## Abstract

Concentrated solar power (CSP) with thermal energy storage (TES) has the potential to achieve grid parity. This can be realized by operating CSP systems at temperatures above 700 °C with high-efficiency sCO_{2} power cycles. However, operating CSP systems at such temperatures poses several challenges, among which the design of solar receivers to accommodate increased thermal loads is critical. To this end, this work explores and optimizes various swirl-inducing internal fin designs for solar receiver tubes. These fin designs not only improve the thermal performance of receiver tubes but also levelize temperature unevenness caused by non-uniform thermal loading. In this work, the geometric parameters of the fin designs are optimized to maximize the Nusselt number with a constraint on the friction factor. This optimization, however, is computationally intensive, requiring hundreds of simulation calls to computational fluid dynamics (CFD) models. To circumvent this problem, surrogate models are used to approximate the simulation outputs needed during the optimization. In addition, this study also examines the fin designs from an entropy generation perspective. To this end, the entropy contributions from thermal and viscous effects are quantitatively compared while varying the operational Reynolds number.

## 1 Introduction

Concentrated solar power (CSP) with thermal energy storage (TES) has emerged as a promising technology for harvesting solar energy. Its ability to store and dispatch heat when needed sets it apart from other renewable energy technologies, such as photovoltaics and wind. Additionally, CSP systems can be easily hybridized with biogas or conventional fuel burners to compensate for irregular solar radiation and to increase power production during times of peak demand [1]. CSP with TES has several advantages over other types of energy storage such as batteries. TES systems, for instance, can store energy at every higher temperature—resulting in higher energy densities and lower costs [2]. This makes CSP with TES ideal for providing baseload power and for supporting grid stability. Moreover, the energy stored in TES systems is in the form of heat—making it particularly attractive for industrial heating.

Solar power towers (SPTs), which use heliostats to focus sunlight onto a central receiver, have gained popularity in recent years because of their potential for high efficiency at temperatures above 700 °C [3,4]. However, uneven solar flux loading, especially at high concentration factors required to reach these temperatures, can lead to large temperature gradients and hot spots on the receivers. Moreover, the low thermal conductivity of molten chloride salt (the candidate heat transfer fluid (HTF) for next-generation SPT plants [4–6]) can result in inadequate heat transfer within the receivers, further exacerbating these concerns.

These thermal issues pose serious safety and reliability challenges for operating such high-temperature CSP systems [7]. For instance, large circumferential temperature gradients can lead to uneven thermal expansion and cause stress fractures/distortions in the receivers. Hot spots, on the other hand, can damage the receiver’s optically selective coating and also act as seed sites for corrosion. To address these challenges, several strategies have been proposed in the literature, including new HTF flow layouts and heliostat aiming methods [7].

A potential solution to these problems is to enhance the convective heat transfer within the SPT receiver tubes using fins [3,7]. This will lower the receiver’s surface temperature, which in turn reduces the heat loss to the ambient air and improves the receiver’s efficiency. These improvements will reduce the length of receiver tubes and make the receiver section compact. While numerous fin designs exist, this application particularly benefits from swirl-inducing helical fins [8]. These fins not only improve the structural and thermal performance of the receivers but also ease temperature unevenness caused by non-uniform solar irradiation. In this work, various helical/twisted fin designs are explored, and their geometric parameters such as height, helical pitch, and number of heads are optimized for thermo-hydraulic performance. These helical fin structures promote fluid mixing and improve heat transfer, but generally at the expense of increased pressure loss. For this reason, the proposed design optimization formulation involves maximization of Nusselt number subjected to a constraint on friction factor.

Although the literature on passive heat transfer enhancement is fairly comprehensive, much of it focuses on characterizing thermo-hydraulic performance without much emphasis on optimizing fin design. This is partly because fin design optimization often requires hundreds, if not thousands, of calls to expensive computational fluid dynamics (CFD) models, and as such employing direct optimization techniques is simply intractable. To circumvent this problem, this work uses Gaussian processes (GPs) as surrogate models to assist the optimization process [9,10]. These surrogate models, once trained on a few evaluations of the CFD model, can approximate the quantities of interest needed during the optimization. Consequently, these inexpensive-to-evaluate surrogate models can be used as proxies instead of CFD models during the optimization.

