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Experimental investigation on heat transfer of n-decane in a vertical square tube under supercritical pressure
International Journal of Heat and Mass Transfer 138 (2019) 631–639
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Experimental investigation on heat transfer of n-decane in a vertical square tube under supercritical pressure Jianqin Zhu a, Chaofan Zhao a, Zeyuan Cheng b, Dasen Lin a, Zhi Tao a, Lu Qiu a,⇑ a b
National Key Laboratory of Science and Technology on Aero-Engine Aero-Thermodynamics, School of Energy and Power Engineering, Beihang University, Beijing 100191, PR China COMAC Beijing Aeronautical Science & Technology Research Institute, Beijing 100191, PR China
a r t i c l e
i n f o
Article history: Received 21 January 2019 Received in revised form 31 March 2019 Accepted 15 April 2019 Available online 25 April 2019 2010 MSC: 00-01 99-00 Keywords: Heat transfer N-decane Supercritical pressure Vertical square tube
a b s t r a c t The internal convective heat transfer characteristics of n-decane at supercritical pressure in a vertical square tube with a hydraulic diameter of 1.8 mm were experimentally investigated. The external wall temperatures of the square tube were measured with welded thermocouples whereas the internal wall temperatures were calculated with space marching method. In this experiment, the operating pressure ranged from 3 to 5 MPa and the heat flux from 100 to 500 kW/m2. Besides, the flow direction in the vertical tube was also switched in order to examine the effects of buoyancy. Moreover, the circumferential uniformity of heat transfer was also discussed. The results showed that the effects of buoyancy on the heat transfer were negligible once the local buoyancy number was less than a threshold. After the threshold, the buoyancy promoted the heat transfer in the downward flow configuration, but weakened the heat transfer in the upward flow one. The reason could be inferred that the buoyancy would influence the intensity of turbulent kinetic energy and the heat transfer near the wall. Finally, new heat transfer correlations for vertical upward and downward flow in square tubes were proposed. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction N-decane, as a typical hydrocarbon fuel, has a critical temperature of 645.5 K and a critical pressure of 2.390 MPa. However, in the application of regenerate cooling, the operation pressure in the channel is around 3.45–6.89 MPa which is higher than the critical pressure of the hydrocarbon fuel [1]. When the fuel flows through the heating channel under supercritical pressure, it could change from liquid state to supercritical state due to the convective heat transfer. In this process, the physical properties of the fuel vary significantly, which plays an important role in influencing the flow and heat transfer. Thus, experimental or numerical investigations have been conducted to study the flow and heat transfer characteristics in a heating tube under supercritical pressure. Yamagata et al. [2] experimentally studied the heat transfer characteristics of water in a circular tube under supercritical pressure. The results showed that the dramatic change of thermal properties at pseudo-critical temperature resulted in a complex relation between heat transfer coefficient and heat flux. Jackson and Hall [3,4] reported that heat transfer deterioration could be generated around the pseudo-critical point. They proposed that ⇑ Corresponding author. E-mail address: [email protected] (L. Qiu). https://doi.org/10.1016/j.ijheatmasstransfer.2019.04.076 0017-9310/Ó 2019 Elsevier Ltd. All rights reserved.
the temperature difference between the main flow and the thin fluid layer adjacent to the wall gave rise to a significant density gradient, which in turn generated the buoyancy flow. Besides, they inferred that the interaction of the buoyancy flow and the main flow could influence the near-wall turbulence production, which affected the convective heat transfer. Later on, Jackson and Hall [3,5,6] developed a semi-empirical theory and deducted a dimensionless buoyancy number Gr=Re2:7 to evaluate the strength of buoyancy force (see the definitions of Gr and Re in Nomenclature). They proposed Gr=Re2:7 ¼ 105 as the threshold of heat transfer deterioration to supercritical water flowing in a upward vertical tube. Aside from the buoyancy effect, thermal acceleration effect is another key factor to heat transfer deterioration with a different mechanism. Shiralkar [7] and Griffith [8,9] experimentally investigated the heat transfer characteristics of supercritical CO2 flow in a circular tube. Even though the flow Reynolds number was relatively high, the heat transfer deterioration occurred in both upward flow and downward flow conditions. In order to quantify the strength of the thermal acceleration effect, McEligot et al. [10] developed a dimensionless number K v as follows,
Kv ¼
4qd 2
Re
lC p T
ð1Þ
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Nomenclature Cp d g Gr h H _ m Nu Pr P q Qx Re T u x
ap C
isobaric specific heat capacity (J/(kg K)) hydraulic diameter (m) gravitational acceleration (m/s2) Grashof number based on bulk temperature, 3 Gr ¼ ðT w T b Þg ap d =m2b convective heat transfer coefficient (W/(m2 K)) enthalpy (J/kg) mass flow rate (g/s) Nusselt number Prandtl number operating pressure (MPa) wall heat flux (W/m2) the heat energy added to the fluid before location x (W) Reynolds number, Re ¼ qb ud=l temperature (K) flow velocity (m/s) the axial distance in the tube (m) thermal expansion coefficient (1/K) the physical property ratio of wall-adjacent fluid against bulk fluid
It was observed that the thermal acceleration could weaken the turbulence intensity and induce relaminarization in the case of K v > 3 106 . In the aforementioned investigations, the working fluid was inorganic substance, such as CO2 and H2O. When it comes to the organic substance, the flow and heat transfer characteristics might be different. Zhou et al. [11] experimentally investigated the heat transfer characteristics of pentane flowed though circular channel under supercritical pressure and developed a heat transfer correlation to include the effects of inlet temperature, wall temperature, mass flux, heat flux, inlet pressure and bulk temperature. The results indicated that the heat transfer deterioration occurred when the fuel temperature approached the pseudo-critical temperature. Liu et al. [12] studied the heat transfer characteristics of n-decane in vertical circular tube under different pressure, mass flux and flow direction. The results showed that the effects of buoyancy and thermal acceleration were negligible in the condition of high inlet flow rate. Zhong et al. [13] experimental investigated the heat transfer characteristics of RP-3 in circular tube under supercritical pressure. In accordance with pure inorganic substance, the dramatic variations of the physical properties gave rise to the heat transfer deterioration for the hydrocarbon fuel. Rectangular channels are widely used in actual regenerative cooling systems due to the manufacturing concerns. Different from the circular tube, non-circular ones have obvious threedimensional effects. Sharabi et al. [14] studied the heat transfer characteristic of CO2 in vertical circular, square, and rectangular tubes under supercritical pressure. They observed that the wall temperature at the corner was always higher than that on the center of the wall, indicating that there was a low heat transfer region in the wall corner. Kim et al. [15,16] compared the heat transfer characteristics of CO2 in vertical circular, triangle, and rectangular tubes under supercritical pressure. The results showed that the rectangular tube was prone to generate heat transfer deterioration compared to the triangle and circular tubes. Sakurai et al. [17] investigated the flow and heat transfer characteristics of CO2 in a transparent rectangular tube. The near-wall flow pattern was captured and was correlated with the convective heat transfer. To sum up, in the application of regenerate cooling, hydrocarbon fuel flows through square heating channels under supercritical pressure. However, most of the related studies focused on the flow
h k
l m q
/
temperature difference between the corner and wall center of the square tube (K) thermal conductivity (W/(m K)) dynamic viscosity (Pa s) kinematic viscosity (m2/s) density (kg/m3) the relative heat transfer difference between the upward and downward flow
Subscripts b bulk cor parameters at the corner of the square tube cen parameters at wall center of the square tube dn downward flow i inner wall in inlet parameter o outer wall up upward flow w wall
and heat transfer in circular tubes under supercritical pressure. Therefore, this work studies the heat transfer characteristic of ndecane in a vertical square tube (di ¼ 1:8 mm) under supercritical pressure. The effects of operating pressure, mass flow rate, and flow direction on wall heat transfer are experimentally investigated. The circumferential heat transfer non-uniformity is examined and discussed as well. Finally, the heat transfer correlations are proposed, in which the local Nusselt number are correlated with physical properties and buoyancy number Gr=Re2:7 .
