# Numerical Analysis on the Heat Transfer Characteristics of Supercritical Water in Vertically Upward Internally Ribbed Tubes

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Models

#### 2.1. Physical Model

_{0}) of 33.40 mm and the maximum inner diameter (d

_{i,max}) of 20.62 mm, the other fixed structural parameters of IRT used in the present studies are listed in Table 2. The heated length of the test section (L) is 2510 mm, with the heat flux from 120 kW/m

^{2}to 250 kW/m

^{2}added on the outer surface. In this model, thermophysical properties is assumed as the function of temperature but ignored the effect of pressure variation. The added heat flux is assumed to be uniformly distributed along the in the whole pipe circumference. The simulation is performed at the pressure of 25 MPa, the mass flux is 600 kg/(m

^{2}·s). The test section was installed vertically with an upward flow. Additionally, another smooth tube with OD/ID 33.40 /20.62 mm is selected as a reference.

#### 2.2. Transport Equations

_{k}and G

_{ω}is the production of turbulence kinetic energy and specific dissipation rate, Y

_{k}and Y

_{ω}represent the dissipation of turbulence kinetic energy and specific dissipation rate.

#### 2.3. Numerical Method

^{+}≤ 1, can get more reliable result. Therefore, 30 boundary layer grids are set in the near-wall region, and the size of the first layer grid is further adjusted according to the preliminary calculated y

^{+}to make sure the final y

^{+}within the range of 0.1~1. Besides, the heat conduction effect in the solid region have been also considered. The conjugated heat transfers between the solid domain and the fluid domain is solved. To maintain a reasonable gird number, a non-uniform structured grid size was adopted, where the same grid size is used in the transition region to match the interfaces at both sides of the inner wall. Additionally, the mesh independence solution is obtained by successively refining the original mesh. Finally, the total grid number in the whole calculated domain meshes is 9.85 million, and the mesh structure is shown in Figure 3.

_{0}, no-slip wall condition is applied to the tube inner surface. The enhanced wall treatment model was adopted for the fluid cell near the wall, which can get more accurate results for supercritical fluids [29]. To get a better convergence, the convective term in momentum and energy equation are firstly solved with the first-order upwind scheme, and then the second-order upwind scheme is employed for solving transport equations after convergence.

#### 2.4. Turbulence Model

^{2}s and heat flux of 400 kW/m

^{2}.

_{pc}= 2152.9 kJ·kg

^{−1}). However, the wall temperature obtained by SST k-ω model agrees well with the experimental data [33]. Therefore, the SST k-ω turbulence model is used for analyzing the turbulent heat transfer in IRTs in the following works.

## 3. Results and Discussions

#### 3.1. The Turbulent Heat Transfer of Supercritical Water in IRTs and STs

_{0}= 120 kW·m

^{−2}), the averaged inner-wall temperature of ST and IRT increases with the fluid bulk enthalpy, but the inner-wall temperature of IRT seem evidently lower than that in ST by 6~7 °C. When the heat flux increased to a higher value (i.e., q

_{0}= 250 kW·m

^{−2}), as shown in Figure 6b, the corresponding averaged inner-wall temperature of ST rapidly rises as the increase of bulk enthalpy, and reaches the peak at H = 1950 kJ/kg, an obvious deteriorated heat transfer occurs in ST at this bulk enthalpy, then inner-wall temperature gradually decreases. However, the averaged inner-wall temperature of IRT varies smoothly without any peaks observed as the bulk enthalpy increases even under the condition with high heat fluxes. Thus, IRTs can enhance heat transfer and effectively avoid the occurrence of HTD.