Though not the primary focus of this manuscript, this research ultimately seeks to manufacture optimally enhanced receiver tubes using additive manufacturing (AM). A key advantage of AM is its ability to create intricate fin geometries that are challenging or outright impossible to achieve using conventional manufacturing methods. AM, in addition, allows for manufacturing these fined receiver tubes as a single component, eliminating the need for welding and reducing the likelihood of material failure.

The paper is organized as follows: Sec. 2 presents the parameterized model of an internally finned receiver tube and its boundary conditions for CFD analysis. Section 3 provides a brief overview of the governing equations and turbulence closure model used in this study. Section 4 provides details of meshing and the numerical schemes used in this work. In Sec. 5, a brief description of thermo-hydraulic performance metrics is provided. Section 6 explains the fin design optimization formulation and the surrogate-assisted method. A parametric analysis of the candidate fin designs is presented in Sec. 7. In Sec. 8, the results from fin design optimization are discussed. Finally, in Sec. 9, the performance of these fin designs is discussed from an entropy generation perspective.

## 2 Solar Power Tower Receiver Tube With Internal Fins

In SPT plants, sun rays are concentrated on to a central receiver using a field of heliostats. These receivers, typically cylindrical in shape, are comprised of several circumferentially arranged tubes. Half of the tubes’ circumference is heated by sun rays reflected from the heliostats, and the other half (sun-shadow side) is typically insulated.

In this study, the flow and heat transfer in an internally finned receiver tube is simulated as a conjugate heat transfer problem by explicitly modeling the solid (tube) and fluid (HTF) regions. To reduce the computational load, only a small portion of the receiver tube length is simulated. The receiver tube’s diameter and wall thickness are selected based on a baseline design from an NREL report [11]. A schematic illustration of the receiver tube simulation model with fins is shown in Fig. 1. The dimensions of the model are listed in Table 1. Notice that the simulation model has a smooth entry segment in front of the finned segment. This smooth region is added purely for computer-aided design (CAD) automation purposes and has nothing to do with the hydrodynamic entrance length. Likewise, a smooth exit segment is added after the finned segment to aid simulation convergence by avoiding back-flow conditions at the outlet. The geometry of the fins is parameterized, allowing them to be optimized based on parameters such as helical pitch and height.

D_{i} | δ_{tube} | L_{f} | L_{en} | L_{ex} |
---|---|---|---|---|

35.3 mm | 1.4 mm | 17D_{i} | 2D_{i} | 6D_{i} |

D_{i} | δ_{tube} | L_{f} | L_{en} | L_{ex} |
---|---|---|---|---|

35.3 mm | 1.4 mm | 17D_{i} | 2D_{i} | 6D_{i} |

As mentioned in Sec. 1, this work ultimately seeks to use selective laser melting technology to manufacture the receiver tubes with optimized internal fins. To this end, a mixture of Inconel 718 and boron (1 wt%) is chosen as the material for the receiver tubes. Inconel 718 is a high-strength nickel-based alloy that has recently become a popular choice for additive manufacturing of high-temperature components [12]. Despite its excellent mechanical and corrosion-resistant properties, Inconel 718 suffers from low absorptivity (≈0.5) and hence would result in inferior optical performance of the receivers. A small percentage of Boron is therefore added to Inconel 718 in an effort to improve its optical performance. Experimental optical testing of 3D printed Inconel 718 + Boron samples is currently underway and will be reported in a later revision of this manuscript. A eutectic molten chloride salt (MgCl_{2}-KCl-NaCl) with mixture wt% 45.98-38.91-15.11 is chosen as the HTF in this study [5]. This salt mixture, in fact, is the proposed HTF for Gen 3 liquid pathway CSP plants [3,4]. Table 2 lists the material properties of the tube and the HTF.

Receiver tube with fins: Inconel 718 + 1 wt% boron | |
---|---|

ρ = 8193.3 kg/m^{3} | λ = 19.5 W/m K |

c_{p} = 550 J/kg K |

Receiver tube with fins: Inconel 718 + 1 wt% boron | |
---|---|

ρ = 8193.3 kg/m^{3} | λ = 19.5 W/m K |

c_{p} = 550 J/kg K |

Heat transfer fluid: Molten chloride salt [5] | |
---|---|

ρ = 1648.9 kg/m^{3} | λ = 0.4392 W/m K |

c_{p} = 1026.4 J/kg K | μ = 3.0505 × 10^{−3} kg/m s |

Heat transfer fluid: Molten chloride salt [5] | |
---|---|

ρ = 1648.9 kg/m^{3} | λ = 0.4392 W/m K |

c_{p} = 1026.4 J/kg K | μ = 3.0505 × 10^{−3} kg/m s |

### 2.1 Boundary Conditions.

For simulation purposes, a solar concentration factor of 500 is assumed (typical for an SPT plant). This translates to a semi-circumferential heat flux of 500 suns (i.e., *q*″ = 500 KW/m^{2}) applied to the sun-facing half of the finned segments’ outer wall. The sun-shadow side of the receiver tube is typically insulated; therefore, an adiabatic condition is imposed. Moreover, the smooth inlet and outlet segments, added to either side of the finned segment solely for computational reasons, are imposed with adiabatic conditions on their outer walls.

