Buoyancy effect on the mixed convection flow and heat transfer of supercritical R134a in heated horizontal tubes

International Journal of Heat and Mass Transfer 144 (2019) 118607

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Buoyancy effect on the mixed convection flow and heat transfer of supercritical R134a in heated horizontal tubes Ran Tian a, Mingshan Wei a, Xiaoye Dai c, Panpan Song a, Lin Shi b,⇑ a

School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Energy and Power Engineering, Tsinghua University, Beijing 100084, China c Department of Chemical Engineering, Tsinghua University, China b

a r t i c l e

i n f o

Article history: Received 22 December 2018 Received in revised form 25 July 2019 Accepted 20 August 2019

Keywords: Numerical study Buoyancy effect Supercritical heat transfer Horizontal flow ORC

a b s t r a c t Thermal non-uniformity in the horizontal mixed convection heat transfer of fluids at the supercritical pressure is a major issue that must be addressed in a trans-critical organic Rankine cycle. However, the heat transfer mechanism is not fully understood. To further investigate the mechanisms of the buoyancy effect and property variations in a horizontal supercritical flow, mixed convection with supercritical pressure R134a is studied numerically herein using the AKN turbulence model. When the buoyancy effect is weak, the difference in the turbulent kinetic energy with ktop < kbottom is the dominating factor resulting in a non-uniform heat transfer. In a strongly buoyancy-affected mixed convection, the flow process is divided into three regions. In region I (Tw increasing section) and region III (gas-like section), ktop < kbottom because of the greater velocity gradient at the bottom; in region II, where Tw,top reaches a peak and subsequently decreases, ktop > kbottom is observed because the newly developed vortexes near the tube top intensifies the turbulence near the top. Heat transfer cases with various tube diameters and pressures are discussed. A stronger buoyancy effect is developed in larger tubes. No new vortex is formed in a 2-mm tube while multiple vortexes are developed in the upper region of 16-mm and 26-mm tubes, providing stronger turbulence for the heat transfer recovery. As the specific heat is sensitive to the pressure variation while the density variation with pressure is moderate, the pressure has less effect on heat transfer in a strong-buoyancy case than in a weak-buoyancy case. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction The utilization of sustainable energy and waste heat can effectively ease the energy crisis and counter potential environmental threats [1,2]. The trans-critical organic Rankine cycle (ORC), i.e., the trans-critical power cycle using organic fluids as working fluids, is a promising technology for low-grade energy utilization that provides higher thermal efficiency and a more compact system [3–5]. An important problem involved in the ORC is the heat transfer of fluids at supercritical pressures. As shown in Fig. 1, during the heat transfer process, the fluids at supercritical pressures undergo significant property variations, leading to complex and unique heat transfer phenomena, such as enhanced and deteriorated heat transfers, thereby affecting the economic and safety performance of the system [7–9]. Supercritical heat transfer is also a typical problem considered in other applications such as refrigeration, heat

⇑ Corresponding author. E-mail address: [email protected] (L. Shi). https://doi.org/10.1016/j.ijheatmasstransfer.2019.118607 0017-9310/Ó 2019 Elsevier Ltd. All rights reserved.

pumps, nuclear reactors, and the cooling technology of spacecraft [10–12]. Heat transfer experiments with supercritical pressure fluids have been conducted extensively, especially for water and CO2 in vertical tubes. Heat transfer enhancement and deterioration owing to property changes and the buoyancy effect are primarily discussed. Pioro and Duffey [13] reviewed over 450 papers and concluded that the majority of experimental data were obtained in vertical circular tubes, and in general, there are three heat transfer models, that is normal heat transfer, deteriorated heat transfer and enhanced heat transfer. The review works of Huang et al. [14] and Cabeza et al. [15] on experimental studies performed using vertical tubes show that the parametric effect (i.e. mass flux, heat flux, pressure and flow direction) on heat transfer have been extensively discussed and relatively consistent conclusions were drawn. However, heat transfer correlations based on one working fluid in circular tubes cannot be applied directly to other working fluids and have difficulty in expanding to other geometries. With the development of the trans-critical ORC, designing a vapor generator requires the understanding of the supercritical

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R. Tian et al. / International Journal of Heat and Mass Transfer 144 (2019) 118607

Nomenclature cp d G Grq/Grth

specific heat (kJ/kg
K) tube diameter (m) mass fluxðkg
m2
s1 Þ buoyancy criterion for horizontal supercritical flow, 

Gr q ¼ g b qw d =kb m2b ,4

Gr th ¼ 3  10 h k p q r R Re T DTw

- 5

 0:5

Pr

 2=3

Re2:75 ð1 þ 2:4 Pr b

1

1=8

Reb

Þ

enthalpy (kJ/kg) turbulent kinetic energy (m2/s2) pressure (MPa) heat flux (kW/m2) radial direction (m) tube diameter (m) Reynolds number temperature (K) wall temperature difference between the top and bottom surface (K) velocity (m/s) pffiffiffiffiffiffiffiffiffiffiffiffi Non-dimensional distance from the wall, =y sw =q=m axial direction of the tube

u y+ z

Greek symbols q density (kg/m3)

