# Numerical Study on Heat Transfer Deterioration of Supercritical n-Decane in Horizontal Circular Tubes

^{*}

## Abstract

**:**

## 1. Introduction

_{c}). Under a supercritical pressure, the thermo-physical properties of fuel, such as density, specific heat capacity, thermal conductivity and dynamic viscosity, change dramatically with temperature, especially when its temperature is near the pseudo-critical temperature (T

_{pc}, corresponds to the maximum specific heat capacity) and the pressure is close enough to the critical pressure [3], and this leads to complex flow and heat transfer phenomena.

_{c}) was below 1.5 and the wall temperature was over the pseudo-critical temperature. Liu et al. [18] also conducted similar experimental studies, an emphatically analyzed the heat transfer and pressure drop characteristics of another type of kerosene inside horizontal circular channels, and concluded that at near-critical pressures, HTD would occur as the film temperature approached the pseudo-critical temperature, accompanied with abnormal phenomena of flow instability and pressure drop deduction, during the corresponding process. As the bulk fluid temperature increased to above the pseudo-critical temperature, the heat transfer and flow stability were regained. Although a certain number of experimental and numerical studies have been conducted to examine the HTD phenomena in supercritical hydrocarbon fuels, more fundamental investigations, especially of the mechanism and critical conditions for the onset of HTD, still need to be further performed.

## 2. Models and Numerical Methods

#### 2.1. Physical Model and Boundary Conditions

^{2}·s) as an example, the length of the heated section (L) may be 4.2, 3.0, 2.1, 1.7, 1.4 or 1.1 m, corresponding to the tubes with different heat fluxes (q) of 200, 300, 400, 500, 600 and 700 kW/m

^{2}, respectively. The inlet and outlet boundary conditions of the computational domain are specified as mass flux inlet and pressure outlet, respectively. The same inlet temperature T

_{in}= 340 K is assigned for all the simulations, and all the outlet bulk fluid temperatures are about 820 K, so as to ensure the n-decane (p

_{c}= 2.1 MPa, T

_{c}= 618 K [19]) experiences the entire process from the over-pressure liquid state to the supercritical state. The range of the outlet pressure is 3–5 MPa.

_{q}) and the threshold Grashof number (Gr

_{th}) is less than 4 × 10

^{−3}based on the entrance parameters, and thus it is similar to a 2D axisymmetric problem. In addition, a certain length are retained in both ends of the heated section to further avoid the effects of inflow and outflow boundary conditions on the numerical results, only the flow and heat transfer characteristics in the bulk fluid temperature range of 425–770 K are concerned.

#### 2.2. Governing Equations

#### 2.3. Thermo-Physical Properties

_{p}), thermal conductivity (λ) and dynamic viscosity (μ), as a function of temperature and pressure, are all computed by the National Institute of Standards and Technology (NIST) Supertrapp software [20], as shown in Figure 2.

#### 2.4. Grids and Numerical Methods

^{+}≤ 5), and the first grid node adjacent to the wall is constructed to satisfy the requirement of y

^{+}≤ 1.

^{−5}, and the residuals of other governing equations are set to be less than 10

^{−7}.

#### 2.5. Grid-Independence Analysis and Numerical Methods Validation

^{2}·s), q = 351 kW/m

^{2}, T

_{in}= 423 K, d = 2 mm, downward flow) [21]. Figure 3 shows variations of the wall temperature and heat transfer coefficient (HTC) with reduced axial length. The HTC is defined as:

_{w}− T

_{b})

_{b}is the bulk fluid temperature (K) calculated as:

^{2}); Subscripts “w” and “b” represent wall and bulk fluid, respectively.

**Figure 3.**Grid-independence analysis and numerical methods validation: (

**a**) wall temperature; and (

**b**) heat transfer coefficient (HTC).

## 3. Results and Discussion

#### 3.1. Effects of Heat Flux and Pressure on the Heat Transfer

^{2}·s), the heat flux ranges from 200 to 700 kW/m

^{2}, while the outlet pressures are 3 and 5 MPa, respectively. Variations of the wall temperature and HTC with bulk fluid temperature at a pressure of 3 MPa are presented in Figure 4.

**Figure 4.**Variations of the wall temperature and HTC with bulk fluid temperature at a pressure of 3 MPa: (

**a**) wall temperature; and (

**b**) HTC.

^{2}, the wall temperature increases linearly with the increase of bulk fluid temperature. Under moderate heat fluxes of 400 and 500 kW/m

^{2}, an interesting phenomenon is discovered in that a slight oscillation in the wall temperature occurs once it exceeds the corresponding pseudo-critical temperature. As the heat flux further increases further, i.e., 600 and 700 kW/m

^{2}, the wall temperature oscillation becomes more intense and two obvious peaks appear, indicating complex flow and heat transfer phenomena, which can be further explained by variation in the HTC.