The heat transfer fluid (i.e., molten salt) is assumed to enter the inlet with a velocity *V*_{in} = 4.5 m/s at a temperature *T*_{in} = 823.15 K and turbulence intensity $Iin=5%$. This inlet velocity corresponds to a Reynolds number ≈ 86,000. The outlet is left to atmospheric pressure, with a back-flow turbulence intensity (*I*_{out}) of 5$%$ and a turbulent viscosity ratio of 10.

## 3 Governing Equations

*ξ*

_{ij}is the mean strain-rate tensor. The notation $\xaf$ and ′ is used here to denote the mean and fluctuating components of the field quantities, respectively. To close the above RANS equations, the Reynolds stress term

*τ*

_{ij}is related to the mean velocity gradients as

*μ*

_{t}is the turbulent viscosity, $k=0.5ui\u2032ui\u2032\xaf$ is the turbulence kinetic energy (TKE), and $\u03f5=\nu \u2202ui\u2032\u2202xj\u2202ui\u2032\u2202xj\xaf$ is the rate of TKE dissipation. The realizable $k\u2212\u03f5$ model is used as the turbulence model here to maintain consistency in results with previous work [15]. For an incompressible flow under no gravity assumption, the transport equations for

*k*and $\u03f5$ are as follows:

*G*

_{k}represents the TKE generation due to mean velocity gradients, and

*ν*is the kinematic viscosity. The values of the constants in the model are: $C2=1.9,\sigma k=1.0,\sigma \u03f5=1.2$. The steady-state energy transport equation for this turbulent flow is as follows:

*E*is the total energy and

*λ*

_{eff}is the effective thermal conductivity defined as

*Pr*

_{t}is the turbulent Prandtl number, which is set to 0.85.

## 4 Meshing and Numerical Method

Design optimization involves several calls to the CFD model while varying fin geometry parameters. This in turn requires CAD and mesh automation. As such, generating a structured mesh, especially in an automated manner, is difficult due to the complex curved surfaces in the geometry. Therefore, a semi-structured hex-dominant mesh with suitable mesh orthogonality and aspect ratio was used in this study. Moreover, the presence of large velocity and temperature gradients in the viscous sublayer and the thermal boundary layer contributes to entropy accumulation near the wall [16]. To properly resolve the flow and temperature fields in these regions, inflation layers are created close to the walls while maintaining *y* + ≤1.

A mesh-convergence study was performed by simulating the reference geometry (tube with continuous helical fins) with various mesh sizes. To this end, a mesh with about 8 million was chosen to achieve an acceptable level of accuracy. Results of the mesh-convergence analysis and model validation can be found in Ref. [15], a precursor to the current study. It should be noted that changing fin geometry affects the local mesh densities. Nevertheless, the mesh settings (such as local/global sizing and curvature capture controls) derived from the convergence analysis of the reference geometry are likely suitable for most geometries in this study. Otherwise, performing mesh convergence for every possible geometry would be intractable.

The steady-state governing equations discretized based on a finite volume formulation were numerically solved in ansys® fluent using the coupled pressure-based solver. The least-squares cell-based approach was used for computing the gradients. Pressure was interpolated using a second-order central difference scheme. A second-order upwind discretization was chosen for computing the convection terms of the flow equations and all other scalar equations [17]. The solution convergence criteria (scaled residuals) were set at 5 × 10^{−4} for all equations except for the energy, which was set at 10^{−8}.

## 5 Thermo-Hydraulic Performance Metrics

This study compares the performance of different swirl-inducing fin designs against three thermo-hydraulic metrics: Friction factor, Nusselt number, and temperature leveling effect. The following section provides a brief description of these metrics and their calculation.