40 1200

800

5

Abbreviations AKN Abe, Kondoh and Nagano turbulence model HTC heat transfer coefficient HTD heat transfer deterioration HTE heat transfer enhancement ORC organic Rankine cycle SHT Supercritical heat transfer

440

140

600

50 3

400

120

desnity(kg/m ) cp(kJ/kg-K)

40

80 60 40

200

30

0

20

60

80

100

120

140

400 380

20

0

40

420

100

viscosity ( Pa-s)

15 10

Subscripts b bulk c critical point top top surface bot, bottom bottom surface pc pseudo-critical point w wall in inner

Tw(K)

20

m

1000 60

25

k

160 70

35 30

dissipation rate (m2s-3) dynamic viscosity (lPa
s) turbulent viscosity (lPa
s) thermal conductivityðW
m1
K1 Þ kinetic viscosity (m2/s)

80

Tpc

(mW/m-K)

e l lt

0

G1500-q20 G1000-q50 G600-q50

360

160

o

T( C)

340 300

Fig. 1. Property changes near the pseudo-critical point of R134a at 4.26 MPa (properties are taken from NIST Refprop [6]).

heat transfer characteristics of organic fluids in horizontal tubes. R134a [16], R245fa [17] and R1233zd(E) [18], the commonly used or the promising working fluids in ORC, were chosen to conduct supercritical heat transfer experiments in vertical tubes. Basic experimental data were obtained and the applicability of various heat transfer correlations were evaluated or new correlations were developed. As heat exchangers are typically horizontally oriented, Tian et al. [19,20] studied the heat transfer characteristics of supercritical pressure R134a in horizontal tubes and modified the threshold value of the buoyancy criteria. Fig. 2 shows the typical heat transfer characteristics of supercritical fluids in a horizontal flow. At a low heat-flux-to-mass-flux ratio (q/G), the wall temperature of forced convection increases slowly with consistent temperature on the top and bottom surfaces [21,22]. With increasing q/G, the circumferential non-uniform temperature distribution is caused by the buoyancy effect owing to the radial density variations, which converts the heat transfer to a mixed convection. The wall temperature difference between the top and bottom surfaces (DTw) enlarges with increasing q/G and a local wall tempera-

320

340

360

380

400

420

440

hb(kJ/kg) Fig. 2. Wall temperatures of horizontal flow with supercritical R134a [19]. (Filled symbols: top surface; hollow symbols: bottom surface; the mass flux G is in kg/m2 s and the heat flux q is in kW/m2).

ture peak appears on the top surface [23,24]. With the fluid enthalpy increasing to higher than the pseudo-critical value (hpc), the DTw declines. The pressure and tube diameter are found to be influential as well. It is reported that the diameter has a negligible effect on the heat transfer at lower q/G when the buoyancy is weak. While, at a higher q/G, the circumferential nonuniformity in the wall temperature is intensified in large tubes and the heat transfer on the top surface is prone to deteriorate in larger tubes [19,25,26]. The heat transfer coefficient (HTC) increases obviously with reduced pressure at a low q/G but the pressure shows a weaker effect on the heat transfer at a larger q/G [19,24,27]. Even though the heat transfer with supercritical pressure fluids are investigated extensively through experiments, the mechanisms for these heat transfer characteristics are still not fully understood.

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R. Tian et al. / International Journal of Heat and Mass Transfer 144 (2019) 118607