^{2}, the total heat transfer process can be divided into three typical regions, and this is closely related to the variations of bulk fluid thermo-physical properties: (1) T

_{b}≤ 614 K. The specific heat capacity rises with the increasing bulk fluid temperature, indicating that the fluid can carry away more heat from the tube wall. On the other hand, the axial velocity increases quickly owning to a continuous decrease in the density and dynamic viscosity, both of which lead to a linear increase of the HTC along the flow direction; (2) 614 K ≤T

_{b}≤ 686 K. When the bulk fluid temperature approaches to its pseudo-critical temperature, the thermo-physical properties begin to change greatly, as a consequence, comprehensive effects of the specific heat capacity, density and dynamic viscosity lead to a significantly enhanced heat transfer. Then, the bulk fluid temperature exceeds the pseudo-critical temperature, and a rapid decrease of the specific heat capacity causes a slight deterioration in heat transfer; (3) T

_{b}≤ 686 K. The density, specific heat capacity and dynamic viscosity change slightly, and merely the variation of the thermal conductivity is beneficial to enhance the heat transfer, so the HTC gradually increases as the bulk fluid temperature increases. Under these operational conditions, the HTC and bulk fluid specific heat capacity have similar variation tendencies. Obviously, the specific heat capacity is the most important factor influencing the heat transfer performance, and a HTC peak appears due to the bulk fluid temperature in the large specific heat capacity region. As the heat flux increases, because of the wall temperature oscillation, the variation of the HTC along the flow direction becomes more complicated, the HTD occurs in both the low and high bulk temperature regions. The same phenomenon has also been observed in an experimental study of the convective heat transfer of n-decane under supercritical pressures [21], as mentioned above. The degree of deteriorated heat transfer deepens gradually with increasing heat flux, featuring a larger corresponding bulk fluid temperature range. Mechanisms of the HTD will be discussed in detail in the following sections, taking q = 200, 600, 700 kW/m

^{2}in Figure 4 as examples.

**Figure 5.**Variations of the wall temperature and HTC with bulk fluid temperature at a pressure of 5 MPa: (

**a**) wall temperature; (

**b**) HTC.

_{D−B}/Nu>2

_{D−B}obtained from the Dittus–Boelter correlation, which is given as follows: where Nu

_{D−B}obtained from the Dittus–Boelter correlation, which is given as follows:

_{b}and Pr

_{b}are the Reynolds number and PrandtI number, respectively, based on properties at the bulk fluid temperature.

_{b}

_{D−B}/ Nu with bulk fluid temperature at a pressure of 3 MPa. As shown in Figure 6, Nu

_{D−B}/ Nu increases with the increase of heat flux, and in general, the maximum value of Nu

_{D−B}/ Nu corresponds to the same position of the minimum HTC point. When the heat flux is 700 kW/m

^{2}, the maximum value of Nu

_{D−B}/ Nu is about 1.4, lower than the critical value proposed by Shiralkar et al. [22]. As the heat flux further increases, significant thermal cracking will occur in the large specific heat capacity region because the wall temperature generally exceeds 870 K. The thermal cracking is an endothermic reaction, can absorb heat through the chemical heat sink, and as a result enhance the heat transfer [23]. Therefore, it can be concluded that the HTD will not cause serious harm to the cooling system under these operational conditions. In addition, Figure 6 also indicates that the Dittus–Boelter expression is no longer applicable for the heat transfer predictions.

_{cal}) and the empirical expressions (Nu

_{emp}) at a pressure of 3 MPa are compared in Figure 7. As shown in Figure 7, at a low heat flux of 300 kW/m

^{2}, the Nusselt number obtained from these seven expressions are in very good agreement with our present numerical results, all of the relative errors are generally within 25%. When the heat flux is at a high level, i.e., 700 kW/m

^{2}, the Bae-Kim expression still has higher precision, and the relative error is less than 20%, meeting the accuracy requirements for engineering calculations. By contrast, the other expressions incur relatively large prediction deviations from the present numerical results, because they cannot predict the HTD phenomena. The maximum relative errors generally reaches around 40%, indicating that these expressions are invalid for heat transfer predictions of supercritical n-decane under high heat fluxes.

**Figure 7.**Comparisons of the Nusselt number from the present numerical calculations and the empirical expressions under a pressure of 3 MPa: (

**a**) q = 300 kW/m

^{2}; and (

**b**) q = 700 kW/m

^{2}.

**Figure 8.**Comparisons of the Nusselt number from the present numerical calculations and the empirical expressions under a pressure of 5 MPa: (

**a**) q = 300 kW/m

^{2}; and (

**b**) q = 700 kW/m

^{2}.