#### 3.2. Distribution of Velocity Components in the Near-Wall Region

_{0}= 120 kW·m

^{−2}); (2) at high heat flux (q

_{0}= 250 kW·m

^{−2}). Due to its differences in spiral ribs/grooves, the parameters at different radial positions corresponding to the grooves (the OV line) and the ribs (the OP line), as seen in Figure 8, are discussed, respectively. Meanwhile, the dimensionless distance (Y = 1 − r/R

_{i,max}) are employed to describe the special heat transfer behaviors in both tubes, where r is the distance from the center of cross-section, and R

_{i,max}represents the maximum inner radius of the tube. Additionally, according to the theory of boundary layer [31], three zones are divided in the boundary layer: if y

^{+}< 5, the region is the viscous sublayer of turbulent flow; 5 < y

^{+}< 70 is the transition region (or buffer layer); and y

^{+}> 70 is the logarithmic layer (or turbulent core region).

^{2}. However, when the heat flux increases to 250 kW/m

^{2}, as seen in Figure 9b, the corresponding axial velocity varies and exhibits a distinct “M-shaped” distribution along the tube diameter direction in ST, where the maximum axial velocity appears at y

^{+}≈ 37 due to the deteriorated heat transfer occurs. However, no HTD observed in IRT at both heat fluxes, the axial velocity in the near-wall region is always lower than that in the center domain. For IRTs, the axial velocity of the supercritical water close to the top of rib is significantly higher than that near the groove due to the shrinkage of the cross-sectional size, thereby the convective heat transfer in the top of rib seems better. Moreover, the axial velocity in IRT is smaller than ST before y

^{+}< 70, which is actually in the viscous sublayer and turbulent transition region. There is also no obvious thermal acceleration exists in the turbulent boundary layer, which means that thermal acceleration suppressed by the existence of the inner rib of the spiral, IRT can avoid (or slow down) the acceleration of the fluid in the boundary layer. As seen from Figure 9b, the tangential velocity of the supercritical water is close to zero because there is no spiral motion in ST, but the tangential velocity in IRT is remarkable and far beyond the corresponding value within ST. A peak of the tangential velocity appears in the groove region. The circumferential motion of the supercritical water in the cross-section of IRT exists but that does not exist in ST. Moreover, as the heat flux increases, the tangential velocity increases in IRT. As seen in Figure 9c, the radial velocity of the supercritical water in the cross-section of the ST is smaller than IRT. A significant radial motion generates in the cross-section of IRT, especially in the groove region, indicating that the fluid radial mixing is more intense in the cross section of IRT.

#### 3.3. Distribution of Thermophysical Properties in the Near-Wall Region

^{+}< 5), therefore, water can absorb a large amount of heat transferred from the wall to the fluid and enhances heat transfer. The specific heat of IRT is always greater than that of ST after reaches the peak value, which means that the heat transfer capacity in IRT is much stronger than that in ST. Once the heat flux achieved to a high value (q = 250 kW/m

^{2}), the occupied region with large specific heat increases significantly, and the corresponding specific heat peak moves to the transition region (5 < y

^{+}< 70), but in the viscous sublayer (y

^{+}< 5), the specific heat is significantly lower than that under the low heat flux, and the profile of specific heat in IRT is very close to ST. As at high heat fluxes, the bulk temperature near the wall rises rapidly and exceed the pseudo-critical temperature the heat transfer capacity in the turbulent viscous sublayer is weakened.

^{+}< 5), but in the region of y

^{+}> 5, the density and thermal conductivity in IRT is significantly lower than the corresponding value in ST. At low heat flux, the local inner temperature of the supercritical water in IRT is much higher than that in ST, resulting in a lower density in IRT. However, under high heat flux, the heat transfer deteriorates in ST because the heating wall is covered by a thin layer with high-temperature but low-density fluids, but for IRTs, heat transfer enhancement occurs, where the ribs make the fluid swirling and promote the heavy fluid migrate to the wall, and simultaneously restrict the light fluid (with low density and poor thermal conductivity) toward the center region.