2. Experiments 2.1. Experimental facilities As shown in Fig. 1(a), the experimental system was composed of four major sub-systems, namely, flow supply sub-system, heating sub-system, cooling sub-system and data acquisition subsystem. The working fluid was extracted from a fuel tank with a plunger pump and then supplied to a pulsation damper to manage the flow oscillation. The mass flow rate was measured with a mass flowmeter, before which a throttle valve were added in order to adjust the flow rate. The working fluid was then supplied to two units of pre-heaters so that the flow temperature could be adjusted before entering the test section. The fuel temperature at the outlet of the preheating system could reach 500 C. In the current work, the test section is a heated square tube placed vertically. The tube wall served as the heat generator since the electrodes were directly connected to the two ends of the square tube. The maximum DC power applied to the tube wall could reach 20 kW. After the test section, the working fluid was cooled down by a cooling system through a heat exchanger. After a relief valve, the working fluid was collected. The operating pressure in the test section was regulated by adjusting the relief valve. As showed in Fig. 1(b), the outer edge of the square tube was 3.0 mm and the wall thickness was 0.6 mm, which gave the inner wall a side length of 1.8 mm. The heated section had a length of 500 mm. In order to supply better inlet and outlet conditions, two 150 mm long unheated straight tube were connected to the inlet and outlet of the test section respectively, which gave a length-to-diameter ratio of 83. The test section was made of 304
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Fig. 1. (a) The schematic of the experimental setup. (b) The SEM image of the cross-section of the vertical square tube.
stainless steel (1Cr18Ni9Ti). The thermocouples were welded on the outer wall of the tube with an interval of 38.46 mm in stream-wise direction. In addition, the external wall temperature T w;cor;o and T w;cen;o were measured by these thermocouples. The corresponding internal wall temperature, T w;cor;i and the T w;cen;i , were calculated by solving the inverse conduction problem. The test section was capsulated by the insulation material Aspen to reduce the heat loss. Since the internal convective heat transfer was strong, the heat loss could be negligible and the outer wall of the tube could be considered as adiabatic. The experimental conditions are listed in Table 1, where the flow inlet temperature was fixed at 473 K in all the cases.
2.2. Coolant and properties As mentioned, the working fluid in the current work was ndecane with a critical pressure and a critical temperature of
Table 1 Experimental conditions. Case
qw (kW/m2)
P (MPa)
_ (g/s) m
Flow direction
A B C D E F G H I J K L
500 500 500 500 300 300 500 500 300 300 500 500
3 4 5 5 5 5 3 4 3 3 5 5
1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.0 1.0 2.0 2.0
Upward Upward Upward Downward Upward Downward Downward Downward Upward Downward Upward Downward
2.39 MPa and 645.5 K, respectively. In order to guarantee the quality of n-decane in the experiments, a gas chromatograph analysis was performed to get the compositions. As shown in Table 2, the purity of n-decane reached 98.7%, which was satisfactory. The physical properties of the working fluid were obtained according to the principle of extended corresponding states (ECS) [18,19] where the unknown physical properties could be determined from a reference material that borne thermodynamic resemblance. Propane is a widely studied hydrocarbon, and a large amount of reliable data are available. Therefore, it could be a good reference material. The density of propane was calculated with the 32-parameter Modified Benedict-Webb-Rubin (MBWR) [20] equation. With propane density, the density of n-decane was obtained based on ECS principle. The transport properties prediction (TAPP) [21,22] was used to calculate the viscosity and thermal conductivity of n-decane. The thermal capacity of n-decane was calculated with the deviation function method [23]. Fig. 2 shows the variations of the physical properties against temperature under different pressures. In addition, the pseudo-critical points are marked on Fig. 2(b). It is noteworthy that the physical properties could be influenced by small impurity in H2O or CO2 [24]. However, regarding the organic working fluid n-decane, the difference between the calculated properties and the data in SUPERTRAPP database (standard database of National Institute of Standards and Technology [25]) was less than 5%. 2.3. Data reduction In order to evaluate the local internal convective heat transfer in the square tube, the heat transfer coefficient, hx , is calculated as follows,
hx ¼
q T w;x T b;x
ð2Þ
Table 2 N-decane chromatographic data. Item name
Analysis item
Liquid wax composition
C9 C10 C11 C12-18
Impurities
Aromatic content Bromine index Total sulfur Total sulfur
Index
Result
98.5%
0.2561 98.7413 0.3026
<0.5% <50) <5.0 +30
<0.5% <30 <5.0 +30
Test method UOP UOP UOP UOP
621 621 621 621
SH/T 0409 ASTM D 2710 SH/T 0253 GB/T 3555
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3. Results and discussion 3.1. Parametrical studies
Fig. 2. The variations of (a) density, (b) specific thermal capacity, (c) thermal conductivity and (d) viscosity against temperature under different pressures.
where T w;x is the local inner wall temperature and T b;x is the local bulk temperature of the main flow in the tube. The local heat transfer coefficient is nondimensionlized to local Nusselt number, Nux , which is defined as follows,
Nux ¼
hx d kb;x
3.1.1. Effects of operating pressure Fig. 3 shows the variations of wall temperature, bulk temperature, and the convective heat transfer coefficient along the tube under different operating pressures. In general, both parameters change significantly. The wall temperature increases almost linearly at the entrance section, a higher system pressure results in a higher wall temperature, which translates into a lower heat transfer coefficient since the heat flux is identical in different cases. Then the slope increases suddenly at the location around x=d ¼ 70, the sudden elevation of wall temperature is more significant in the case of lower pressure (case A), which generates a significant heat transfer deterioration. In the vicinity of the point x=d ¼ 130, the wall temperature reaches the maximum in all the cases. However, the differences of the wall temperature among the three cases are enlarged in the peak region. The highest wall temperature reaches around 900 K and 830 K in the case of the 3 MPa and 5 MPa, respectively. After the peak, an abrupt wall temperature decrease is observed before the location x=d ¼ 200. In the last section of the tube, the gradual increase of wall temperature is reestablished, and the heat transfer reduces accordingly. It could be inferred that the pseudo-critical point is approached when the wall temperature is suddenly elevated in the range from x=d ¼ 70 to 120. In this case, the significant variations of the thermal properties are anticipated. Since the heat energy is added to the bulk flow through the liquid layer adjacent to the wall, a relatively high temperature gradient should be established in the direction normal to the wall. In the vicinity of the pseudo-critical point, a moderate
ð3Þ
where d is the hydraulic diameter of the tube cross section and kb;x is the conductivity evaluated with local bulk temperature. As aforementioned, only the external wall temperature T w;cor;o and T w;cen;o was measured at the corner and wall center (see Fig. 1(b)). Therefore, a space marching method [26] was used to calculate the inner wall temperature T w;cor;i and T w;cen;i . This method was presented to solve the multidimensional inverse heat conduction problems based on the control volume method. According to the calculation, the maximum temperature difference of outer wall and inner wall is 3.8 K. The bulk temperature is calculated from the heat flux and the inlet temperature with the enthalpy balance,
T b;x ¼ H1
Qx þ HðT in Þ _ m
ð4Þ
where H is the enthalpy of working fluid, which is a function of flow temperature. The function H1 is the reverse process to calculate flow temperature from enthalpy. In the current study, the temperature difference between inner wall and bulk temperature is larger than 20 K. Since the uncertainty of bulk temperature and inner wall temperature are 0.6 K and 1.05 K, the maximum error of the temperature difference is around 8.2%. According to the error propagation theory, the uncertainty of the bulk temperature is mainly depended on the accuracy of the inlet temperature measurements since the electrical heating power could be precisely measured. It should be noted that the systematic error is not included. Besides, the deviation of the heat flux is around 1.49% of the value. The uncertainties of h and Nu are 8.8% and 9.5%, respectively.
Fig. 3. (a) The distributions of wall temperature and bulk temperature. (b) The variations of heat transfer coefficient under different pressures.
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temperature change is corresponding to a large property variation. However, the bulk temperature never reaches the pseudo-critical temperature. According to the measurements, the outlet temperature is 668 K, 677 K and 679 K in Cases A, B and C, respectively. In order to illustrate the phenomenon clearer, the following parameters are defined. Comparing the fluid density, heat capacity, and thermal conductivity that are evaluated with the inner wall temperature to that are evaluated with the bulk temperature, a series of parameter ratios can be determined.