To better understand the supercritical heat transfer mechanisms, computational fluid dynamics analysis has been employed to provide the detailed information for the flow structure and thermal field. For vertical flow, even though the heat transfer mechanisms of supercritical heat transfer are complicated, the general picture has been developed [28,29]. Owing to abrupt variations in properties, the large density gradient near the wall leads to strong buoyancy forces. The flow in the viscous layer accelerates strongly due to the buoyancy force and the flow in the core region retards relatively [30]. Subsequently, an M-shaped velocity profile appears and reduces the total and the turbulent shear stress in the buffer layer [31,32]. Consequently, the turbulence production is suppressed, thereby resulting in the partial laminarization of the flow followed by the localized impairment in heat transfer [33,34]. For the horizontal flow, the buoyancy force is no longer collinear with the flow direction in the mixed convection, which makes the interactions between the heat transfer and flow more complex. However, the horizontal flow is still considered less. To the best of the authors’ knowledge, only Chu and Laurien [35] conducted a direct numerical simulation (DNS) for horizontal flow and reported that due to buoyancy, flow stratification was developed with lowdensity fluid gathering in the upper part of the horizontal tube, and that the turbulence production near the top wall was restrained due to the reduced velocity gradient. However, owing to computation cost limitation, DNS is currently limited to flows with low Reynolds numbers, small diameters, and limited length-to-diameter ratios (z/d), not able to analyze the complete process of heat transfer deterioration and recovery. RANS models are adopted more often [36–46]. However, many of the numerical studies of horizontal supercritical heat transfer focused on expanding the operating parameter ranges through simulation with analyzing the wall temperatures, heat transfer coefficients, and buoyancy criteria [36–40], rather than discussing detailed property distributions and flow structure information, which is more valuable for understanding the heat transfer mechanisms. In some papers, the flow structure and thermal field were analyzed, for instance, Wang et al. [41] compared the streamline and average heat transfer coefficient at various heat fluxes of supercritical CO2 and found that the heat transfer deteriorated at a higher heat flux because part of the flow separates from the secondary circulation and flows upwards instead of circulating downwards. Gao and Bai [42] suggested that the interactive influence between secondary flows and thermophysical properties indicated important effects on heat transfer. The formation of a new vortex near the top was observed but the effect of the vortex on heat transfer was not discussed further. Unfortunately, in these studies contradictory results appeared regarding the flow structure and turbulence characteristics. Chu and Laurien [35] demonstrated that with z/d < 35, the turbulent kinetic energy on the top surface is lower than that on the bottom owing to the secondary flow. However, Lei et al. [43] and Cheng et al. [44] reported the opposite results for tubes of a larger z/d, that is the turbulence on the top is stronger. Chu and Laurien [35] and Lei et al. [43] found that new vortexes of the secondary flow developed near the top surface, while the results of Han et al. [45] indicated the formation of new vortexes in the middle

Non-heating

of the tube. These contradictions reveal the deficiency in understanding the underlying mechanisms, therefore, more systematic investigation is needed. Accordingly, this paper aims to provide insights into the interactions of property variations and buoyancy effects on horizontal flow through numerical analysis. Typical horizontal supercritical heat transfer cases with different buoyancy intensity were investigated. The roles of property variations and flow structure modification due to buoyancy force are clarified in generating the difference in heat transfer ability between the bottom and top surfaces; The cause(s) of heat transfer deterioration and recovery are clarified by analyzing the distribution features of turbulent kinetic energy and the evolution of secondary flow of the whole process including the heat transfer deterioration and the subsequent recovery; At last, effects mechanisms of tube diameter and pressure on the heat transfer were studied from the perspective of interaction relation of property variations and buoyancy effects. 2. Numerical methodology 2.1. Physical model As the flow in horizontal tubes is asymmetrical in the axial, circumferential, and radial directions, a three-dimensional calculation domain was used, as shown in Fig. 3. The finite volume method was used to solve the continuity, momentum, and energy equations. Various turbulent models were tested in this study, and the detailed information on the turbulence models can be found in Ref. [46–53]. The geometry of the calculated model was set the same as the test section used in the experiment [19]: 10.3 mm in diameter and 2.5 m in heated length with 0.5 m non-heating section at the inlet. Experiments with various inlet temperatures were conducted to obtain data in a wider bulk enthalpy range. In the numerical simulation, different heated lengths were set (see Table 1) for various operating conditions to overlap a wider enthalpy range, to ensure that the outlet bulk enthalpy of the fluid was higher than hpc. By examining the fluid velocity profile in the non-heating section, a non-heating section of 50d was set to obtain the fully developed flow before heating. The Ansys solver Fluent was used to solve the equations. The same working fluid used in the experiments was adopted in the simulation, that is R134a, a typically used working fluid in the ORC. The supercritical pressure state of the fluid is kept by setting

Table 1 Tube diameters and the corresponding heated length in the simulation. Inner diameter (mm)

Heated length (m)

2 10.3

0.9 4.5 7.5 5.5 9.0

16 26

Top (0o)

Heating section

Vertical radial

inlet

g Bottom (180o) Fig. 3. Physical model.

R. Tian et al. / International Journal of Heat and Mass Transfer 144 (2019) 118607

the fluid properties at specific supercritical pressure. The effect of pressure on the physical properties was neglected as the pressure drop along the tube was very small, so only the effect of temperature on the physical properties was considered. The physical properties of R134a can be found from refprop [6] and were integrated into Fluent by a piecewise linear function. As the properties change significantly and the temperature gradient is large in the fluid domain, the Boussinesq approximation cannot be used. In the present simulation, where a non-zero gravity field and temperature gradient are present simultaneously, the turbulence models account for the buoyancy by the term of generation of k due to buoyancy. The SIMPLEC scheme was used for pressure–velocity coupling. The second-order upwind method was used to solve the momentum, turbulent kinetic energy, turbulent dissipation rate, and energy equations. A constant source term was set in the solid wall instead of using a constant heat flux as the thermal boundary condition because the heat conduction in the solid tube affects the predicted wall temperature greatly [54]. The outer surface of the solid wall was set as thermal isolation. The inlet and outlet boundary conditions of the fluid domain were specified as mass flux inlet and pressure outlet. The calculation is regarded as converged when the residuals are within 105 with the monitored parameters, such as the fluid outlet temperature and velocity unchanged.

