# Numerical Study on Flow and Heat Transfer Characteristics of Trapezoidal Printed Circuit Heat Exchanger

^{*}

## Abstract

**:**

_{2}power cycle, with the advantages of a large specific surface area and compact structure. Its tiny and complex flow channel structure brings enhanced heat transfer performance, while increasing pressure drop losses. It is, thus, important to balance heat transfer and flow resistance performances with the consideration of sCO

_{2}as the working agent. Herein, three-dimensional models are built with a full consideration of fluid flow and heat transfer fields. A trapezoidal channel is developed and its thermal–hydraulic performances are compared with the straight, the S-shape, and the zigzag structures. Nusselt numbers and the Fanning friction factors are analyzed with respect to the changes in Reynolds numbers and structure geometric parameters. A sandwiched structure that couples two hot channels with one cold channel is further designed to match the heat transfer capacity and the velocity of sCO

_{2}flows between different sides. Through this novel design, we can reduce the pressure drop by 75% and increase the regenerative efficiency by 5%. This work can serve as a solid reference for the design and applications of PCHEs.

## 1. Introduction

_{2}) power cycle. Fluid flow and heat transfer characteristics in PCHE are significantly different from those in traditional shell-and-tube heat exchangers, and its thermal–hydraulic performance has a great impact on the efficiency and output power of the sCO

_{2}power cycle due to the huge heat recovery.

_{2}as the working fluid. Ishizuka et al. [1] studied the heat transfer and pressure drop characteristics of PCHE on the sCO

_{2}cycle experimental loop and proposed empirical formulas for pressure loss, and local and overall heat transfer coefficients, with the Reynolds number range of 2400–6000 and 5000–13,000 at the hot and cold side, respectively. Nikitin et al. [2] studied the heat transfer and pressure drop characteristics of a zigzag PCHE. As a result, the experimentally measured overall heat transfer coefficient of PCHE was in the range of 300–650 W/m

^{2}. Ngo et al. [3,4] studied the thermo-hydraulic characteristics of zigzag and S-shape PCHE using carbon dioxide as the working fluid, and proposed that the Nusselt number of zigzag PCHE was 24–34% higher than that of the S-shape, with a four to five times higher pressure drop under the same Reynolds number. Kim et al. [5,6,7] carried out comprehensive numerical and experimental research on zigzag PCHE using He, water, carbon dioxide, and their mixed working fluids. The correlations of the Fanning friction factor and Nusselt number with respect to the channel angle, pitch length, and hydraulic diameter in the zigzag PCHE were obtained. Mylavarapu et al. [8] designed and manufactured two straight-channel PCHEs on the high-temperature helium test platform. The experimental correlations of the Nusselt number and the Fanning friction factor were obtained, and it was also found that the laminar to turbulent transition region appeared earlier in the PCHE than in the circular tube, corresponding to a Reynolds number of 1700. Baik et al. [9] built a cycle test platform and performed a heat transfer performance test using sCO

_{2}and water. The Fanning friction factor and heat transfer correlations were proposed, respectively. Chu et al. [10] also designed a sCO

_{2}and water heat transfer platform and tested the performance of a straight channel PCHE. Experiments found that the heat transfer performance of sCO

_{2}was 1.2–1.5 times better than that of water. Zhang et at. [11] tested a 100-kW novel airfoil fin PCHE as a cooler, and also conducted the numerical analysis. The airfoil fin showed a comparative heat transfer rate with only 1/6 pressure drop of the zigzag structure.