Cq ¼
qw qb
ð5Þ
CC ¼
C p;w C p;b
ð6Þ
Ck ¼
kw kb
ð7Þ
Fig. 4 illustrates the variations of the three different parameters against the dimensionless location under different operating pressures. An apparent diversity of heat capacity ratio as well as thermal conductivity ratio among different cases is observed in the range between x=d ¼ 70 and 170. The maximum difference is observed around the point x=d ¼ 120. The heat capacity ratio is around 1.0 in the case of P = 5 MPa, whereas it reduces to 0.05 in the case of P = 3 MPa. Similarly, the thermal conductivity ratio reduces from 0.9 to 0.4 when the system pressure changes from 5 MPa to 3 MPa. A reduction of the density ratio with pressure can be observed around the pseudo-transient point, but it is not as significant as the other two parameters. A small value of thermal capacity ratio or thermal conductivity ratio means a weak thermal energy transfer capacity of the fluid in the vicinity of the heated wall. The significant reduction of the heat capacity and thermal conductivity of the wall-adjacent flow should be responsible to the significant wall heat transfer deteriorations in the middle of the channel.
635
and vertical downward flow. Fig. 5 compares the variations of the wall temperature, bulk temperature, and heat transfer coefficient in the vertical tube with two different flow directions and two different heating powers. As shown in Fig. 5(a), in the cases of high heat flux, unlike the peak of the wall temperature in the middle of the upward flow channel, the wall temperature increases monotonously in the downward flow channel. Despite the huge difference in the middle section of the channel, the measured wall temperature in the two cases are almost identical in the entrance and outlet region of the tube. The maximum wall temperature difference reaches around 120 K. Therefore, it could be concluded that the buoyancy has negative effects on heat transfer in the vertical upward flow configuration. For the upward flow, the direction of the buoyancy force of the wall-adjacent flow is in accordance with the main flow direction, which may result in a significant change of the velocity profile. According to Ref. [27], the distortion of the flow velocity and shear stress profile resulted in a low turbulence generation in a zone, which led to the heat transfer deterioration. The velocity change induced shear stress reduction could abate the intensity of the turbulent kinetic energy, which in turn gave rise to a heat transfer deterioration. For downward flow, on the contrary, the buoyancy force counteracts the main flow and enhances the heat transfer. Unlike the high-heat-flux cases, the wall temperature does not exceed the pseudo-transient point in Cases E and F. Therefore, there should not be significant thermal property gradient in each cross-section of the tube. Fig. 5(b) shows that the heat transfer difference between the vertical upward and downward flow is negligible in the cases of q ¼ 300 kW/m2, indicating that the buoyancy does not play an important role in this scenario. Thus, it would be helpful if a criterion could be generalized as an index to describe
3.1.2. Effects of flow direction As aforementioned, the buoyancy effect could play an important role in a vertical tube since the density varies across the channel. However, the scenario is different for the vertical upward flow
Fig. 4. The distributions of the physical property ratios along the tube under different pressures.
Fig. 5. (a) The distributions of wall temperature and bulk temperature. (b) The variations of heat transfer coefficient with different flow directions and different heating powers.
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whether the heat transfer deterioration could be generated in the vertical upward flow. 3.1.3. Heat transfer deterioration threshold The buoyant effect in supercritical flow is complex. Therefore, there are many literatures [3–5,28–30] regarding the parameters to evaluate the buoyancy effects. According to the review of Huang [31], the parameter Gr=Re2:7 was the best to fit the experimental
due to the thicker boundary layer at corner, which reduces the convective energy transportation. Secondly, according to the physical properties with temperature (see Fig. 2 (c)), the fluid thermal conductivity of the flow adjacent to the corner wall is much lower than that is close to the center wall, which results in a larger thermal resistance. The corner-to-center inner wall temperature difference h ¼ T w;cor;i T w;cen;i is employed to quantify the heat transfer
data. Therefore, a threshold based on the parameter Gr=Re2:7 is proposed in this paper. Given that the buoyancy effects in vertical upward flow and downward flow are different, comparing the heat transfer between those two cases could be a good method to shown the significance of the factor. Thus, a new parameter is defined as follows,
/¼
Nudn Nuup Nudn þ Nuup
ð8Þ
It quantifies the relative difference of the Nusselt number between vertical downward and upward configurations. As shown in Fig. 6, five different sets of data measured under different operating pressure and mass flow rate are plotted. Each set of data are collected in the same case at multiple spatial locations. It shows that there is a strong relation between the parameter / and Gr=Re2:7 . For example, when the buoyancy number Gr=Re2:7 is less than 1:6 105 , the parameter / is close to zero, and the heat transfer difference in vertical downward and upward flow configurations are almost identical. However, at a higher buoyancy number, a significant change of the parameter / is observed. When the buoyancy number is larger than 3 105 , the heat transfer difference is around 25–45%. 3.2. Heat transfer non-uniformity Since the square tube is not axis-symmetrical, the heat transfer variation in circumferential direction could be significant. Fig. 7 illustrates the variations of the inner wall temperature at the corner and center, T w;cor;i and T w;cen;i , along the tube. In general, the temperature on those two points varies in the same trend. However, the T w;cor;i is 10–30 K higher than the T w;cen;i . The reason could be inferred as follows. Firstly, the flow velocity of the fluid at the corner is lower than the flow velocity of fluid at the wall center
Fig. 6. Variation of parameter / against Gr=Re2:7 in a variety of working conditions.
Fig. 7. The wall temperature distributions of center points and corner points.
Fig. 8. The temperature difference distributions between corner points and center points of Case C and D.
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non-uniformity in the circumferential direction. Fig. 8 compares the variations of h in the upward flow and downward flow conditions. Three different regions of the tube flow could be defined, namely, x=d < 40; 40 < x=d < 140, and x=d > 140. The difference of h between upward flow and downward flow is calculated as Dh ¼ hup hdn , which indicates the intensity of the buoyancy effect. In the first region (x=d < 40), the difference between upward and downward flow is negligible (Dh 0). The reason could be inferred that the entrance effect suppresses the effects of buoyancy due to the thin thermal boundary layer. In the second region (40 < x=d < 140), the parameter Dh increases along the tube apparently. In the upward tube, the corner-to-center temperature difference hup increases continuously until the peak is reached at hup ¼ 28 K. However, the parameter decreases in the downward flow configuration. Therefore, the buoyancy plays an important role in Section 2. In the third part of the tube (x=d > 140), the parameter Dh decreases gradually, indicating that the buoyancy effect is weakened. The corner-to-center temperature differences in both upward and downward flow configurations are decreasing along the tube. However, the variation of h is more apparent in the case of upward flow. 3.3. Heat transfer correlations For the supercritical flow in a tube, the heat transfer correlations are important for potential engineering applications. However, unlike the circular tubes, a few studies were conducted on square tubes, and only numbered correlations have been reported. Therefore, it is necessary to establish the heat transfer correlations for square tube. 3.3.1. Comparison with published correlations In this section, the experimental data are compared with the predictions made by the related existing correlations in the literatures. Three representative correlations that were proposed by Jackson et al. [3], Watts and Chou [32], and Krasnoshchekov et al. [33] are chosen to predict Nu with the experimental conditions in this paper. As shown in Table 3, the heat transfer correlations of Jackson et al. [3] were the implicit expressions. Watts and Chou [32] developed their correlations based on Dittus-Boelter equation explicitly. Krasnoshchekov et al. [33] considered the correction terms of density and thermal capacity based on the correlation proposed by Petukhov et al. [34]. The calculations of U and Bo⁄ in Table 3 can be found in Refs. [3,32]. It is noteworthy that the aforementioned three correlations were generalized based on the old thermal properties for H2O and CO2, they should be corrected [35] to meet the IF-97 Standard. Fig. 9 shows the comparisons of the predictions of the existing correlations and our measurements. Since we employed n-decane as the working fluid, which could be pyrolyzed at high temperature, the experimental data were not used in the comparison if the maximum wall temperature was higher than 800 K. It can be seen that the existing correlations overestimate the heat transfer in general, especially for the center points. A large deviation is observed.