2.2. Mesh independence analysis Three-dimensional body-fitted hexahedral meshes were generated by ICEM, as shown in Fig. 4. Mesh independence was carefully tested with a 1.5 m-long heated tube. The mesh parameters are listed in Table 2. To accurately capture the flow information of the near-wall supercritical fluid, it is necessary to set sufficient nodes in the boundary layer. The dimensionless distance from the wall of the first layer y+ decreases to less than 1 from mesh A to B. Meshes C and D are obtained by setting more nodes in the circumferential and axial direction, respectively. The grid E contains more nodes in both the axial and circumferential directions. As shown in Fig. 5, the wall temperatures calculated with meshes A and B indicated an obvious difference and the wall temperatures predicted by meshes B–E are almost the same. In addition, it is found that a smaller axial mesh size exhibits a better convergence

460 440 420

Tw(K)

4

400

A B C D E

380 360 340 0.5

1.0

1.5

2.0

z(m)

2.5

Fig. 5. Predicted wall temperatures with various meshes. (AKN model, G = 600 kg/m2 s, q = 50 kW/m2; solid line: bottom surface; dashed line: top surface).

speed. Thus, mesh D was chosen considering its calculation accuracy and computing cost. 2.3. Simulated experimental cases and turbulence model The experimental data were obtained from the test facility built in Tsinghua University. The experimental results and heat transfer characteristics analysis can be found in the authors’ previous work [19,20]. To study the effect mechanisms of property variation, buoyancy force, diameter, and pressure, various experiments were chosen in this study, as listed in Table 3. A primary issue involved in the numerical study of supercritical heat transfer is the choice of turbulence model. Various turbulence models has been compared in the literature and the RNG, AKN, and SST models indicated better prediction accuracy in horizontal flow [36–45]. In this study, various models were tested and the predicted wall temperatures were compared with the experimental data in Figs. 6 and 7. For the case with weak buoyancy (G = 1000 kg/m2 s, q = 30 kW/m2), all the models provided similar results that well predicted the wall temperature. Meanwhile, for the heat transfer case with strong buoyancy effect (G = 600 kg/m2 s, q = 50 kW/m2), the AKN model demonstrated the best agreement with the data. As one of the

Y Z X Fig. 4. The mesh used in the simulation.

Table 2 The mesh parameters (0.5 m non-heating section and 1.5 m heating section). Mesh

Width of the first layer (mm)

y+

Circumferential mesh spacing (°)

Axial mesh size (mm)

Total nodes

A B C D E

0.01 0.001 0.001 0.001 0.001

1.1–6.0 0.1–0.55 0.1–0.58 0.1–0.6 0.1–0.56

9 9 4.5 9 4.5

5 5 5 2.5 2.5

4.78  105 6.74  105 1.47  106 1.25  106 2.74  106

5

R. Tian et al. / International Journal of Heat and Mass Transfer 144 (2019) 118607 Table 3 The operating parameters of various experimental cases simulated in this paper. Cases

P (MPa)

G (kg/m2 s)

q (kW/m2)

din (mm)

Buoyancy group

4.26 4.26 4.26 4.12, 4.26, 4.87 4.12, 4.26, 4.87

1000 600 600 1000 1000

30 50 50 30 100

10.3 10.3 2, 10.3, 16, 26 10.3 10.3

Diameter group Pressure group

400

400

390

390

380

380

370

realizable kRNG standard kkSST AKN

360 350 340 330 300

320

340

360

380

Tw,bottom(K)

Tw,top(K)

experiment

experiment

370 realizable kRNG standard kkSST AKN

360 350 340 330 300

400

hb(kJ/kg)

320

340

360

380

400

hb(kJ/kg)

(a)

(b)

Fig. 6. Comparison between the experimental and predicted wall temperature with din = 10.3 mm, p = 4.26 MPa, G = 1000 kg/m2 s, q = 30 kW/m2, (a) top surface, (b) bottom surface.

460

460

440

440

420

420

Tw,bottom(K)

Tw,top(K)

Experiment

400 380 360 340

standard k-e RNG realizable k-e

SST AKN LS

standard k-e RNG realizable k-e

SST AKN LS

400 380 360

Experiment

340

300 320 340 360 380 400 420 440 460

300 320 340 360 380 400 420 440 460

hb(kJ/kg)

hb(kJ/kg)

(a)

(b)

Fig. 7. Comparison between the experimental and predicted wall temperature with din = 10.3 mm, p = 4.26 MPa, G = 600 kg/m2 s, q = 50 kW/m2 (a) top surface, (b) bottom surface.

low-Reynolds number turbulence model, the AKN model is characterized with introducing the Kolmogorov velocity scale (ue), instead of the friction velocity (us), to account for the near-wall and low-Reynolds-number effects and adopting a composite time scale, which could better model the buoyancy-driven mixed layer [47]. Thus, the AKN model was employed in the following study.