_{2}zigzag PCHE model and performed related numerical simulations. It was found that the outlet temperature and pressure obtained from the simulation agreed well with the experimental results of Ishizuka et al. In addition, they proposed a structure of airfoil ribs, which can reduce the pressure drop loss to one-twelfth of the zigzag channel, while ensuring the heat exchange performance. Bartel et al. [16] conducted a series of comparative studies on the zigzag flow channel PCHE and found that the best heat transfer performance was obtained at a pitch angle of 15°, with the increase in pressure drop loss acceptable relative to the straight-channel PCHE. Khan et al. [17] performed a three-dimensional steady-state heat transfer simulation of zigzag PCHE at four different inclination angles of 0°, 5°, 10°, and 15° and four different Reynolds numbers of 350, 700, 1400, and 2100, which found that there were flow enhancement and secondary flow areas inside the zigzag channel. Kim et al. [18] used the ANSYS CFX to explore and verify the existing zigzag PCHE correlations, and expanded the applicable Reynolds number range to 2000–58,000 based on the Ishizuka’s experimental correlations. Baik et al. [19] numerically studied the S-shape PCHE and analyzed the influence of the channel amplitude and period on the heat transfer performance. Chen et al. [20] compared the performance of four types of NACA 00XX airfoil structures with zigzag and found that the airfoil structure can significantly reduce the flow pressure drop loss while maintaining heat transfer performance. Aneesh et al. [21] used helium working fluid to study the heat transfer performance of zigzag, sine-shaped, and trapezoidal flow channel structures and found that trapezoidal PCHE has the highest heat transfer performance and maximum pressure. Ren et al. [22] developed a new local heat transfer correlation based on a generalized mean temperature difference (GMTD) method in a horizontal semicircular straight channel of PCHE, which predicts 93% of the data, with errors of less than ±15%. Lv et al. [23] proposed three new hybrid flow channel structures, which combined the S-shape in the high-, medium-, and low-temperature sections of the straight channel, respectively. The results showed that the type C (with the wavy section used in the low-temperature region) was the best, with a maximum pressure drop reduction of 23%, and the heat transfer coefficient increased by 2.6 times higher compared to type A (with the wavy section used in the high-temperature region).

_{2}cycle and lack comparison of different channel structures. Among these structures, the trapezoidal structure presented better heat transfer performance; however, it suffers from a larger pressure drop. To overcome this problem, numerical models of the trapezoidal flow channel are developed, and its thermal and hydraulic performance with sCO

_{2}as the medium is studied. The dimensionless Nusselt numbers and Fanning friction factors are analyzed with respect to the changes in Reynolds numbers and structure geometric parameters. An optimized structure is further proposed to match the heat transfer capacity and velocity of hot and cold flows of sCO

_{2}at different temperatures and pressures.

## 2. Model and Methods

#### 2.1. Model and System

_{2}Brayton cycle and related temperature–entropy (T-s) diagram are presented in Figure 1 [24]. The high-temperature and high-pressure sCO

_{2}from heat resource (point 1) enters the turbine for power generation. After the temperature and pressure reduced (point 2), the medium enters the regenerator to transfer the residual heat to the cold-side working fluid. In addition, after the cooling device, the sCO

_{2}with its temperature close to the critical point (point 3) enters the compressor to increase pressure. Finally, the high-pressure medium (point 4) returns to point 1 state by heat recovery and resource heating. Between the cooling and heating processes, a huge amount of heat exchange through the regenerator is needed, making the performance of PCHE vital to the system efficiency.

_{2}critical point (304.25 K, 7.38 MPa). The inlet mass flow rate range on both sides is from 4.82 × 10

^{−4}to 14.45 × 10

^{−3}kg/s.

#### 2.2. Grid and Independence Verification

- 1
- The continuous medium flows uniformly in every channel of the PCHE.
- 2
- The total mass flow of sCO
_{2}is distributed equally in each hot/cold channel because the flow resistance is the same. - 3
- The inlet temperature and pressure of all hot/cold channels are the same and identical to the hot/cold pipe of the PCHE.
- 4
- The effect of pressure changes on the CO
_{2}properties is neglectable in the flowing process, as the pressure loss is much smaller than the working pressure.

#### 2.3. Properties of sCO_{2}

_{2}are calculated at pressures of the hot side (7.6 MPa) and the cold side (20.2 MPa), respectively. The thermal and hydraulic properties, including the density, the specific heat, the thermal conductivity, and the dynamic viscosity, can be described as polynomial functions of temperature using MATLAB software between 350 K and 750 K. The purpose is to ensure the accuracy of physical property parameters while reducing the computational resources generated by quoting the FLUENT’s built-in database. All the data are obtained from NIST Reference Database, and the fitting correlations are listed in Table 4.

#### 2.4. Calculation Method

_{2}.