### 5.1 Friction Factor.

*D*

_{i}length (see Fig. 3)—computed as the difference in the area-average pressure between the ends of this 5

*D*

_{i}region. For the fin designs of interest, the HTF tends to reach a fully developed state by the time it reaches this 5

*D*

_{i}region. Notice that, to simplify the friction factor calculation, the internal diameter of the tube is used in Eq. (9) instead of the hydraulic diameter.

### 5.2 Nusselt Number.

*λ*is the thermal conductivity of the fluid, and

*h*is the convective heat transfer coefficient calculated as

*A*

_{h}is the surface area on which the heat flux is applied (

*A*

_{h}= 0.5

*πD*

_{o}

*L*

_{f}),

*A*

_{fl}is the internal wall area of the heated side of the tube assuming there are no fins (

*A*

_{fl}= 0.5

*πD*

_{i}

*L*

_{f}),

*T*

_{wall}is the average internal wall temperature, and

*T*

_{bulk}is bulk temperature of the HTF

*A*

_{fl}) in this study to compute the Nusselt number.

### 5.3 Temperature Leveling.

*θ*is the non-dimensional average temperature of the tube’s outer wall defined as

*θ*

_{wall}signifies a better temperature leveling effect, and usually fins that generate strong swirl flow tend to result in lower Δ

*θ*

_{wall}values.

## 6 Design Optimization of Fins Based on Thermo-Hydraulic Performance

### 6.1 Optimization Formulation.

The turbulence and swirl induced by the fins result in better fluid mixing, which in turn improves the convective heat transfer [15]. However, the flow obstruction caused by the fins increases the pressure loss in the tubes. This increased pressure drop requires higher pumping power to maintain the flowrate, thereby diminishing the benefits of increased thermal performance. As such, an optimization formulation that tries to maximize the heat transfer performance of the fins while constraining the pressure loss is employed in this work. This formulation is shown as

**x**is a vector of the fin’s geometric design parameters, and

**x**

_{min}and

**x**

_{max}are the bounds. Notice that the objective function and constraint are scaled with the Nusselt and friction factor of a smooth tube (Nu

_{s}and

*f*

_{s}).

### 6.2 Surrogate Assisted Optimization Approach.

CFD simulations of the fin designs are computationally intensive. In fact, each simulation in this study took an average of 6+ h on a machine with 90 cores. Therefore, direct optimization approaches, which typically require hundreds of calls to the CFD model, are computationally intractable. To circumvent this issue, Kriging models (also called Gaussian processes) are used to approximate the CFD predictions needed during the optimization. Once trained using a few CFD evaluations as part of a computational design of experiment, these surrogate models can be coupled to an optimizer and called numerous times without incurring the cost of CFD simulations.

### 6.3 Kriging.

*m*(

**x**) is a trend (e.g., a constant or a linear function), and

*Z*(

**x**) is a stationary zero-mean Gaussian process with covariance

*R*(

*x*_{i},

*x*_{j}) is the correlation kernel function—which in this work is modeled as a squared exponential function with

*l*as its length scale hyper-parameter. The hyper-parameters of the Kriging model (i.e.,

*σ*

_{Z}and

*l*) are found by maximum likelihood. Kriging has an important advantage over other surrogate models in that it provides the variance (i.e., uncertainty) of its predictions.

## 7 Candidate Fin Designs and Their Thermo-Hydraulic Performance

In this study, five different designs of swirl-inducing helical fins are examined. These helical fins, besides increasing the surface area for heat transfer, also induce swirling or rotation in the flow. The secondary flows generated by this swirl effect will improve fluid mixing and thereby reduce thermal gradients [15]. Helical fin designs are therefore ideal candidates for heat transfer enhancement in concentrated solar receiver tubes.

The thermo-hydraulic performance (i.e., friction factor, Nusselt number, and the temperature leveling effect) of these fin designs is analyzed as a function of their design parameters. This section briefly describes the candidate fin designs and summarizes their performance characteristics via trend plots obtained by post-processing the surrogate built for optimization.

### 7.1 Continuous Helical Fins.

Continuous helical fins are often the first choice when it comes to passively introducing swirl in a pipe flow. The design of continuous helical fins, shown in Fig. 4(a), is defined geometrically by its height and pitch. Here, pitch refers to the length of one helix turn measured parallel to the tube’s axis. Higher pitch results in a slow-turning helical fin with a low helix angle. Listed in Table 3 are the range of values (i.e., optimization bounds) for these geometric parameters.