3. Results and discussion In this section, the simulated results of the cases in Table 3 are discussed. The analysis of the cases in the buoyancy group aims to reveal the mechanisms of the buoyancy effect and property variations in the horizontal supercritical flow. Cases in the diameter group expand the diameter range in the experiment, where only

two tubes of different diameters were tested in the experiments. Varying the operating pressure at two different q/G is targeted at determining the pressure effect mechanisms.

3.1. Mechanisms of heat transfer enhancement and deterioration 3.1.1. Heat transfer case with weak buoyancy effect The heat transfer case with G = 1000 kg/m2 s and q = 30 kW/m2 is characterized with a weak buoyancy effect, where obvious but small DTw exists, and the wall temperature increases moderately without a temperature peak. To demonstrate the influence of buoyancy, parameters are analyzed by considering and not considering gravity, as shown in Fig. 8. When gravity is not considered, the top and bottom wall

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R. Tian et al. / International Journal of Heat and Mass Transfer 144 (2019) 118607

10000

Experiment top bottom

with gravity top bottom

370

360

8000

2

HTC (W/m K)

Tw,in(K)

380

g=0 top bottom

6000 no gravity

4000 2000

350 0

200

400

600

300

800

320

340

360

380

400

420

440

hb(kJ/kg)

z/d

(a)

(b)

30 700

with gravity g=0

top

3

density(kg/m )

cp(kJ/kg)

600

top

20

bottom

bottom

10

constant cp

500

400

with gravity g=0

300

0 1E-3

0.01

0.1

1

R-r (mm)

10

0

1

2

3

R-r (mm)

(c)

4

5

6

(d)

2.0

0.020

bottom

with gravity g=0

bottom

1.6

0.015 top

top

0.010

2

2

k(m /s )

u(m/s)

1.2

0.8

constant property

0.005 constant property

0.4

with gravity g=0

0.000

0.0 0

1

2

3

4

5

6

1E-3

0.01

0.1

R-r (mm)

R-r (mm)

(e)

(f)

1

10

Fig. 8. Predicted parameters with and without gravity. din = 10.3 mm, p = 1.05pc, G = 1000 kg/m2 s, q = 30 kW/m2, (a) wall temperature, (b) heat transfer coefficient (blue line: bottom HTC if only cp is considered, black line: top HTC if only cp is considered), (c) specific heat in vertical radial direction, (d) density in vertical radial direction, (e) axial velocity in vertical radial direction, (f) turbulent kinetic energy in vertical radial direction. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

temperatures (Tw,top, Tw,bot) are consistent, lying between the Tw,top and Tw,bot of the full gravity case, and indicating that the gravity effect increases the heat transfer ability on the bottom while impairs that on the top surface. The vertical radial distributions of the properties and flow parameters at z/d = 630, where the bulk enthalpy reaches hpc, are plotted in Fig. 8(c–f). The specific heat (cp) in the no-gravity case is much higher than the constant value of cp, thereby resulting in the high HTC peak near hpc. When gravity is applied, the Tw,top and Tw,bot separate from each other. cp near the top surface is higher while cp near the bottom is lower than the condition without gravity. From the perspective of the cp effect, the HTC of the top surface would be higher than that of the bottom (shown by the black and blue lines, respectively, in Fig. 8b);

however, this is opposite to the actual experimental HTC distribution trend in Fig. 8(b). Compared with the no-gravity case, density near the bottom is slightly higher than that near the top (Fig. 8d), then the fluid near the bottom wall accelerates while the velocity near on the top decreases (Fig. 8e). As a result, the turbulence in the bottom part is intensified, causing the heat transfer coefficients to increase and exceed that of the no-gravity case, as shown by the blue arrow in Fig. 8(b). For the upper part, the restrained turbulence results in the reduced heat transfer coefficients as shown by the black arrow. Consequently, in this case with the weak buoyancy effect, the variation in cp leads to the intensified heat transfer compared with the heat transfer with constant properties, instead of the dominating factor leading to the heat transfer non-

R. Tian et al. / International Journal of Heat and Mass Transfer 144 (2019) 118607

7

Property variations + Buoyancy

Property variation

Forced convection with constant properties

Forced convection with property variations

Buoyancy

Mixed convection with property variations

Fig. 9. Heat transfer patterns with supercritical pressure fluids.