Table 3 Existing correlations in the literatures. Reference Jackson [3] Watts and Chou [32] Krasnoshchekov [33]
Correlations 2 0:46 Nu ¼ 1 8 104 Bo Nu f 0:35 0:55 qw Pr U Nub ¼ 0:021Re0:8 b b qb n 0:3 Nub ¼ Nu0 ðqw =qb Þ C p =C p;b
Nu Nuf
Fig. 9. Comparisons of the experimental data verses the correlation predicted results (a) for the corner points and (b) for the wall-center points.
In order to quantify the prediction error, the mean absolute percentage deviation (MAPD), root-mean-square deviation (RMSD), mean relative deviation (MRD) and error band (R30) of the three selected correlations are shown in Table 4. These residuals are calculated without distinguishing the flow directions (upward or downward flow) and locations (wall corner or wall center). It shows that the correlation developed by Krasnoshchekov et al. [33] is the best one to make the heat transfer prediction among the three correlations. However, MRD and RMSD are calculated
Table 4 Four correlations predictive statistics. Correlations
MAPD/%
MRD/%
RMSD/%
R30/%
Jackson [3] Watts and Chou [32] Krasnoshchekov et al. [33] Current work
48.52 48.50 30.38 10.87
44.88 47.49 22.29 0.19
53.97 52.75 39.05 13.42
42.0 32.83 72.86 99.7
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Table 5 The correlation coefficients. Flow direction
Position
c0
c1
c2
c3
c4
c5
c6
Upward
Center Corner
2.41 4.27
0.93 0.95
0.94 0.62
2.19 1.77
1.42 1.16
9.74 6.1
0.04 0.09
Downward
Center Corner
0.55 0.56
1.10 0.57
0.33 0.10
0.26 0.11
0.17 2.26
0.70 1.60
0.02 0.19
4. Conclusion The heat transfer characteristics of n-decane in a vertical square tube under supercritical pressure were experimental investigated. The temperature of inner wall corner and inner wall center was calculated with the space marching method based on the outer wall temperature measured by thermocouples. Besides, the flow direction in the vertical tube was switched in order to examine the effects of buoyancy. Moreover, the circumferential uniformity of heat transfer was discussed. The conclusions could be summarized as follows. (1) It was tested that the Gr=Re2:7 ¼ 1:6 105 was a threshold for n-decane in a vertical tube, after which the buoyancy effect was prominent. (2) Due to the square cross-section, there was a heat transfer non-uniformity in circumferential direction. The buoyancy promoted or suppressed the non-uniformity in the upward or downward flow configuration. (3) An empirical correlation for convective heat transfer of ndecane in a vertical square tube under supercritical pressure was proposed. It was able to make the heat transfer prediction with an error less than 30% for 99.7% of the data. Fig. 10. Comparison of experimental data and predicted results of correlation developed in this paper.
to be 22.3% of and 39.1% respectively, indicating a poor prediction precision. There are 72.9% data predicted with Krasnoshchekov correlation are within 30% band. All the three listed correlations are inaccurate to predict Nusselt number for the supercritical convective flow heat transfer in the vertical square tube. Therefore, it is necessary to develop the heat transfer correlations for supercritical n-decane in a vertical square tube. 3.3.2. Proposed heat transfer correlations According to the above discussions, the heat transfer is influenced by the variations of the physical properties and the buoyancy effect. Therefore, a heat transfer correlation expressed as follows is proposed for vertical square tube. The corresponding constants are listed in Table 5.
Nu ¼ c0
Nuc01
qw qb
!c 5 c2 c3 c4 c6 lw kw Cp Gr lb kb C p;b Re2:7 ð9Þ
Nu0 ¼
ð f =8ÞReb Pr pffiffiffiffiffiffiffiffi 2 0:76 þ 12:7 f =8 Pr3 1
f ¼ ½1:82 lgðReÞ 1:64
2
ð10Þ
ð11Þ
Fig. 10 shows the comparisons between the predictions and measurements. The accuracy of the correlation is relatively good. According to the analysis, 99.7% predicted data fall in the error band of 30%.
Acknowledgements The authors gratefully acknowledge funding support from the Program for National Natural Science Foundation of China (51876005). Conflict of interest None. References [1] T. Edwards, S. Zabarnick, Supercritical fuel deposition mechanisms, Industr. Eng. Chem. Res. 32 (12) (1993) 3117–3122. [2] K. Yamagata, K. Nishikawa, S. Hasegawa, T. Fujii, S. Yoshida, Forced convective heat transfer to supercritical water flowing in tubes, Int. J. Heat Mass Transf. 15 (12) (1972) 2575–2593. [3] J.D. Jackson, W.B. Hall, Forced convection heat transfer to fluids at supercritical pressure, 1979. [4] W.B. Hall, J.D. Jackson, Laminarization of a Turbulent Pipe Flow by Buoyancy Forces, in: ASME, New York, 1969, pp. 66. [5] J. Jackson, W.B. Hall, Influences of buoyancy on heat transfer to fluids flowing in vertical tubes under turbulent conditions, Turbul. Forced Convect. Channels Bund. 2 (1979) 613–640. [6] J. Jackson, M. Cotton, B. Axcell, Studies of mixed convection in vertical tubes, Int. J. Heat Fluid Flow 10 (1) (1989) 2–15. [7] J. Lee, P. Hejzlar, P. Saha, P. Stahle, M. Kazimi, D. McEligot, Deteriorated turbulent heat transfer (dtht) of gas up-flow in a circular tube: experimental data, Int. J. Heat Mass Transf. 51 (13-14) (2008) 3259–3266. [8] B.S. Shiralkar, P. Griffith, Deterioration in heat transfer to fluids at supercritical pressure and high heat fluxes, J. Heat Transf. 91 (1) (1969) 27–36. [9] B. Shiralkar, P. Griffith, The effect of swirl, inlet conditions, flow direction, and tube diameter on the heat transfer to fluids at supercritical pressure, J. Heat Transf. 92 (3) (1970) 465–471. [10] D. McEligot, C. Coon, H. Perkins, Relaminarization in tubes, Int. J. Heat Mass Transf. 13 (2) (1970) 431–433. [11] W. Zhou, W. Bao, J. Qin, Y. Qu, Deterioration in heat transfer of endothermal hydrocarbon fuel, J. Therm. Sci. 20 (2) (2011) 173–180.
J. Zhu et al. / International Journal of Heat and Mass Transfer 138 (2019) 631–639 [12] B. Liu, Y. Zhu, J.J. Yan, Y. Lei, B. Zhang, P.X. Jiang, Experimental investigation of convection heat transfer of n-decane at supercritical pressures in small vertical tubes, Int. J. Heat Mass Transf. 91 (2015) 734–746. [13] F. Zhong, X. Fan, G. Yu, J. Li, C.J. Sung, Heat transfer of aviation kerosene at supercritical conditions, J. Thermophys. Heat Transf. 23 (3) (2009) 543–550. [14] M. Sharabi, W. Ambrosini, S. He, J. Jackson, Prediction of turbulent convective heat transfer to a fluid at supercritical pressure in square and triangular channels, Ann. Nucl. Energy 35 (6) (2008) 993–1005. [15] J.K. Kim, H.K. Jeon, J.S. Lee, Wall temperature measurement and heat transfer correlation of turbulent supercritical carbon dioxide flow in vertical circular/ non-circular tubes, Nucl. Eng. Des. 237 (15-17) (2007) 1795–1802. [16] J.K. Kim, H.K. Jeon, J.S. Lee, Wall temperature measurements with turbulent flow in heated vertical circular/non-circular channels of supercritical pressure carbon-dioxide, Int. J. Heat Mass Transf. 50 (23–24) (2007) 4908–4911. [17] K. Sakurai, H.S. Ko, K. Okamoto, H. Madarame, Visualization study for pseudoboiling in supercritical carbon dioxide under forced convection in rectangular channel, in: Proceedings of the First International Symposium on Supercritical Water-Cooled Reactors, Design and Technology, 2000. [18] T.W. Leland, P.S. Chappelear, The corresponding states principle review of current theory and practice, Industr. Eng. Chem. 60 (7) (1968) 15–43. [19] B.E. Poling, J.M. Prausnitz, J.P. O’connell, et al., The Properties of Gases and Liquids, vol. 5, Mcgraw-hill, New York, 2001. [20] B. Younglove, J.F. Ely, Thermophysical properties of fluids. II. Methane, ethane, propane, isobutane, and normal butane, J. Phys. Chem. Ref. Data 16 (4) (1987) 577–798. [21] J.F. Ely, H. Hanley, Prediction of transport properties. 1. Viscosity of fluids and mixtures, Industr. Eng. Chem. Fundam. 20 (4) (1981) 323–332. [22] T.S. Storvick, S.I. Sandler, et al., Phase equilibria and fluid properties in the chemical industry, Am. Chem. Soc. (1977). [23] B.E. Poling, J.M. Prausnitz, J.P. O’connell, The Properties of Gases and Liquids, Mcgraw-hill, New York, 2001. [24] V.A. Kurganov, Y.A. Zeigarnik, I.V. Maslakova, Heat transfer and hydraulic resistance of supercritical-pressure coolants. Part I: Specifics of thermophysical properties of supercritical pressure fluids and turbulent heat transfer under heating conditions in round tubes (state of the art), Int. J. Heat Mass Transf. 55 (11–12) (2012) 3061–3075.