_{2}, $l$ is the length of trapezoidal flow channel, and $u$ is the bulk velocity. As the density of sCO

_{2}changes in both sides with the heat exchange and there exist differences in the velocities and the dynamic pressures, $\Delta {p}_{f}$ should be calculated as shown in Equation (8):

## 3. Results and Discussion

#### 3.1. Thermal and Hydraulic Performance

#### 3.2. Comparison between the Trapezoidal and Previous Channel Structures

^{−3}kg/s. The four models only have differences in geometric shapes to evaluate the performance of the new trapezoidal structure and the previous structures.

_{2}. However, the trapezoidal channel suffers from the largest flow resistance, as shown in Figure 6b, where its Fanning friction factor is 5 times, 2.5 times, and 1.2 times higher than that of the straight, the S-shape, and the zigzag channels, respectively.

#### 3.3. Optimization of Trapezoidal PCHE

^{−3}kg/s, 9.63 × 10

^{−4}kg/s, and 7.23 × 10

^{−4}kg/s, corresponding to the Reynolds number of 33,366, 22,244, and 16,683, respectively. As for the hot side flow, it is the same with the cold side in the double-channel structure, and half of the cold side in the sandwich structure because there are two hot channels. Significant reductions in pressure drop loss in the hot channel are obtained, where the values are 75.4% (from 154.81 kPa to 38.37 kPa) and 74.7% (from 39.65 kPa to 10.05 kPa) at 42 kPa and 11 kPa pressure loss cases in the cold side, respectively. The sandwich structure also shows a higher regenerative efficiency, as shown in Figure 8, where the values are increased by 5% at all same length and mass flow conditions.

## 4. Conclusions

_{2}as the medium were studied. Key factors, including the convective heat transfer coefficient, the regenerative efficiency, and the dimensionless Nusselt numbers, were calculated and analyzed with respect to the changes in geometric parameters, such as the flow channel length, the trapezoidal bottom angle, and the straight length of trapezoidal upper. The Fanning friction factors were further calculated to describe the frictional resistance of different cases. Comparisons of the heat transfer performance and the pressure loss were also discussed between the current structure and previous flow channel types in the literature. We found the Nusselt numbers increased approximately linearly with the increase in Reynolds numbers, but kept almost a constant value at different flow lengths, indicating the independence of heat transfer performance from the flow length. Besides, the thermal and hydraulic performance of the trapezoidal channel changed uniformly with the geometric parameters, where both the Nusselt numbers and Fanning friction factors could be enhanced with the increase in bottom angle and the decrease in upper straight length. Among the four channels, the trapezoidal structure presented the largest increase in Nusselt number in both the hot side and the cold side, and its Fanning friction factor of the trapezoidal channel was 5, 2.5, and 1.2 times higher than that of the straight, S-shaped, and zigzag channels, respectively. By comparing the heat transfer and flow resistances, the overall performance of the trapezoidal structure was found to be close to the zigzag structure, and much better than the S-shaped and the straight structures. Based on the above findings, a sandwiched structure was designed with a couple of hot channels corresponding to one cold channel to optimize the pressure loss in the hot channel and enhance the regenerative efficiency. Through this novel design, the regenerative efficiency could be increased by more than 5% and the pressure drop loss of the hot channel could be reduced by about 75% to only 40 kPa.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Effects of (

**a**) the Reynolds numbers, (

**b**) the trapezoidal bottom angle, and (

**c**) the straight length on the Nusselt numbers.

**Figure 4.**Effects of (

**a**) the trapezoidal bottom angle and (

**b**) the straight length on the Fanning factors.

**Figure 6.**(

**a**) The Nusselt numbers and (

**b**) the Fanning friction factors of the trapezoidal, the zigzag, the S-shape, and the straight channels.

Structure | Medium | Temperature (℃) | Pressure (MPa) | Flow Rate (kg/h) | Deficiency | Ref. |
---|---|---|---|---|---|---|

Straight | Helium | Hot side: 208–790 Cold side: 85–390 | Hot side: 1.0–2.7 Cold side:1.0–2.7 | 15–49 | No sCO_{2}Low pressure One channel only | [8] |