#### 3.4. Distribution of Turbulent Kinetic Energy within Boundary Layer

^{+}< 5). However, for IRTs, the variation of turbulent kinetic energy profile is complex. Under y

^{+}< 20, the turbulent kinetic energy in IRT is lower than in ST, but in the transition region (5 < y

^{+}< 70), especially in 20 < y

^{+}< 70, the turbulent kinetic energy in IRT is significantly greater than that in ST, and the maximum value appears at the top of ribs. Interestingly, under high heat flux, as shown in Figure 11b, the turbulent kinetic energy on the cross-section of ST reaches a minimum value at y

^{+}≈ 37, and then recovers, another peak appears at y

^{+}≈ 80. Additionally, the turbulent kinetic energy of IRT reaches the maximum at the same location (y

^{+}≈ 37). Known from Figure 9, the corresponding axial velocity V reaches the maximum value in here, where the radial gradient (∂V/∂r) is sharply reduced, resulting in a sharp decrease of the turbulent shear stress. Such special variation of turbulence intensity in the transition region reflects that their heat transfer in such region plays an important role. For IRTs, due to the strong disturbance caused by the ribs, the radial velocity and turbulent kinetic energy in the corresponding position of IRT enhanced and reached the maximum value, so that the minimum value of the turbulent kinetic energy is successfully eliminated that is observed in ST under the condition of high heat flux, which enhances the heat transfer and effectively avoids the occurrence of HTD.

#### 3.5. Effect of Rib Structure on Heat Transfer in IRTs

^{−2}·s

^{−1}, and heat flux of 120 and 250 kW·m

^{−2}.

_{o}=120 kW·m

^{−2}), when the rib lift angle increase from 40 to 60, the averaged HTC decreases at least 10%. However, at high heat flux, the heat transfer coefficient gradually decreases along with the bulk enthalpy. With the increase in lift angle, the corresponding heat transfer performance in IRT is weakened and the difference in heat transfer coefficient at various lift angle is even decreased by 80% on average. The higher the lift angle, the lower the HTC. As seen in Figure 12b, with the height of the rib increases, the averaged heat transfer coefficient increases because the disturbance of ribs enhances the heat transfer. When the rib height (e) increases from 0.8 mm to 1.25 mm, the averaged HTC increases 20% under these operating conditions. Evidently, the increase of the rib height is favorable for the heat transfer enhancement of the supercritical water in the internally ribbed tube. As the spiral ribs can deeply destory turbulent boundary layer, intensify the turbulent mixing under the effect of large height ribs. It can be seen from Figure 12c that as the circumferential rib width increases, the averaged heat transfer coefficient decreases, where the corresponding heat transfer of the supercritical water in IRTs is weakened as the increases of rib width. When the rib width decreases from 6.0 mm to 3.58 mm, the corresponding HTC increases 20% under these operating conditions. Moreover, as seen in Figure 12d, when the number of ribs (m) increases from 2 to 6, the averaged inner-wall temperature gradually increases 2 °C, but the corresponding HTC decreases 58% under these operating conditions. However, when the threads number is relatively small, such as m changes from 2 to 4, the heat transfer coefficient essentially unchanged, but if the threads number increases from 4 to 6, the heat transfer coefficient change remarkable. Such a situation can be explained by the following reasons. By increasing the threads number, the ratio between the groove width and rib width is reduced, causing the heat transfer in IRTs weakened.