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[25] J. Ely, M. Huber, Nist thermophysical properties of hydrocarbon mixtures database (supertrapp), NIST Standard Reference Database 4. [26] J. Taler, W. Zima, Solution of inverse heat conduction problems using control volume approach, Int. J. Heat Mass Transf. 42 (6) (1999) 1123–1140. [27] V.A. Kurganov, Y.A. Zeigarnik, I.V. Maslakova, Heat transfer and hydraulic resistance of supercritical-pressure coolants. Part II: experimental data on hydraulic resistance and averaged turbulent flow structure of supercritical pressure fluids during heating in round tubes under normal and deteriorated heat transfer conditions, Int. J. Heat Mass Transf. 58 (1-2) (2013) 152–167. [28] S.H. Liu, Y.P. Huang, G.X. Liu, J.F. Wang, L.K.H. Leung, Improvement of buoyancy and acceleration parameters for forced and mixed convective heat transfer to supercritical fluids flowing in vertical tubes, Int. J. Heat Mass Transf. 106 (2017) 1144C1156. [29] D.E. Kim, M.H. Kim, Experimental study of the effects of flow acceleration and buoyancy on heat transfer in a supercritical fluid flow in a circular tube, Nucl. Eng. Des. 240 (2010) 3336C3349. [30] V.S. Protopopov, Generalized correlations for the local heat-transfer coefficients turbulent flows of water and carbon dioxide at supercritical pressure in a uniform heated circular tube, Teplofizika Vysokikh Temp. 15 (1977) 815C821. [31] D. Huang, W. Li, A brief review on the buoyancy criteria for supercritical fluids, Appl. Therm. Eng. 131 (2018) 977–987. [32] M.J. Watts, Mixed convection heat transfer to supercritical pressure water, in: Proceedings of the 7th International Heat Transfer Conference, Munchen, W. Germany, 1982, pp. 6–10,. [33] E. Krasnoshchekov, V. Protopopov, About heat transfer in flow of carbon dioxide and water at supercritical region of state parameters, Therm. Eng. 10 (1960) 94. [34] B. Petukhov, P. Kirillov, About heat transfer at turbulent fluid flow in tubes, Therm. Eng. 4 (1958) 63–68. [35] V.A. Kurganov, Y.A. Zeigarnik, I.V. Maslakova, Heat transfer and hydraulic resistance of supercritical pressure coolants. Part III: generalized description of SCP fluids normal heat transfer, empirical calculating correlations, integral method of theoretical calculations, Int. J. Heat Mass Transf. 67 (2013) 535–547.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Experimental investigation on heat transfer of n-decane in a vertical square tube under supercritical pressure Jianqin Zhu a, Chaofan Zhao a, Zeyuan Cheng b, Dasen Lin a, Zhi Tao a, Lu Qiu a,⇑ a b
National Key Laboratory of Science and Technology on Aero-Engine Aero-Thermodynamics, School of Energy and Power Engineering, Beihang University, Beijing 100191, PR China COMAC Beijing Aeronautical Science & Technology Research Institute, Beijing 100191, PR China
a r t i c l e
i n f o
Article history: Received 21 January 2019 Received in revised form 31 March 2019 Accepted 15 April 2019 Available online 25 April 2019 2010 MSC: 00-01 99-00 Keywords: Heat transfer N-decane Supercritical pressure Vertical square tube
a b s t r a c t The internal convective heat transfer characteristics of n-decane at supercritical pressure in a vertical square tube with a hydraulic diameter of 1.8 mm were experimentally investigated. The external wall temperatures of the square tube were measured with welded thermocouples whereas the internal wall temperatures were calculated with space marching method. In this experiment, the operating pressure ranged from 3 to 5 MPa and the heat flux from 100 to 500 kW/m2. Besides, the flow direction in the vertical tube was also switched in order to examine the effects of buoyancy. Moreover, the circumferential uniformity of heat transfer was also discussed. The results showed that the effects of buoyancy on the heat transfer were negligible once the local buoyancy number was less than a threshold. After the threshold, the buoyancy promoted the heat transfer in the downward flow configuration, but weakened the heat transfer in the upward flow one. The reason could be inferred that the buoyancy would influence the intensity of turbulent kinetic energy and the heat transfer near the wall. Finally, new heat transfer correlations for vertical upward and downward flow in square tubes were proposed. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction N-decane, as a typical hydrocarbon fuel, has a critical temperature of 645.5 K and a critical pressure of 2.390 MPa. However, in the application of regenerate cooling, the operation pressure in the channel is around 3.45–6.89 MPa which is higher than the critical pressure of the hydrocarbon fuel [1]. When the fuel flows through the heating channel under supercritical pressure, it could change from liquid state to supercritical state due to the convective heat transfer. In this process, the physical properties of the fuel vary significantly, which plays an important role in influencing the flow and heat transfer. Thus, experimental or numerical investigations have been conducted to study the flow and heat transfer characteristics in a heating tube under supercritical pressure. Yamagata et al. [2] experimentally studied the heat transfer characteristics of water in a circular tube under supercritical pressure. The results showed that the dramatic change of thermal properties at pseudo-critical temperature resulted in a complex relation between heat transfer coefficient and heat flux. Jackson and Hall [3,4] reported that heat transfer deterioration could be generated around the pseudo-critical point. They proposed that ⇑ Corresponding author. E-mail address: [email protected] (L. Qiu). https://doi.org/10.1016/j.ijheatmasstransfer.2019.04.076 0017-9310/Ó 2019 Elsevier Ltd. All rights reserved.
