
• The machine learning results suggest that a trained ML model based on the deep neural network (DNN) can estimate 99% of the Nusselt number and friction factor data with 90% confidence. Therefore, trained ML models based on the artificial neural network (ANN) that are capable of including numerous degrees of freedom for intricate data structures are well suited for thermohydraulic predictions in the pre-cooler’s operating regime.