Author | Correlation | Fluid |
---|---|---|

Bae-Kim [24] | CO_{2} | |

where, | ||

Gnielinski [11] | H_{2}O, CO_{2} | |

where, | ||

f = (0.79ln(Re_{b}) − 1.64)^{−2} | ||

Sieder-Tate [25] | H_{2}O, CO_{2} | |

Bishop et al. [26] | , | H_{2}O |

where x_{L} is the distance from the initial point of heating, m. | ||

Mccarthy-Wolf [27] | H_{2} | |

Taylor [27] | H_{2} | |

Giovanetti et al. [28] | RP-1 |

#### 3.2. Mechanisms of the HTD

_{i}, E

_{i}(i = 1–4) are the corresponding tube cross-sections of the initial point of deteriorated heat transfer and the ending point of heat transfer recovered, respectively, as clearly depicted in Figure 4b. As shown in Figure 10, under a low heat flux of 200 kW/m

^{2}, there is no abnormal distribution of the turbulent kinetic energy near the wall, and which increases monotonically along the flow direction. Under high heat fluxes of 600 and 700 kW/m

^{2}, the near-wall turbulent kinetic energy first decreases and then increases along the tube axial direction, corresponding to the deteriorated heat transfer and heat transfer recovered in the low bulk fluid temperature region. This phenomenon indicates that the flow in the near-wall region has already transited from turbulent-like to laminar-like. The flow laminarization near the wall was generally caused by the strong buoyancy force in upward flow or thermal acceleration in downward flow [6,7,8,9,10], but for horizontal channels, so far, it has not yet been a consistent conclusion. During the related heat transfer process, i.e., I

_{i}–E

_{i}(i = 1,2), it can be found from Figure 4a that the wall temperature is slightly higher than the pseudo-critical temperature, thus the fluid temperature in the near-wall region is close to the pseudo-critical temperature. Therefore, it can be considered that the near-wall fluid is subjected to a fluid acceleration effect due to large drops of the density with temperature, and thus the turbulent heat transfer near the wall is significantly suppressed [29], which resulting in the occurrence of HTD.

_{w}) with bulk fluid temperature. As shown in Figure 11, the wall shear stress and near-wall turbulent kinetic energy show a similar variation trend corresponding to the deteriorated heat transfer and heat transfer recovered in the low bulk fluid temperature region. This means that the frictional resistance has been subjected to a certain extent to weakening during the related heat transfer process. Clearly, this phenomenon is caused by the flow acceleration near the wall. As the wall temperature increases, the pseudo-critical point gradually moves away from the wall, the near-wall turbulent kinetic energy starts to slowly recover due to weakening of the flow acceleration, and thus the heat transfer is improved.

**Figure 10.**Variations of the near-wall turbulent kinetic energy with bulk fluid temperature under different heat fluxes.

**Figure 11.**Variations of the wall shear stress with bulk fluid temperature under different heat fluxes.

_{r}

_{-ave}, positive toward the wall) with bulk fluid temperature at three different heat fluxes, as mentioned above. As shown in Figure 12, under a low heat flux of 200 kW/m

^{2}, the HTD does not appear and the averaged radial velocity is close to zero during the entire heat transfer process. As the heat flux increases, i.e., 600 and 700 kW/m

^{2}, the averaged radial velocity shows a larger variation, and two obvious waveforms, first negative and then positive, are observed corresponding to the deteriorated heat transfer and heat transfer recovered in the high bulk fluid temperature region. This phenomenon indicates that this HTD is necessarily linked with the abnormal distribution of radial velocity. Meng et al. [11,12] considered that the radial velocity oscillation had a negative impact on the heat transfer, and would lead to HTD and a slight oscillation of HTC, but its action mechanism on HTD has not yet received a detailed explanation.

**Figure 12.**Variations of the averaged radial velocity with bulk fluid temperature under different heat fluxes.

^{2}, and the following explanation can be offered for effect mechanism of the radial velocity on heat transfer: firstly, the radial velocity is pointing to the center of tube cross-section, which means that the fluid of cross-section has a trend to flow toward the tube center. Thus, the mixing of hot and cold fluids near the wall is weakened, and the wall cannot be effectively cooled, which resulting in a rapid deterioration of heat transfer along the tube axial direction. Then, the radial velocity begins to change direction to the wall, from the near-wall region gradually extending to the entire radial extent. That is to say, the fluid of the cross-section has a trend to flow toward the wall. As a consequence, the mixing of near-wall fluids is enhanced, and the heat transfer is gradually improved along the flow direction. The demarcation point between the two processes is essentially when the bulk fluid temperature is about the pseudo-critical temperature, but the deterioration and oscillation of turbulent heat transfer in supercritical hydrocarbon fuels, e.g., methane and n-heptane, and the radial velocity direction were not necessarily linked in Hua’s study [11,12].