the temperature difference between the main flow and the thin fluid layer adjacent to the wall gave rise to a significant density gradient, which in turn generated the buoyancy flow. Besides, they inferred that the interaction of the buoyancy flow and the main flow could influence the near-wall turbulence production, which affected the convective heat transfer. Later on, Jackson and Hall [3,5,6] developed a semi-empirical theory and deducted a dimensionless buoyancy number Gr=Re2:7 to evaluate the strength of buoyancy force (see the definitions of Gr and Re in Nomenclature). They proposed Gr=Re2:7 ¼ 105 as the threshold of heat transfer deterioration to supercritical water flowing in a upward vertical tube. Aside from the buoyancy effect, thermal acceleration effect is another key factor to heat transfer deterioration with a different mechanism. Shiralkar [7] and Griffith [8,9] experimentally investigated the heat transfer characteristics of supercritical CO2 flow in a circular tube. Even though the flow Reynolds number was relatively high, the heat transfer deterioration occurred in both upward flow and downward flow conditions. In order to quantify the strength of the thermal acceleration effect, McEligot et al. [10] developed a dimensionless number K v as follows,
Kv ¼
4qd 2
Re
lC p T
ð1Þ
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Nomenclature Cp d g Gr h H _ m Nu Pr P q Qx Re T u x
ap C
isobaric specific heat capacity (J/(kg K)) hydraulic diameter (m) gravitational acceleration (m/s2) Grashof number based on bulk temperature, 3 Gr ¼ ðT w T b Þg ap d =m2b convective heat transfer coefficient (W/(m2 K)) enthalpy (J/kg) mass flow rate (g/s) Nusselt number Prandtl number operating pressure (MPa) wall heat flux (W/m2) the heat energy added to the fluid before location x (W) Reynolds number, Re ¼ qb ud=l temperature (K) flow velocity (m/s) the axial distance in the tube (m) thermal expansion coefficient (1/K) the physical property ratio of wall-adjacent fluid against bulk fluid
It was observed that the thermal acceleration could weaken the turbulence intensity and induce relaminarization in the case of K v > 3 106 . In the aforementioned investigations, the working fluid was inorganic substance, such as CO2 and H2O. When it comes to the organic substance, the flow and heat transfer characteristics might be different. Zhou et al. [11] experimentally investigated the heat transfer characteristics of pentane flowed though circular channel under supercritical pressure and developed a heat transfer correlation to include the effects of inlet temperature, wall temperature, mass flux, heat flux, inlet pressure and bulk temperature. The results indicated that the heat transfer deterioration occurred when the fuel temperature approached the pseudo-critical temperature. Liu et al. [12] studied the heat transfer characteristics of n-decane in vertical circular tube under different pressure, mass flux and flow direction. The results showed that the effects of buoyancy and thermal acceleration were negligible in the condition of high inlet flow rate. Zhong et al. [13] experimental investigated the heat transfer characteristics of RP-3 in circular tube under supercritical pressure. In accordance with pure inorganic substance, the dramatic variations of the physical properties gave rise to the heat transfer deterioration for the hydrocarbon fuel. Rectangular channels are widely used in actual regenerative cooling systems due to the manufacturing concerns. Different from the circular tube, non-circular ones have obvious threedimensional effects. Sharabi et al. [14] studied the heat transfer characteristic of CO2 in vertical circular, square, and rectangular tubes under supercritical pressure. They observed that the wall temperature at the corner was always higher than that on the center of the wall, indicating that there was a low heat transfer region in the wall corner. Kim et al. [15,16] compared the heat transfer characteristics of CO2 in vertical circular, triangle, and rectangular tubes under supercritical pressure. The results showed that the rectangular tube was prone to generate heat transfer deterioration compared to the triangle and circular tubes. Sakurai et al. [17] investigated the flow and heat transfer characteristics of CO2 in a transparent rectangular tube. The near-wall flow pattern was captured and was correlated with the convective heat transfer. To sum up, in the application of regenerate cooling, hydrocarbon fuel flows through square heating channels under supercritical pressure. However, most of the related studies focused on the flow
h k
l m q
/
temperature difference between the corner and wall center of the square tube (K) thermal conductivity (W/(m K)) dynamic viscosity (Pa s) kinematic viscosity (m2/s) density (kg/m3) the relative heat transfer difference between the upward and downward flow
Subscripts b bulk cor parameters at the corner of the square tube cen parameters at wall center of the square tube dn downward flow i inner wall in inlet parameter o outer wall up upward flow w wall
and heat transfer in circular tubes under supercritical pressure. Therefore, this work studies the heat transfer characteristic of ndecane in a vertical square tube (di ¼ 1:8 mm) under supercritical pressure. The effects of operating pressure, mass flow rate, and flow direction on wall heat transfer are experimentally investigated. The circumferential heat transfer non-uniformity is examined and discussed as well. Finally, the heat transfer correlations are proposed, in which the local Nusselt number are correlated with physical properties and buoyancy number Gr=Re2:7 .
2. Experiments 2.1. Experimental facilities As shown in Fig. 1(a), the experimental system was composed of four major sub-systems, namely, flow supply sub-system, heating sub-system, cooling sub-system and data acquisition subsystem. The working fluid was extracted from a fuel tank with a plunger pump and then supplied to a pulsation damper to manage the flow oscillation. The mass flow rate was measured with a mass flowmeter, before which a throttle valve were added in order to adjust the flow rate. The working fluid was then supplied to two units of pre-heaters so that the flow temperature could be adjusted before entering the test section. The fuel temperature at the outlet of the preheating system could reach 500 C. In the current work, the test section is a heated square tube placed vertically. The tube wall served as the heat generator since the electrodes were directly connected to the two ends of the square tube. The maximum DC power applied to the tube wall could reach 20 kW. After the test section, the working fluid was cooled down by a cooling system through a heat exchanger. After a relief valve, the working fluid was collected. The operating pressure in the test section was regulated by adjusting the relief valve. As showed in Fig. 1(b), the outer edge of the square tube was 3.0 mm and the wall thickness was 0.6 mm, which gave the inner wall a side length of 1.8 mm. The heated section had a length of 500 mm. In order to supply better inlet and outlet conditions, two 150 mm long unheated straight tube were connected to the inlet and outlet of the test section respectively, which gave a length-to-diameter ratio of 83. The test section was made of 304
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Fig. 1. (a) The schematic of the experimental setup. (b) The SEM image of the cross-section of the vertical square tube.
stainless steel (1Cr18Ni9Ti). The thermocouples were welded on the outer wall of the tube with an interval of 38.46 mm in stream-wise direction. In addition, the external wall temperature T w;cor;o and T w;cen;o were measured by these thermocouples. The corresponding internal wall temperature, T w;cor;i and the T w;cen;i , were calculated by solving the inverse conduction problem. The test section was capsulated by the insulation material Aspen to reduce the heat loss. Since the internal convective heat transfer was strong, the heat loss could be negligible and the outer wall of the tube could be considered as adiabatic. The experimental conditions are listed in Table 1, where the flow inlet temperature was fixed at 473 K in all the cases.
2.2. Coolant and properties As mentioned, the working fluid in the current work was ndecane with a critical pressure and a critical temperature of
Table 1 Experimental conditions. Case
qw (kW/m2)
P (MPa)
_ (g/s) m
Flow direction
A B C D E F G H I J K L
500 500 500 500 300 300 500 500 300 300 500 500
3 4 5 5 5 5 3 4 3 3 5 5
1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.0 1.0 2.0 2.0
Upward Upward Upward Downward Upward Downward Downward Downward Upward Downward Upward Downward
2.39 MPa and 645.5 K, respectively. In order to guarantee the quality of n-decane in the experiments, a gas chromatograph analysis was performed to get the compositions. As shown in Table 2, the purity of n-decane reached 98.7%, which was satisfactory. The physical properties of the working fluid were obtained according to the principle of extended corresponding states (ECS) [18,19] where the unknown physical properties could be determined from a reference material that borne thermodynamic resemblance. Propane is a widely studied hydrocarbon, and a large amount of reliable data are available. Therefore, it could be a good reference material. The density of propane was calculated with the 32-parameter Modified Benedict-Webb-Rubin (MBWR) [20] equation. With propane density, the density of n-decane was obtained based on ECS principle. The transport properties prediction (TAPP) [21,22] was used to calculate the viscosity and thermal conductivity of n-decane. The thermal capacity of n-decane was calculated with the deviation function method [23]. Fig. 2 shows the variations of the physical properties against temperature under different pressures. In addition, the pseudo-critical points are marked on Fig. 2(b). It is noteworthy that the physical properties could be influenced by small impurity in H2O or CO2 [24]. However, regarding the organic working fluid n-decane, the difference between the calculated properties and the data in SUPERTRAPP database (standard database of National Institute of Standards and Technology [25]) was less than 5%. 2.3. Data reduction In order to evaluate the local internal convective heat transfer in the square tube, the heat transfer coefficient, hx , is calculated as follows,
hx ¼
q T w;x T b;x
ð2Þ
Table 2 N-decane chromatographic data. Item name
Analysis item
Liquid wax composition
C9 C10 C11 C12-18
Impurities
Aromatic content Bromine index Total sulfur Total sulfur
Index
Result
98.5%
0.2561 98.7413 0.3026
<0.5% <50) <5.0 +30
<0.5% <30 <5.0 +30
Test method UOP UOP UOP UOP
621 621 621 621
SH/T 0409 ASTM D 2710 SH/T 0253 GB/T 3555
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3. Results and discussion 3.1. Parametrical studies
Fig. 2. The variations of (a) density, (b) specific thermal capacity, (c) thermal conductivity and (d) viscosity against temperature under different pressures.