P_{fin} | H_{fin} | |
---|---|---|

Min | 35.3 mm (1D_{i}) | 3.52 mm (0.1D_{i}) |

Max | 105.9 mm (3D_{i}) | 14.12 mm (0.4D_{i}) |

P_{fin} | H_{fin} | |
---|---|---|

Min | 35.3 mm (1D_{i}) | 3.52 mm (0.1D_{i}) |

Max | 105.9 mm (3D_{i}) | 14.12 mm (0.4D_{i}) |

Figure 5(a) illustrates the improvement in heat transfer achieved by the continuous helical fins, with the enhancement becoming more pronounced as the fin height increases and/or fin pitch decreases. However, this improvement comes at the cost of increased pressure drop, as shown in Fig. 5(b). To facilitate easier interpretation, the Nusselt number and friction factor values in these plots are presented as ratios relative to the smooth tube values. Moreover, Fig. 5(c) displays the effect of the fins on temperature leveling in terms of Δ*θ*_{wall}. These figures clearly indicate that taller fins with a lower pitch provide better heat transfer and temperature leveling but lead to higher pressure losses.

The increased thermal performance due to these fins can be attributed to two main factors: a strong swirl/circumferential flow and a longer effective flow length due to the helical flow path. As such

increasing

*H*_{fin}increases the strength of the helical flow stream by restricting the axial flow path.decreasing

*P*_{fin}increases the effective flow length as lower pitch results in more helical turns.

### 7.2 Interrupted Helical Fins.

These fins can be visualized as a design formed by cutting angular slots into a continuous helical fin along the tube axis. These slots allow axial flow to pass through them—resulting in superior fluid mixing between circumferential and axial flow streams. In this study, the fins of this kind are designed with six slots, each with a size of *θ*_{slot}, cut circumferentially at uniform spacing (see Fig. 4(b)). Table 4 lists the range of design parameter values for this fin design.

P_{fin} | H_{fin} | θ_{slot} | |
---|---|---|---|

Min | 35.3 mm (1D_{i}) | 3.52 mm (0.1D_{i}) | 20 deg |

Max | 105.9 mm (3D_{i}) | 14.12 mm (0.4D_{i}) | 40 deg |

P_{fin} | H_{fin} | θ_{slot} | |
---|---|---|---|

Min | 35.3 mm (1D_{i}) | 3.52 mm (0.1D_{i}) | 20 deg |

Max | 105.9 mm (3D_{i}) | 14.12 mm (0.4D_{i}) | 40 deg |

Figure 6 presents the thermo-hydraulic performance of this fin design as a function of its design parameters. The interrupted design should intuitively facilitate easier axial flow through the slots, but in fact, a higher pressure loss is observed compared to the continuous helical fin design. CFD flow analysis suggests that this increased pressure loss is the result of circumferential flow colliding with the axial flow (through the slots) at periodic intervals. Though these fins generate significantly higher turbulence and fluid mixing, their thermal performance is found to be no better than the continuous helical fin design.

### 7.3 Multi-Head Helical Fins.

In this design, numerous short helical fins, each with a height and width equal to the tube’s wall thickness, are arranged at uniform spacing on the inner surface of the tube. These fins are designed to introduce a thin layer of circumferential flow close to the wall without causing much disruption to the axial flow. Figure 4(c) shows a CAD model illustrating this fin design and its design parameters. Listed in Table 5 are the range of values (i.e., optimization bounds) for the geometric parameters.

P_{fin} | N_{fin} | |
---|---|---|

Min | 35.3 mm (1D_{i}) | 4 |

Max | 105.9 mm (3D_{i}) | 8 |

P_{fin} | N_{fin} | |
---|---|---|

Min | 35.3 mm (1D_{i}) | 4 |

Max | 105.9 mm (3D_{i}) | 8 |

The Nusselt number, friction factor, and temperature leveling effect of this fin design are shown in Fig. 7. These results show that, compared to the pitch *P*_{fin}, the number of fin replications *N*_{fin} has very little effect on the thermo-hydraulic performance of the design. Additionally, this design has lower friction factor values than the other four fin designs, but with less heat transfer enhancement (Nu/Nu_{s} ≤ 1.8).

### 7.4 Wedge-Shaped Double Helix Fins.

Tubes with periodic converging-diverging sections are known to exhibit enhanced heat transfer characteristics owing to recurring disruption and redevelopment of the boundary layer [21]. The wedge cross section double-helix fins extend this idea to helical designs. In this design, two fins, each with a wedge-shaped cross section, are arranged in a double helix fashion (see Fig. 4(d)). This design, when viewed along the longitudinal cross section of the tube, resembles that of a periodically converging-diverging tube and hence can potentially have similar boundary layer behavior. Table 6 lists the range of design parameter values for this fin design.