440

Tw,in(K)

420 400 380 360

with gravity bottom top

340 0

100

200

300

g=0 bottom top 400

500

z/d Fig. 10. Wall temperature predicted by AKN model with and without gravity, din = 10.3 mm, p = 1.05pc, G = 600 kg/m2 s, q = 50 kW/m2.

uniformity. The difference in turbulent mixing intensity owing to buoyancy is the major factor responsible for the difference in the heat transfer ability of the top and bottom surfaces. The present study aims to clarify the identification of heat transfer enhancement and heat transfer deterioration of supercritical fluids in a horizontal flow. In general, the heat transfer enhancement is characterized with the heat transfer coefficients peak, and heat transfer deterioration is characterized with a wall temperature peak, which is rather subjective. The heat transfer patterns and their relationships are illustrated in Fig. 9. At the supercritical pressure, the conventional forced convection with

z/d=50

z/d=150

constant properties is transformed into forced convection with property variations. According to the analysis above, this is because of the high specific heat caused the heat transfer on the top and bottom surfaces in forced convection (buoyancy free) to be improved uniformly compared to the condition with constant properties. This is called the first-kind, property-induced enhancement, in which case the heat transfer is circumferentially uniform. When the buoyancy takes effect, the forced convection changes to a mixed convection. The top heat transfer is deteriorated while the bottom is enhanced due to the restrained and intensified turbulence, respectively, compared with the forced convection with property variations. Even though the buoyancy effect leads to the redistribution of cp in this process, the effect of flow structure dominates, thus overwhelming the influence of cp. In this stage, the enhancement and deterioration of heat transfer are defined based on the forced convection with property variations. The heat transfer enhancement owing to buoyancy is termed the second-kind enhancement. Therefore, when discussing the enhancement and deterioration of supercritical heat transfer, the comparison benchmark should be clarified clearly.

3.1.2. Heat transfer case with strong buoyancy effect With the heat flux increasing and the mass flux decreasing, the buoyancy becomes stronger in the case of G = 600 kg/m2 s and q = 50 kW/m2, which is characterized by the typical Tw,top peak and large DTw (Fig. 10). Fig. 11 demonstrates the development of flow profiles along the flow direction, including the fluid density, temperature, and axial velocity before (z/d = 50), at (z/d = 150), and after (z/d = 200, 240) the Tw,top peak. Less flow stratification is shown near the inlet (z/d = 50) with the local high temperature and low density fluid distributed near the top region of the tube.

z/d=200

z/d=240

Density

Temperature

Velocity

Fig. 11. Flow field of case G = 600 kg/m2 s, q = 50 kW/m2 at various axial locations, density (kg/m3), temperature (K), and axial velocity (m/s).

8

R. Tian et al. / International Journal of Heat and Mass Transfer 144 (2019) 118607

indicating that the heat transfer ability contributed by cp is impaired compared to the buoyancy free flow. The density near the top is much smaller than that near the bottom and the overall density decreases with z/d increasing because of heating. Consequently, the velocity distribution in Fig. 12(d) deviates from the classical log-law profile (as shown by the unheated profile) because the fluid accelerates in the near-wall region, leading to an M-shaped velocity [30]. Subsequently, the distribution of turbulent kinetic energy k is modified as shown in Fig. 12(e). As the fluid velocity continues increasing along the flow direction owing to the density decrease, the velocity gradient increases and consequently, the turbulence of both the top and bottom fluids are intensified along the flow direction. Even though k continues increasing along the flow direction, the heat transfer on the top surface undergoes deterioration and recovery, not in the same tendency with the k distribution. In addition, it is natural to assume that the turbulence near the bottom is stronger than

At the Tw peak, the density and temperature are strongly stratified owing to the buoyancy effect with fluid accelerating at the lower part. After the Tw peak, the flow stratification weakens gradually with two high-velocity centers appearing. To analyze the flow condition in the boundary layer in detail, distributions of various parameters along the vertical diameter are shown in Fig. 12. Both the Tw,top and Tw,bot exceed Tpc and the fluid near the top part gradually reach Tpc along the downstream direction with the pseudo-critical point moving towards the tube center. Correspondingly, the cp peak of the top part fluid moves into the core flow region, away from the near-wall layer. Meanwhile, the high specific heat of the bottom fluid locates within y+ < 30. Only when the high specific heat is concentrated in the near wall region, can the heat transfer be enhanced; thus, to some extent, the stronger heat transfer ability on the bottom surface benefits from the higher specific heat. However, compared with the cp without gravity, both the top and bottom cp decrease,

bottom

440

Tpc

380

z/d bottom unheated 50 150 200 240 top unheated 50 150 200 240

+

y =30

25

cp(kJ/kg-K)

400

without gravity (z/d=150)

30

unheated 50 150 200 240

420

T(K)

35

top

z/d

20 15 10 5

360

0 340 -6

-4

-2

0

2

4

6

1E-3

0.01

0.1

y(mm)

R-r(mm)

(a)