where T w;x is the local inner wall temperature and T b;x is the local bulk temperature of the main flow in the tube. The local heat transfer coefficient is nondimensionlized to local Nusselt number, Nux , which is defined as follows,
Nux ¼
hx d kb;x
3.1.1. Effects of operating pressure Fig. 3 shows the variations of wall temperature, bulk temperature, and the convective heat transfer coefficient along the tube under different operating pressures. In general, both parameters change significantly. The wall temperature increases almost linearly at the entrance section, a higher system pressure results in a higher wall temperature, which translates into a lower heat transfer coefficient since the heat flux is identical in different cases. Then the slope increases suddenly at the location around x=d ¼ 70, the sudden elevation of wall temperature is more significant in the case of lower pressure (case A), which generates a significant heat transfer deterioration. In the vicinity of the point x=d ¼ 130, the wall temperature reaches the maximum in all the cases. However, the differences of the wall temperature among the three cases are enlarged in the peak region. The highest wall temperature reaches around 900 K and 830 K in the case of the 3 MPa and 5 MPa, respectively. After the peak, an abrupt wall temperature decrease is observed before the location x=d ¼ 200. In the last section of the tube, the gradual increase of wall temperature is reestablished, and the heat transfer reduces accordingly. It could be inferred that the pseudo-critical point is approached when the wall temperature is suddenly elevated in the range from x=d ¼ 70 to 120. In this case, the significant variations of the thermal properties are anticipated. Since the heat energy is added to the bulk flow through the liquid layer adjacent to the wall, a relatively high temperature gradient should be established in the direction normal to the wall. In the vicinity of the pseudo-critical point, a moderate
ð3Þ
where d is the hydraulic diameter of the tube cross section and kb;x is the conductivity evaluated with local bulk temperature. As aforementioned, only the external wall temperature T w;cor;o and T w;cen;o was measured at the corner and wall center (see Fig. 1(b)). Therefore, a space marching method [26] was used to calculate the inner wall temperature T w;cor;i and T w;cen;i . This method was presented to solve the multidimensional inverse heat conduction problems based on the control volume method. According to the calculation, the maximum temperature difference of outer wall and inner wall is 3.8 K. The bulk temperature is calculated from the heat flux and the inlet temperature with the enthalpy balance,
T b;x ¼ H1
Qx þ HðT in Þ _ m
ð4Þ
where H is the enthalpy of working fluid, which is a function of flow temperature. The function H1 is the reverse process to calculate flow temperature from enthalpy. In the current study, the temperature difference between inner wall and bulk temperature is larger than 20 K. Since the uncertainty of bulk temperature and inner wall temperature are 0.6 K and 1.05 K, the maximum error of the temperature difference is around 8.2%. According to the error propagation theory, the uncertainty of the bulk temperature is mainly depended on the accuracy of the inlet temperature measurements since the electrical heating power could be precisely measured. It should be noted that the systematic error is not included. Besides, the deviation of the heat flux is around 1.49% of the value. The uncertainties of h and Nu are 8.8% and 9.5%, respectively.
Fig. 3. (a) The distributions of wall temperature and bulk temperature. (b) The variations of heat transfer coefficient under different pressures.
J. Zhu et al. / International Journal of Heat and Mass Transfer 138 (2019) 631–639
temperature change is corresponding to a large property variation. However, the bulk temperature never reaches the pseudo-critical temperature. According to the measurements, the outlet temperature is 668 K, 677 K and 679 K in Cases A, B and C, respectively. In order to illustrate the phenomenon clearer, the following parameters are defined. Comparing the fluid density, heat capacity, and thermal conductivity that are evaluated with the inner wall temperature to that are evaluated with the bulk temperature, a series of parameter ratios can be determined.
Cq ¼
qw qb
ð5Þ
CC ¼
C p;w C p;b
ð6Þ
Ck ¼
kw kb
ð7Þ
Fig. 4 illustrates the variations of the three different parameters against the dimensionless location under different operating pressures. An apparent diversity of heat capacity ratio as well as thermal conductivity ratio among different cases is observed in the range between x=d ¼ 70 and 170. The maximum difference is observed around the point x=d ¼ 120. The heat capacity ratio is around 1.0 in the case of P = 5 MPa, whereas it reduces to 0.05 in the case of P = 3 MPa. Similarly, the thermal conductivity ratio reduces from 0.9 to 0.4 when the system pressure changes from 5 MPa to 3 MPa. A reduction of the density ratio with pressure can be observed around the pseudo-transient point, but it is not as significant as the other two parameters. A small value of thermal capacity ratio or thermal conductivity ratio means a weak thermal energy transfer capacity of the fluid in the vicinity of the heated wall. The significant reduction of the heat capacity and thermal conductivity of the wall-adjacent flow should be responsible to the significant wall heat transfer deteriorations in the middle of the channel.
635
and vertical downward flow. Fig. 5 compares the variations of the wall temperature, bulk temperature, and heat transfer coefficient in the vertical tube with two different flow directions and two different heating powers. As shown in Fig. 5(a), in the cases of high heat flux, unlike the peak of the wall temperature in the middle of the upward flow channel, the wall temperature increases monotonously in the downward flow channel. Despite the huge difference in the middle section of the channel, the measured wall temperature in the two cases are almost identical in the entrance and outlet region of the tube. The maximum wall temperature difference reaches around 120 K. Therefore, it could be concluded that the buoyancy has negative effects on heat transfer in the vertical upward flow configuration. For the upward flow, the direction of the buoyancy force of the wall-adjacent flow is in accordance with the main flow direction, which may result in a significant change of the velocity profile. According to Ref. [27], the distortion of the flow velocity and shear stress profile resulted in a low turbulence generation in a zone, which led to the heat transfer deterioration. The velocity change induced shear stress reduction could abate the intensity of the turbulent kinetic energy, which in turn gave rise to a heat transfer deterioration. For downward flow, on the contrary, the buoyancy force counteracts the main flow and enhances the heat transfer. Unlike the high-heat-flux cases, the wall temperature does not exceed the pseudo-transient point in Cases E and F. Therefore, there should not be significant thermal property gradient in each cross-section of the tube. Fig. 5(b) shows that the heat transfer difference between the vertical upward and downward flow is negligible in the cases of q ¼ 300 kW/m2, indicating that the buoyancy does not play an important role in this scenario. Thus, it would be helpful if a criterion could be generalized as an index to describe
3.1.2. Effects of flow direction As aforementioned, the buoyancy effect could play an important role in a vertical tube since the density varies across the channel. However, the scenario is different for the vertical upward flow
Fig. 4. The distributions of the physical property ratios along the tube under different pressures.
Fig. 5. (a) The distributions of wall temperature and bulk temperature. (b) The variations of heat transfer coefficient with different flow directions and different heating powers.