P_{fin} | W_{fin base} | H_{fin} | |
---|---|---|---|

Min | 35.3 mm (1D_{i}) | 4.1 mm | 3.53 mm (0.1D_{i}) |

Max | 105.9 mm (3D_{i}) | 17.65 mm | 14.12 mm (0.4D_{i}) |

P_{fin} | W_{fin base} | H_{fin} | |
---|---|---|---|

Min | 35.3 mm (1D_{i}) | 4.1 mm | 3.53 mm (0.1D_{i}) |

Max | 105.9 mm (3D_{i}) | 17.65 mm | 14.12 mm (0.4D_{i}) |

The thermo-hydraulic performance of this fin design is shown in Fig. 8. These plots show that this design can enhance heat transfer by up to three times within a 50-fold increase in pressure drop. Moreover, this is the only design among five candidate designs that exhibits a maximum Nu/Nu_{s} at a design parameter location that does not correspond to the largest pressure drop.

### 7.5 Helical Tapes.

There has been extensive research on helical tape inserts with various vortex generator attachments and perforation patterns [22]. In this work, the fin design shown in Fig. 4(e) is referred to as helical tapes for lack of a better name. In fact, this design is a variant of continuous helical fins with fin height equal to the tube’s internal radius. As a result, this design leads to a tube design with completely separated spiral flow channels. The number of flow channels is double that of the helical tapes. The design parameter ranges of this design are listed in Table 7.

P_{tape} | N_{tape} | |
---|---|---|

Min | 35.3 mm (1D_{i}) | 1 |

Max | 105.9 mm (3D_{i}) | 3 |

P_{tape} | N_{tape} | |
---|---|---|

Min | 35.3 mm (1D_{i}) | 1 |

Max | 105.9 mm (3D_{i}) | 3 |

The thermo-hydraulic performance of helical tape design is shown in Fig. 9 as a function of its design parameters. As anticipated, an increase in the number of tapes leads to a higher Nusselt number. Nonetheless, this increase comes at a considerable expense of pressure drop, particularly when the pitch of the tapes is low. Nevertheless, helical tapes offer superior temperature leveling performance as they constrain the flow to pass through swirling channels between the tapes.

## 8 Optimization Results

When analyzing the fin designs using two different metrics, namely Nusselt numbers and friction factors, it becomes clear that there is no notion of so-called “best fin configuration” that outperforms all other designs. Instead, there are multiple optimal configurations, each with its own unique set of trade-offs between heat transfer enhancement and pressure loss [15]. To this end, several optimizations, each maximizing Nusselt number with a different friction factor threshold, are carried out in this study. Each of these optimization problems produces an optimal fin configuration with maximum heat transfer (Nu) for a given friction factor threshold. The sequential quadratic programming, a gradient-based nonlinear optimization method, is used to solve these optimization problems. It is important to note here that during the optimization process, only the surrogate approximations of the objective and constraint functions are used—not the actual CFD model. The results from these optimization problems are listed in Table 8 for each of the five candidate fin designs.