#### 3.6. Optimal Rib Structures in IRTs

## 4. Conclusions

- IRTs can enhance the heat transfer to supercritical water as a result of the generation of obvious spiral flow, which produces significant circumferential and radial motion in the cross-section. At low heat fluxes, the temperature of IRT is lower than that in ST by 6~7 °C. At high heat fluxes, deteriorated heat transfer occurs in smooth tube, but not happens in IRTs, the maximum temperature difference is 36 °C. The heat transfer enhancement is more pronounced in the pseudocritical region, where the ratio between IRT and ST is about 1.81, but the ratio is only 1.21 in the low enthalpy region.
- In the cross-section, axial velocity suppressed, but tangential and radial velocity increases as a result of disturbance of spiral ribs, the velocity deviation between IRT and ST is about 20–50%. At low heat flux, the specific heat of supercritical water in IRT is about 3% greater than that of ST within the viscous sublayer (y
^{+}< 5), resulting in a better heat transfer capability. However, at high heat flux, the heat transfer deteriorates occurs in ST because the heating wall is covered by a thin layer with high-temperature but low-density, low-thermal conductivity (a 20% reduction) fluids, but for IRTs, heat transfer enhancement occurs, where the ribs make the fluid swirling and promote the heavy fluid migrate to the wall, and simultaneously restrict the light fluid (with low density and poor thermal conductivity) toward the center region. - In IRTs, a higher turbulent kinetic energy observed in the transition region of the turbulent boundary layer. At high heat flux, the turbulent kinetic energy in ST got a minimum value (=0.0025 m
^{2}⋅s^{−2}) in the transition region of the turbulent boundary layer, and the axial velocity at this position reaches the maximum value (=1.7 m⋅s^{−1}), which is a major cause to the occurrence of HTD. Due to the spiral flow induced by the internal rib, IRT can enhance the turbulent kinetic energy in the transition region and avoid the occurrence of the minimum value of the turbulent kinetic energy. IRT avoids (or postpones) the occurrence of HTD in supercritical water at high heat flux. - With the increase in the lift angle of ribs, the HTC decreases; with the increase in the rib height, the HTC increases. As the circumferential rib width increases, the heat transfer coefficient decreases. When the threads number is relatively small (m ≤ 2), the heat transfer coefficient essentially unchanged, but if the threads number is high (m ≥ 4), the heat transfer coefficient change remarkable.
- An optimal rib structure has been obtained. When the diameter is 33.4 mm, lift angle is 50 degrees, height is 0.58 mm, width is 3.5 mm, threads number is 6, the corresponding overall performance is the best, which can be applied in engineering.

## 5. Further Scope

_{2}) Bryton power cycle.

_{2}) Bryton power cycle has been considered as one of the promising energy conversion systems, which also faces extremely high temperture and heat transfer deterioration issue in future. Adapting an optimal rib structure internally ribbed tube can effectively enhance heat transfer and avoids (or postpones) the occurrence of HTD at high heat fluxes, which is potential to use in such advanced energy systems.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

_{Cp} | specific heat at constant pressure, J/kg K |

D | diameter, mm |

e | height, mm |

f | frictional coefficient |

G | mass flux, kg/m^{2}s |

g | gravitational acceleration, m/s^{2} |

H | enthalpy, kJ·kg^{−1} |

h | heat transfer coefficient, kW·m^{−2}·K^{−1} |

k | turbulent kinetic energy, m^{2}·s^{−2} |

L | length, m |

m | threads |

Nu | Nusselt number |

P | Pressure, MPa |

p | pitch, mm |

Pr | Prandtl number |

q | heat flux, kW·m^{−2} |

R | radius, mm |

r | distance, mm |

t | Time, s |

Re | Reynolds number |

S | width, mm |

Greek Letters | |

α | lift angel, ° |

ΔP | pressure drop, kPa |

β | thermal expansion coefficient, 1/°C |

δ | thickness, mm |

ε | turbulent dissipation rate, m^{2}/s^{3} |

η | Performance |

λ | Thermal conductivity, W·m^{−1}·K^{−1} |

μ | dynamic viscosity, Pa s |

μ_{t} | turbulent eddy viscosity, Pa s |

k | turbulent kinetic energy, kg⋅m/s^{2} |

ρ | density, kg/m^{3} |

ω | Specific dissipation, 1/s |

Y | Non-Dimensional Distance; (1-r/R_{i,max}) |

Subscripts or superscripts | |

b | bulk |

i, j, k | i, j, k components |

pc | pseudocritical |

t | turbulent |

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**Figure 1.**Schematic of internally ribbed tube (IRT) and its Structural Parameters in the Cross-section (where e is the rib height, S is the rib width, α is the rib lift angle, and α is defined as the angle between the spiral direction of the rib and the cross section of the pipe. Under the same pipe diameter, the larger the lift-angle of the thread, the larger the rib pitch).