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whether the heat transfer deterioration could be generated in the vertical upward flow. 3.1.3. Heat transfer deterioration threshold The buoyant effect in supercritical flow is complex. Therefore, there are many literatures [3–5,28–30] regarding the parameters to evaluate the buoyancy effects. According to the review of Huang [31], the parameter Gr=Re2:7 was the best to fit the experimental
due to the thicker boundary layer at corner, which reduces the convective energy transportation. Secondly, according to the physical properties with temperature (see Fig. 2 (c)), the fluid thermal conductivity of the flow adjacent to the corner wall is much lower than that is close to the center wall, which results in a larger thermal resistance. The corner-to-center inner wall temperature difference h ¼ T w;cor;i T w;cen;i is employed to quantify the heat transfer
data. Therefore, a threshold based on the parameter Gr=Re2:7 is proposed in this paper. Given that the buoyancy effects in vertical upward flow and downward flow are different, comparing the heat transfer between those two cases could be a good method to shown the significance of the factor. Thus, a new parameter is defined as follows,
/¼
Nudn Nuup Nudn þ Nuup
ð8Þ
It quantifies the relative difference of the Nusselt number between vertical downward and upward configurations. As shown in Fig. 6, five different sets of data measured under different operating pressure and mass flow rate are plotted. Each set of data are collected in the same case at multiple spatial locations. It shows that there is a strong relation between the parameter / and Gr=Re2:7 . For example, when the buoyancy number Gr=Re2:7 is less than 1:6 105 , the parameter / is close to zero, and the heat transfer difference in vertical downward and upward flow configurations are almost identical. However, at a higher buoyancy number, a significant change of the parameter / is observed. When the buoyancy number is larger than 3 105 , the heat transfer difference is around 25–45%. 3.2. Heat transfer non-uniformity Since the square tube is not axis-symmetrical, the heat transfer variation in circumferential direction could be significant. Fig. 7 illustrates the variations of the inner wall temperature at the corner and center, T w;cor;i and T w;cen;i , along the tube. In general, the temperature on those two points varies in the same trend. However, the T w;cor;i is 10–30 K higher than the T w;cen;i . The reason could be inferred as follows. Firstly, the flow velocity of the fluid at the corner is lower than the flow velocity of fluid at the wall center
Fig. 6. Variation of parameter / against Gr=Re2:7 in a variety of working conditions.
Fig. 7. The wall temperature distributions of center points and corner points.
Fig. 8. The temperature difference distributions between corner points and center points of Case C and D.
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non-uniformity in the circumferential direction. Fig. 8 compares the variations of h in the upward flow and downward flow conditions. Three different regions of the tube flow could be defined, namely, x=d < 40; 40 < x=d < 140, and x=d > 140. The difference of h between upward flow and downward flow is calculated as Dh ¼ hup hdn , which indicates the intensity of the buoyancy effect. In the first region (x=d < 40), the difference between upward and downward flow is negligible (Dh 0). The reason could be inferred that the entrance effect suppresses the effects of buoyancy due to the thin thermal boundary layer. In the second region (40 < x=d < 140), the parameter Dh increases along the tube apparently. In the upward tube, the corner-to-center temperature difference hup increases continuously until the peak is reached at hup ¼ 28 K. However, the parameter decreases in the downward flow configuration. Therefore, the buoyancy plays an important role in Section 2. In the third part of the tube (x=d > 140), the parameter Dh decreases gradually, indicating that the buoyancy effect is weakened. The corner-to-center temperature differences in both upward and downward flow configurations are decreasing along the tube. However, the variation of h is more apparent in the case of upward flow. 3.3. Heat transfer correlations For the supercritical flow in a tube, the heat transfer correlations are important for potential engineering applications. However, unlike the circular tubes, a few studies were conducted on square tubes, and only numbered correlations have been reported. Therefore, it is necessary to establish the heat transfer correlations for square tube. 3.3.1. Comparison with published correlations In this section, the experimental data are compared with the predictions made by the related existing correlations in the literatures. Three representative correlations that were proposed by Jackson et al. [3], Watts and Chou [32], and Krasnoshchekov et al. [33] are chosen to predict Nu with the experimental conditions in this paper. As shown in Table 3, the heat transfer correlations of Jackson et al. [3] were the implicit expressions. Watts and Chou [32] developed their correlations based on Dittus-Boelter equation explicitly. Krasnoshchekov et al. [33] considered the correction terms of density and thermal capacity based on the correlation proposed by Petukhov et al. [34]. The calculations of U and Bo⁄ in Table 3 can be found in Refs. [3,32]. It is noteworthy that the aforementioned three correlations were generalized based on the old thermal properties for H2O and CO2, they should be corrected [35] to meet the IF-97 Standard. Fig. 9 shows the comparisons of the predictions of the existing correlations and our measurements. Since we employed n-decane as the working fluid, which could be pyrolyzed at high temperature, the experimental data were not used in the comparison if the maximum wall temperature was higher than 800 K. It can be seen that the existing correlations overestimate the heat transfer in general, especially for the center points. A large deviation is observed.
Table 3 Existing correlations in the literatures. Reference Jackson [3] Watts and Chou [32] Krasnoshchekov [33]
Correlations 2 0:46 Nu ¼ 1 8 104 Bo Nu f 0:35 0:55 qw Pr U Nub ¼ 0:021Re0:8 b b qb n 0:3 Nub ¼ Nu0 ðqw =qb Þ C p =C p;b
Nu Nuf
Fig. 9. Comparisons of the experimental data verses the correlation predicted results (a) for the corner points and (b) for the wall-center points.
In order to quantify the prediction error, the mean absolute percentage deviation (MAPD), root-mean-square deviation (RMSD), mean relative deviation (MRD) and error band (R30) of the three selected correlations are shown in Table 4. These residuals are calculated without distinguishing the flow directions (upward or downward flow) and locations (wall corner or wall center). It shows that the correlation developed by Krasnoshchekov et al. [33] is the best one to make the heat transfer prediction among the three correlations. However, MRD and RMSD are calculated
Table 4 Four correlations predictive statistics. Correlations
MAPD/%
MRD/%
RMSD/%
R30/%
Jackson [3] Watts and Chou [32] Krasnoshchekov et al. [33] Current work
48.52 48.50 30.38 10.87
44.88 47.49 22.29 0.19
53.97 52.75 39.05 13.42
42.0 32.83 72.86 99.7
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J. Zhu et al. / International Journal of Heat and Mass Transfer 138 (2019) 631–639
Table 5 The correlation coefficients. Flow direction
Position
c0
c1
c2
c3
c4
c5
c6
Upward
Center Corner
2.41 4.27
0.93 0.95
0.94 0.62
2.19 1.77
1.42 1.16
9.74 6.1
0.04 0.09
Downward
Center Corner
0.55 0.56
1.10 0.57
0.33 0.10
0.26 0.11
0.17 2.26
0.70 1.60
0.02 0.19
4. Conclusion The heat transfer characteristics of n-decane in a vertical square tube under supercritical pressure were experimental investigated. The temperature of inner wall corner and inner wall center was calculated with the space marching method based on the outer wall temperature measured by thermocouples. Besides, the flow direction in the vertical tube was switched in order to examine the effects of buoyancy. Moreover, the circumferential uniformity of heat transfer was discussed. The conclusions could be summarized as follows. (1) It was tested that the Gr=Re2:7 ¼ 1:6 105 was a threshold for n-decane in a vertical tube, after which the buoyancy effect was prominent. (2) Due to the square cross-section, there was a heat transfer non-uniformity in circumferential direction. The buoyancy promoted or suppressed the non-uniformity in the upward or downward flow configuration. (3) An empirical correlation for convective heat transfer of ndecane in a vertical square tube under supercritical pressure was proposed. It was able to make the heat transfer prediction with an error less than 30% for 99.7% of the data. Fig. 10. Comparison of experimental data and predicted results of correlation developed in this paper.
to be 22.3% of and 39.1% respectively, indicating a poor prediction precision. There are 72.9% data predicted with Krasnoshchekov correlation are within 30% band. All the three listed correlations are inaccurate to predict Nusselt number for the supercritical convective flow heat transfer in the vertical square tube. Therefore, it is necessary to develop the heat transfer correlations for supercritical n-decane in a vertical square tube. 3.3.2. Proposed heat transfer correlations According to the above discussions, the heat transfer is influenced by the variations of the physical properties and the buoyancy effect. Therefore, a heat transfer correlation expressed as follows is proposed for vertical square tube. The corresponding constants are listed in Table 5.
Nu ¼ c0
Nuc01
qw qb
!c 5 c2 c3 c4 c6 lw kw Cp Gr lb kb C p;b Re2:7 ð9Þ
Nu0 ¼
ð f =8ÞReb Pr pffiffiffiffiffiffiffiffi 2 0:76 þ 12:7 f =8 Pr3 1
f ¼ ½1:82 lgðReÞ 1:64
2
ð10Þ
ð11Þ
Fig. 10 shows the comparisons between the predictions and measurements. The accuracy of the correlation is relatively good. According to the analysis, 99.7% predicted data fall in the error band of 30%.
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