Continuous helical fins | |||
---|---|---|---|

f/f_{s} threshold | 10.00 | 20.00 | 25.00 |

Nu/Nu_{s} | 1.93 | 2.20 | 2.48 |

H_{fin} | 13.84 mm | 8.41 mm | 14.12 mm |

P_{fin} | 63.30 mm | 35.30 mm | 40.74 mm |

Continuous helical fins | |||
---|---|---|---|

f/f_{s} threshold | 10.00 | 20.00 | 25.00 |

Nu/Nu_{s} | 1.93 | 2.20 | 2.48 |

H_{fin} | 13.84 mm | 8.41 mm | 14.12 mm |

P_{fin} | 63.30 mm | 35.30 mm | 40.74 mm |

Interrupted helical fins | |||
---|---|---|---|

f/f_{s} threshold | 20.00 | 40.00 | 55.00 |

Nu/Nu_{s} | 1.83 | 2.17 | 2.21 |

H_{fin} | 11.31 mm | 11.52 mm | 13.29 mm |

θ_{slot} | 20.10 deg | 20.00 deg | 21.79 deg |

H_{fin} | 57.25 mm | 35.75 mm | 35.30 mm |

Interrupted helical fins | |||
---|---|---|---|

f/f_{s} threshold | 20.00 | 40.00 | 55.00 |

Nu/Nu_{s} | 1.83 | 2.17 | 2.21 |

H_{fin} | 11.31 mm | 11.52 mm | 13.29 mm |

θ_{slot} | 20.10 deg | 20.00 deg | 21.79 deg |

H_{fin} | 57.25 mm | 35.75 mm | 35.30 mm |

Multi-head helical fins | |||
---|---|---|---|

f/f_{s} threshold | 3.00 | 6.50 | 8.00 |

Nu/Nu_{s} | 1.40 | 1.65 | 1.74 |

P_{fin} | 86.50 mm | 60.94 mm | 36.47 mm |

N_{fin} | 8 | 8 | 8 |

Multi-head helical fins | |||
---|---|---|---|

f/f_{s} threshold | 3.00 | 6.50 | 8.00 |

Nu/Nu_{s} | 1.40 | 1.65 | 1.74 |

P_{fin} | 86.50 mm | 60.94 mm | 36.47 mm |

N_{fin} | 8 | 8 | 8 |

Wedge-shaped double helix fins | |||
---|---|---|---|

f/f_{s} threshold | 20.00 | 30.00 | 40.00 |

Nu/Nu_{s} | 2.18 | 2.45 | 2.71 |

P_{fin} | 56.58 mm | 46.18 mm | 36.10 mm |

H_{fin} | 12.74 mm | 13.10 mm | 13.61 mm |

W_{fin base} | 4.10 mm | 4.10 mm | 4.10 mm |

Wedge-shaped double helix fins | |||
---|---|---|---|

f/f_{s} threshold | 20.00 | 30.00 | 40.00 |

Nu/Nu_{s} | 2.18 | 2.45 | 2.71 |

P_{fin} | 56.58 mm | 46.18 mm | 36.10 mm |

H_{fin} | 12.74 mm | 13.10 mm | 13.61 mm |

W_{fin base} | 4.10 mm | 4.10 mm | 4.10 mm |

Helical tapes | |||
---|---|---|---|

f/f_{s} threshold | 10.00 | 30.00 | 50.00 |

Nu/Nu_{s} | 2.00 | 2.50 | 2.71 |

P_{tape} | 95.59 mm | 57.80 mm | 37.55 mm |

N_{tape} | 3 | 3 | 1 |

Helical tapes | |||
---|---|---|---|

f/f_{s} threshold | 10.00 | 30.00 | 50.00 |

Nu/Nu_{s} | 2.00 | 2.50 | 2.71 |

P_{tape} | 95.59 mm | 57.80 mm | 37.55 mm |

N_{tape} | 3 | 3 | 1 |

The interrupted helical fin design, even after rigorous optimization, has lower thermo-hydraulic performance than the continuous helical fin design. This can be observed by comparing its optimal Nusselt numbers (in Table 8) to those of continuous helical fin design (for comparable friction factor thresholds). Apart from the multi-head helical fins, all candidate designs exhibit thermal enhancements greater than 2 (i.e., Nu/Nu_{s} > 2), although with variable pressure losses. This represents a substantial improvement in heat transfer, especially at such high Reynolds numbers (Re ≈ 86,000). Although the heat transfer improvement by multi-head helical fins is lower, it is still a viable choice when the priority is to choose a design resulting in lower frictional losses.

## 9 Entropy Generation Analysis

*s*

_{fr}) is numerically split into two parts: direct dissipation $(s\xaffr)$ in the mean flow field and the turbulent dissipation (

*s*′

_{fr}) due to fluctuating velocities. These entropy contributions per unit volume are given as

*s*

_{ht}) also has two components: the entropy due to mean temperature gradients $(s\xafht)$ and the entropy due to gradients of temperature fluctuations (

*s*′

_{ht}). These quantities are given as

*s*′

_{ht}and

*s*′

_{fr}) by relating them to the turbulent dissipation rate $(\u03f5)$ and the mean temperature $(T\xaf)$

*α*=

*λ*/(

*ρc*

_{p}) is the thermal diffusivity and

*α*

_{t}is the turbulent thermal diffusivity. Within the solid region, the entropy generation originates only from the presence of thermal gradients, and there is also no notion of fluctuational thermal gradients. Finally, the total entropy generation

*S*

_{total}in the system is obtained by integrating the sum of all entropy contributions over the fluid and solid regions

*S*

_{ht}and

*S*

_{fr}are the heat transfer and frictional contributions to the total entropy, respectively.