Straight | Hot side: sCO _{2}Cold side: water | Hot side: 37–102 Cold side: - | Hot side: 8–11 Cold side: - | sCO_{2}:150–650 | Not sCO_{2} heat exchangeLow temperature One channel only | [10] |

Zigzag | CO_{2} | Hot side: 280–300 Cold side: 90–108 | Hot side: 2.2–3.2 Cold side: 6.5–10.5 | 40–80 | Low pressure under supercritical state One channel only | [2] |

Zigzag S-shape | CO_{2} | Hot side: 120 Cold side: 35–55 | Hot side: 6 Cold side: 7.7–12 | 40–150 | Low pressure at hot side | [3,4] |

Zigzag | Hot side: He-CO _{2}Cold side: water | Hot side: 104.5–217.6 Cold side: 23.6–25.2 | Hot side: 1.17–1.72 Cold side: 0.104–0.306 | He-CO_{2}:154–329 Water: 389–1966 | Mixed sCO_{2}Low pressure Low temperature One channel only | [5,6,7] |

Zigzag | Hot side: CO _{2}Cold side: water | Hot side: 26–43 Cold side: 15 | Hot side: 7.3–8.6 Cold side: - | - | Not sCO_{2} heat exchangeTranscritical phase exists Low temperature One channel only | [9] |

Airfoil | Hot side: CO _{2}Cold side: water | Hot side: 70–110 Cold side: 16–25 | Hot side: 7.6–9.0 Cold side: Approx. 0.1 | CO_{2}:500–1800 Water: Approx. 3000 | Not sCO_{2} heat exchangeLow temperature One channel only | [11] |

Structure | Medium | Temperature (℃) | Pressure (MPa) | Flow Rate (kg/h) | Deficiency | Ref. |
---|---|---|---|---|---|---|

Straight | Hot side: CO _{2}Cold side: water | Hot side: 40–100 Cold side: 14–50 | Hot side: 7.5/8.1 Cold side: 0.101325 | Hot side: 2.22–8.87 Cold side: 22.17–26.60 | Not sCO_{2} heat exchangeLow temperature One channel only | [22] |

Straight S-shape | sCO_{2} | 101.85 | 8 | 1.44 | Not a heat exchanger One side only Low temperature | [23] |

Zigzag Airfoil | CO_{2} | Hot side: 279.9 Cold side: 107.9 | Hot side: 2.52 Cold side: 8.28 | Hot side: 0.52 Cold side: 1.13 | Low pressure under supercritical state Low temperature | [15] |

Zigzag | Helium | Hot side: 800 Cold side: 520 | Hot side: 7 Cold side: 7.97 | 450 | No sCO_{2}One channel only | [16] |

Zigzag | CO_{2} | Hot side: 303.3 Cold side: 61.9 | Hot side: 1.9 Cold side: 1.9 | - | Low pressure under supercritical state Low temperature One channel only | [17] |

Zigzag | CO_{2} | Hot side: 280 Cold side: 108 | Hot side: 3.2 Cold side: 10.5 | 30–400 | Low pressure under supercritical state Low temperature One channel only | [18] |

Zigzag Airfoil | CO_{2} | Hot side: 279.9 Cold side: 107.9 | Hot side: 2.52 Cold side: 8.28 | Hot side: 3.12 Cold side: 3.4 | Low pressure under supercritical state Low temperature One channel only | [20] |

S-shape | Hot side: LNG flue gas Cold side: sCO _{2} | Hot side: 650 Cold side: 224 | Hot side: 0.1 Cold side: 13.6 | - | Not sCO_{2} heat exchangeLow pressure Low temperature One channel only | [19] |

S-shape | CO_{2} | Hot side: 280 Cold side: 108 | Hot side: 2.5 Cold side: 7.4 | 64.7 | Low pressure under supercritical state Low temperature One channel only | [12,13,14] |

Zigzag S-shaped Trapezoidal | Helium | Hot side: 900 Cold side: 540 | Hot side: 3 Cold side: 3 | 10–50 | No sCO_{2}Low pressure | [21] |

Serial Number | Grid Number | The Hot Outlet Temperature (K) | GCI |
---|---|---|---|