(b)

1200

1

10

top

bottom

1.2 3

Density (kg/m )

1000

u (m/s)

800 600

z/d 400 200 -6

-4

unheated 50 150 200 240 -2

0.8

z/d

0.4

unheated 50 150 200 240

0.0

0

2

4

-6

6

y(mm)

-4

-2

0

2

4

6

y(mm)

(c)

(d) 0.012

2

2

k(m /s )

0.009

0.006

0.003

z/d bottom unheated 50 150 200 240 top unheated 50 150 200 240

0.000 1E-3

0.01

0.1

1

10

R-r (mm)

(e) Fig. 12. Parameter distributions in vertical radial direction of case G = 600 kg/m2 s, q = 50 kW/m2, (a) fluid temperature, (b) specific heat, (c)density, (d) axial velocity, and (e) turbulent kinetic energy.

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that near the top such that the heat transfer coefficients on the bottom is larger. However, Fig. 12(e) shows that the turbulent kinetic energy near the top is stronger than that near the bottom except for the location of z/d = 50. To further investigate the variation in k, Fig. 13 presents the distribution of k near y+ = 30 for both the top and bottom surfaces along the whole flow path together with the wall temperatures. It can be divided into three regions according to the relative magnitudes of the top turbulent kinetic energy, ktop, and the bottom turbulent kinetic energy, kbottom. In the first region where z/d < 100, ktop < kbottom, corresponding to the increasing Tw section. Similar results have been reported in DNS and RNAS studies [35,41] where the simulated tubes are short with z/d < 100. In region II where 100 < z/d < 340, ktop > kbottom, corresponding to the Tw,top peak and the heat transfer recovery section. In region III where z/d > 340, ktop is lower than kbottom again, corresponding to the second Tw,top increase section. This behavior of Tw and k can be explained from the secondary flow, as shown in Fig. 14. Unlike the heat transfer convection with single phase fluid of constant property where the secondary flow due to density changes can be negligible, the buoyancy force due to dramatic density variation can lead to strong secondary flow in heat transfer of supercritical pressure fluids [55,56]. The streamline of secondary flow at various axial locations is plotted in Fig. 14. A pair of symmetrical vortexes appear at z/d < 100 (before the Tw peak) and the vortexes move downward to the bottom along the flow

Tw,top

0.08

Tw,bottom 0.07

400

0.05 0.04

380

I

III

II

0.03

360

3.2. Diameter effect on supercritical heat transfer 2

0.06

2

420

k(m /s )

Tw(K)

440

0.02

340

0.01 k (top) k (bottom)

320 0

100

200

300

400

direction. In this region, as the radial velocity gradient near the bottom surface is larger, the turbulence on the bottom is stronger, i.e., ktop < kbottom. However, both the ktop and kbottom are at a lower level in this region, and ktop exhibits an obvious decrease after heating is applied; thus, the wall temperature increases fast and the lower ktop leads to a higher top wall temperature. When z/d > 100, k continues rising in the streamwise direction because of the increasing velocity gradient owing to flow acceleration; more importantly, two new vortexes appear at the upper part of the tube. The newly developed vortexes enhance the turbulence of the top part, resulting in ktop increasing and even exceeding kbottom; hence, the heat transfer on the top surface recovers with the top wall temperature increasing more slowly and subsequently decreasing. In the downstream flow, when the bulk fluid temperature gradually exceeds Tpc, the density gradient on the cross section declines (Fig. 11), which weakens the buoyancy effect. Consequently, the pair of vortexes near the top disappear gradually and the original major vortexes return to the middle of the tube. With the flow structure of the top and bottom part of the tube restoring to symmetry again, the heat transfer recovers to being uniform such that the top and bottom wall temperatures tend to be consistent. Therefore, in a strongly buoyancy-influenced case, the evolution of the secondary flow is critical to the heat transfer performance. It can be seen that the relative magnitude of the turbulent kinetic energy of the top and bottom of the tube varies gradually along the flow so that discussion in different z/d ranges can lead to different conclusions, which explains the contradiction about turbulent kinetic energy distribution in literature [35,43–45].

0.00 500

z/d Fig. 13. Turbulent kinetic energy near y+ = 30 for both the top and bottom surfaces and the wall temperatures.

z/d=50

z/d=100

z/d=250

z/d=300

In the experimental studies, the range of pipe diameter is limited owing to the limitations of the experimental conditions, time, and cost. Therefore, in this section, the effect of tube diameter on supercritical heat transfer is analyzed in a wider diameter range through numerical simulation. Through literature review, it is summarized that the diameter of the test section ranges in 1.27– 38.1 mm for water and ranges in 0.099–29 mm for CO2 [20], thus the simulated tube diameters are selected to be within this range. In order to exclude the thermal acceleration effect to analyze the buoyancy effect clearly and avoid the complicated interaction of thermal acceleration and buoyancy, the lower limit of the tube diameter was chosen as 2 mm [57]. The upper limit of 26 mm

z/d=150

z/d=350

z/d=200

z/d=400

Y X Fig. 14. Evolution of secondary flow at various axial locations of case G = 600 kg/m2 s, q = 50 kW/m2.