### 9.1 Entropy Generation Versus Reynolds Number.

In this section, the entropy generation contributions from heat transfer and viscous effects are analyzed while varying the Reynolds number. This study determines the Reynolds number (or operational flowrate) leading to the minimum total entropy generation for a given fin design. Note that entropy generation versus Reynolds number may differ between designs, even if the designs perform similarly thermo-hydraulically. The reason for this is that the distribution/magnitude of temperature and velocity gradients within the tube (responsible for entropy generation) can differ significantly depending on the geometry of the fins.

Improving heat transfer, either by increasing the Reynolds number or by adding fins, will decrease the entropy generation. This is because improving heat transfer reduces local thermal non-equilibrium, which in turn lowers thermal gradients responsible for entropy production. In contrast, frictional loss and its entropy generation contribution increase when the Reynolds number is increased or fins are employed. However, it is important to note that the frictional entropy only starts dominating once the operational flowrate goes past the optimal Reynolds number. Therefore, any heat transfer improvement obtained by increasing the operational Reynolds number past the optimal Reynolds number comes with a greater cost of pumping power.

The entropy generation associated with heat transfer can be orders of magnitude greater than the frictional entropy, especially at lower Reynolds numbers. Therefore, at low Reynolds numbers below the optimal Reynolds number, heat transfer enhancement is always beneficial to lower the entropy production even at an additional cost from increased friction loss.

Helical fin designs that were previously optimized at Re = 86,000 are now analyzed from an entropy generation perspective while varying the Reynolds number. The results from this analysis are presented in Figs. 10–14 along with their Nusselt number and friction factor ratios. Notice that the entropy generation splits from the mean and fluctuating field quantities are also shown in these figures. The Reynolds number at which the dominant entropy generation mechanism changes from thermal-related to friction-related is shown as a vertical dashed line passing through the intersection of $S\xaffr$ and $S\xafht$ on these plots.

For the sake of completeness, a plot of the total entropy generation (*S*_{total}) versus Reynolds number ($ReDi$) curves for the five fin designs is provided in Fig. 15. Notice that, among these five curves, only the multi-head helical fin seems to exhibit minimum total entropy close to the Reynolds number of interest to this study (≈86,000).

## 10 Conclusion

Thermally enhanced SPT receiver tubes with internal helical fins are analyzed from thermo-hydraulic and entropy generation perspectives. In this context, five candidate fin designs are optimized to maximize heat transfer under various pressure loss constraints. Kriging surrogates are used to efficiently approximate the thermo-hydraulic quantities needed during these optimizations. Results for these optimizations have shown that four of the five candidate designs can improve heat transfer more than twofold (even at the high Reynolds number ≈86,000 relevant to this study). Additionally, the effectiveness of these fin designs for leveling circumferential temperatures is also assessed. Helical tapes are found to be particularly effective for this purpose, capable of reaching low Δ*θ*_{wall} (≈10) but at a high friction costs (*f*/*f*_{s} ≈ 130).

Striking an effective balance between the thermo-hydraulic quantities (Nu, *f*, Δ*θ*_{wall}) is often challenging. To this end, the fin designs are analyzed from an entropy generation perspective. This allows one to assess the thermal (Nu, Δ*θ*_{wall}) and viscous (*f*) trade-offs in terms of their relative contributions to total entropy generation. In addition, the results of this analysis also indicate that, for three of the five candidate designs, the operational Reynolds number (≈86,000) greatly exceeds the flowrate expected to achieve minimum total entropy generation.

In the future, this study will be combined with the ongoing research on structural and optical aspects of thermally enhanced receiver tubes [26]. In addition, the optimization will be performed for a range of Reynolds numbers instead of a fixed one. The overarching goal of this effort is to additively manufacture and test optimally designed enhanced receiver tubes for next-generation CSP systems.

## Acknowledgment

The authors are grateful for the support from the U.S. Department of Energy’s Office of Energy Efficiency and Renewable Energy (EERE) under the Solar Energy Technologies Office (SETO) Award No. DE-EE0009380.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## References

_{2}Eutectic Molten Salt as a Next-Generation High-Temperature Heat Transfer Fluids in Concentrated Solar Power Systems

_{2}for Application as High Temperature Heat Transfer Fluids

*Ansys® Academic Research Fluent, Theory Guide*, Release 19.3.