**Figure 5.**Turbulence model verification results (exp. Data from [33]).

**Figure 6.**The inner-wall temperature in pseudocritical region. (

**a**) Heat transfer enhancement; (

**b**) Heat transfer deterioration.

**Figure 10.**Comparison of the thermophysical properties distribution of supercritical water in IRT and ST.

Author | m | p/mm | e/mm | S/mm | α/° | D/mm |
---|---|---|---|---|---|---|

Ackerman [2] | 6 | 21.84 | — | — | — | 18.03 |

Griem [4] | — | — | — | — | — | 24 |

Cheng [6,7] | 4 | 10.04 | 0.58 | 4.61 | 49.3 | 11.69 |

Cheng [8,9,10] | 4 | 21 | 0.81 | 4.8 | 61.15 | 15.24 |

Wang [11,12,13] | 4 | 21.55 | 0.85 | 4.8 | 60 | 15.24 |

Wang [14] | 6 | 11.61 | 1.20 | 4.8 | 50 | 17.63 |

Pan [15] | 4 | 19 | 0.92 | 4.62 | 50.5 | 19.4 |

Yang [16] | 4 | 22.7 | 0.85 | 5.3 | 54 | 20.3 |

Taklifi [17] | 6 | 10.8 | 1.2 | 4.77 | 45 | 19 |

Shen [18] | 4 | 18.1 | 1.24 | 6.2 | 50 | 19.1 |

Structural Parameters | Fixed Rib Structure | Variable Rib Structures |
---|---|---|

Outside diameter (d_{0}) | 33.40 mm | 25.4, 28.6, 33.4, 38 mm |

Maximum inside diameter (d_{i,max}) | 20.62 mm | 20.62 mm |

Length (L) | 2510 mm | 2510 mm |

Number of threads (m) | 6 | 1, 2, 4, 6 |

Rib width (S) | 3.58 mm | 3.5, 4.3, 4.8, 6.0 mm |

Rib height (e) | 0.8 mm | 0.58, 0.85, 1.0, 1.2 mm |

Rib lift angle (α) | 30° | 30, 40, 50, 60° |

Pitch (p) | 12.87 mm | 12.87 mm |

Lead (m × p) | 77.2 mm | 51.48, 77.22, 102.96 mm |

Criteria | Diameter (mm) | Lift Angle (°) | Height (mm) | Width (mm) | Threads (-) |
---|---|---|---|---|---|

h | 33.4 | 30 | 0.58 | 3.5 | 6 |

f | 38 | 60 | 0.58 | 3.5 | 1 |

η | 33.4 | 50 | 0.58 | 3.5 | 6 |

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**MDPI and ACS Style**

Lei, X.; Guo, Z.; Peng, R.; Li, H.
Numerical Analysis on the Heat Transfer Characteristics of Supercritical Water in Vertically Upward Internally Ribbed Tubes. *Water* **2021**, *13*, 621.
https://doi.org/10.3390/w13050621

**AMA Style**

Lei X, Guo Z, Peng R, Li H.
Numerical Analysis on the Heat Transfer Characteristics of Supercritical Water in Vertically Upward Internally Ribbed Tubes. *Water*. 2021; 13(5):621.
https://doi.org/10.3390/w13050621

**Chicago/Turabian Style**

Lei, Xianliang, Ziman Guo, Ruifeng Peng, and Huixiong Li.
2021. "Numerical Analysis on the Heat Transfer Characteristics of Supercritical Water in Vertically Upward Internally Ribbed Tubes" *Water* 13, no. 5: 621.
https://doi.org/10.3390/w13050621