1 | 3,780,000 | 473.37 | $GC{I}_{1,2}=2.42\%$ |

2 | 2,780,000 | 478.32 | $GC{I}_{2,3}=3.66\%$ |

3 | 2,080,000 | 485.94 |

Hot Side | |
---|---|

Physical parameters | Correlations |

Density (kg/${\mathrm{m}}^{3}$) | $\rho =8.5283\times {10}^{-9}{T}^{4}-2.0952\times {10}^{-5}{T}^{3}+0.01942{T}^{2}-8.175T+1403$ |

Specific heat (J/(kg K)) | ${C}_{p}=-5.6476\times {10}^{-10}{T}^{5}+1.6674\times {10}^{-6}{T}^{4}-1.9573\times {10}^{-3}{T}^{3}+1.1431{T}^{2}-332.24T+\text{39,583}$ |

Thermal conductivity (W/(mK)) | $k=7.16\times {10}^{-5}T+0.00069$ |

Viscosity (Pa s) | $\mu =3.65\times {10}^{-8}T+6.65\times {10}^{-6}$ |

Cold side | |

Physical parameters | Correlations |

Density (kg/${\mathrm{m}}^{3}$) | $\rho =8.878\times {10}^{-8}{T}^{4}-2.0795\times {10}^{-4}{T}^{3}+0.018288{T}^{2}-72.032T+\text{10,976}$ |

Specific heat (J/(kg K)) | ${C}_{p}=1.7498\times {10}^{-12}{T}^{6}-7.3206\times {10}^{-9}{T}^{5}+1.2479\times {10}^{-5}{T}^{4}-0.01116{T}^{3}+5.5477{T}^{2}-1459.5T+\text{160,650}$ |

Thermal conductivity (W/(mK)) | $k=-4.3538\times {10}^{-14}{T}^{5}+1.2744\times {10}^{-10}{T}^{4}-1.4888\times {10}^{-7}{T}^{3}+8.6829\times {10}^{-5}{T}^{2}-0.025249T+2.9656$ |

Viscosity (Pa s) | $\mu =2.2325\times {10}^{-19}{T}^{6}-7.5867\times {10}^{-16}{T}^{5}+\text{10,702}\times {10}^{-12}{T}^{4}-8.0234\times {10}^{-10}{T}^{3}+3.3745\times {10}^{-7}{T}^{2}-7.5534\times {10}^{-5}T+7.0617\times {10}^{-3}$ |

PEC-cold | Trapezoid/Zigzag | Trapezoid/S-shape | Trapezoid/Straight |

0.978 | 1.154 | 1.157 | |

PEC-hot | Trapezoid/Zigzag | Trapezoid/S-shape | Trapezoid/Straight |

1.034 | 1.211 | 1.214 |

Reynolds Number of the Cold Side | Pressure Loss (kPa) | |||
---|---|---|---|---|

Double-Channel Structure | Sandwich Structure | |||

Cold Side | Hot Side | Cold Side | Hot Side | |

33,366 | 42.62 | 154.81 | 41.94 | 38.37 |

22,244 | 19.73 | 71.11 | 18.60 | 17.77 |

16,683 | 11.07 | 39.65 | 11.04 | 10.05 |

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**MDPI and ACS Style**

Ji, Y.; Xing, K.; Cen, K.; Ni, M.; Xu, H.; Xiao, G.
Numerical Study on Flow and Heat Transfer Characteristics of Trapezoidal Printed Circuit Heat Exchanger. *Micromachines* **2021**, *12*, 1589.
https://doi.org/10.3390/mi12121589

**AMA Style**

Ji Y, Xing K, Cen K, Ni M, Xu H, Xiao G.
Numerical Study on Flow and Heat Transfer Characteristics of Trapezoidal Printed Circuit Heat Exchanger. *Micromachines*. 2021; 12(12):1589.
https://doi.org/10.3390/mi12121589

**Chicago/Turabian Style**

Ji, Yuxuan, Kaixiang Xing, Kefa Cen, Mingjiang Ni, Haoran Xu, and Gang Xiao.
2021. "Numerical Study on Flow and Heat Transfer Characteristics of Trapezoidal Printed Circuit Heat Exchanger" *Micromachines* 12, no. 12: 1589.
https://doi.org/10.3390/mi12121589