**Figure 13.**Radial velocity distributions corresponds to the deteriorated heat transfer and heat transfer recovered processes in the high bulk fluid temperature region under a heat flux of 700 kW/m

^{2}.

_{i}–E

_{i}(i = 3,4), as the bulk fluid temperature approaches to the pseudo-critical temperature, the tube cross-section with a larger drastic density variation in its radial direction. Especially in the near-wall region, a large volume of low-density fluid (gas-like) accumulates due to the relatively high fluid temperature.

**Figure 14.**Variations of the near-wall mass flux with bulk fluid temperature under different heat fluxes.

#### 3.3. Critical Condition for the Onset of HTD

^{2}·s).

Author | Correlation | Fluid |
---|---|---|

Yamagata et al. [30] | q_{cr} = 0.2 G^{1.2} | H_{2}O |

Styrikovich et al. [31] | q_{cr} = 0.58 G | H_{2}O |

Kim et al. [9] | q_{cr} = 0.0002 G^{2} | CO_{2} |

Urbano et al. [15] | q_{cr} = (0.0432p + 0.0314)G | CH_{4} |

Zhou et al. [16] | q_{cr} = 0.0000855 G^{2} + 0.1368 G_{p} + 4.1 p^{2} − 0.162 G + 37p − 42.6 | C_{5}H_{12} |

_{cr}appears when min (dT

_{w}/dx) = 0 is satisfied, which is used in the present study. Meanwhile, under the same pressure, a linear relation is found between the critical heat flux and mass flux. When the pressure is 3 MPa, a corresponding critical condition for the onset of HTD is (q/G)

_{cr}= 0.268 kJ/kg. The heat transfer will be improved with the increasing pressure, hence, under the pressures of 4 and 5 MPa, the corresponding critical conditions for the onset of HTD are increased to (q/G)

_{cr}= 0.493 kJ/kg and (q/G)

_{cr}= 0.718 kJ/kg, respectively. Figure 16 further presents variation of (q/G)

_{cr}with the pressure. Note that (q/G)

_{cr}is also a linear function of the pressure, i.e., (q/G)

_{cr}= 0.225p − 0.407.

**Figure 15.**Comparisons of the critical heat flux from the present numerical calculations and the empirical expressions.

_{cr}= (0.225p − 0.407)G

_{c}≤ 2.369, 400 kg/(m

^{2}·s) ≤ G ≤ 2000 kg/(m

^{2}·s). Within this range, the relative error between the calculated values and numerical results is within 4.12%. For a given pressure and mass flux, the HTD will occur when the heat flux is higher than the critical heat flux obtained from this expression.

## 4. Conclusions

- (1)
- Two different types of HTD phenomena could occur when the wall temperature exceeds the corresponding pseudo-critical temperature under the composite operational conditions of low pressure and high heat flux. Increasing the pressure would effectively alleviate and eliminate the HTDs. The Bae-Kim expression which has higher prediction accuracy when the HTD occurs can be better used for convective heat transfer predictions of n-decane at supercritical pressures.
- (2)
- The onset of HTD ocurrs because of the drastic density variations in the near-wall and bulk fluid regions, respectively. The HTD with low bulk fluid temperature occurs when the wall temperature is slightly higher than the pseudo-critical temperature, which is caused by the fluid acceleration. The near-wall flow is laminarized and the wall shear stress significantly decreases during the related heat transfer process. The HTD with relatively high bulk fluid temperature occurs in the large specific heat capacity region, mainly due to the pressure imbalance. During the corresponding heat transfer process, the pressure imbalance leads to the radial velocity oscillation, further affects the mixing ability of near-wall fluids, and the wall surface is filmed with a layer of illiquid hot fluid, thus it can be considered as a pseudo-film boiling deteriorated heat transfer phenomenon.
- (3)
- Based on the numerical analysis, a new correction for critical heat flux of HTD, which is applicable to the supercritical n-decane, has been successfully developed.

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Wang, Y.; Li, S.; Dong, M.
Numerical Study on Heat Transfer Deterioration of Supercritical n-Decane in Horizontal Circular Tubes. *Energies* **2014**, *7*, 7535-7554.
https://doi.org/10.3390/en7117535

**AMA Style**

Wang Y, Li S, Dong M.
Numerical Study on Heat Transfer Deterioration of Supercritical n-Decane in Horizontal Circular Tubes. *Energies*. 2014; 7(11):7535-7554.
https://doi.org/10.3390/en7117535

**Chicago/Turabian Style**

Wang, Yanhong, Sufen Li, and Ming Dong.
2014. "Numerical Study on Heat Transfer Deterioration of Supercritical n-Decane in Horizontal Circular Tubes" *Energies* 7, no. 11: 7535-7554.
https://doi.org/10.3390/en7117535