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was chosen based on the comprehensive consideration of the commonly used pipe diameter range and the calculation cost. The predicted wall temperatures with the tube diameter ranging from 2 mm to 26 mm of G600-q50 are shown in Fig. 15. The predicted wall temperatures agree well with the experimental data in 10.3-mm and 16-mm tubes. The bottom surface temperatures of the different tubes are almost identical even though the tube diameter varies in a wide range. In contrast, the top wall temperature increases significantly with the tube diameter. The circumferential temperature of a 2-mm tube is nearly uniform with the maximum DTw of less than 2 K. While in a 26-mm tube, the maximum DTw reaches 150 K. The buoyancy criterion for horizontal flow, Grq/Grth, in Fig. 16 reveals that for the same operating condition, the buoyancy effect is intensified significantly in larger tubes. The threshold value of Grq/Grth for organic fluids in a horizontal flow was 2mm 10mm 16mm 26mm

550

Tw,in(K)

500 450 400 experiment

350 300

330

360

16 mm 10.3 mm

390

420

450

hb(kJ/kg) Fig. 15. Wall temperatures of tubes with various inner diameters. G = 600 kg/m2 s, q = 50 kW/m2 (solid line and solid symbols: top surface; dashed line, and hollow symbols: bottom surface).

determined to be 5 [20], indicating that the heat transfer in tubes with diameters ranging between 10 and 26 mm are affected significantly by the buoyancy effect. The value of Grq/Grth in the 2-mm tube drops to lower than 20, i.e., near the threshold value, indicating an almost negligible buoyancy effect. Therefore, a more severe buoyancy effect in a larger tube results in worse heat transfer on the top. Fig. 17 illustrates the secondary flow structure at locations where the wall temperature peaks appear (hb = 350 kJ/kg) and the turbulent kinetic energy near y+ = 30 is as shown in Fig. 18. In the 2-mm tube, vortexes of the secondary flow almost distribute in the middle of the tube, being symmetrical up and down. The ktop and kbottom are close to each other with kbottom always being slightly larger than ktop. Consequently, the heat transfer in this tube is nearly circumferentially uniform. Two additional smaller vertexes developed in the 10.3-mm tube in the top region, as discussed in section 3.1.2. With the diameter increasing to 16 mm and 26 mm, more vertexes are developed at the top part of the tube and the upper area occupied by newly developed vortexes expands with increasing tube diameter, providing stronger turbulence for the heat transfer to recovery. The ktop in region II is higher in the larger tube, as shown in Fig. 18, such that the higher top wall temperature of the larger tube can decrease to the similar level with that of the smaller tube. As shown in Fig. 15, even though the calculated tube length is not long enough, the wall temperatures of various tubes tend to be coincident when the bulk enthalpy increases to be far enough from hpc (larger than 450 kJ/kg). The flow structure and turbulent kinetic energy near the bottom part of the tubes are similar; thus, the bottom wall temperatures are nearly the same. 3.3. Pressure effect on the heat transfer Previous experimental results have revealed that the pressure has an obvious effect on the heat transfer at small q/G conditions,

800

400 200

bottom top 2mm 10mm 16mm 26mm

0.03 2 2

Grq/Grth

600

0.04

k (m /s )

2mm 10mm 16mm 26mm

0.02 0.01

threshold value

0.00

0 300 320 340 360 380 400 420 440

0

hb(kJ/kg)

200

300

400

500

z/d

Fig. 16. Buoyancy criterion of Grq/Grth of tubes with various diameters. G = 600 kg/ m2 s, q = 50 kW/m2.

din=2 mm

100

din=10.3 mm

Fig. 18. The turbulent kinetic energy near y+ = 30 of tubes with various diameters, G = 600 kg/m2, q = 50 kW/m2 (solid line: bottom, dashed line: top).

din=16 mm

din=26 mm

Y X Fig. 17. Streamline of secondary flow of various diameter tubes (the cross sections are taken at hb = 350 kJ/kg).

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while it has a weak effect on large q/G heat transfer conditions. This section aims to explain why the pressure affects the supercritical heat transfer as such by investigating the flow field and property distributions. Fig. 19 compares the parameters of two heat transfer cases with different ratios of q/G. For the lower q/G heat transfer condition with G1000-q30, in which case the buoyancy effect is weak, a remarkable increase in the heat transfer coefficient is observed with declining pressure. The specific heat at 1.02 pc is much larger than that at other higher pressures, especially within y+ < 30. In contrast to the significant difference in cp, the turbulent kinetic energy at various pressures undergoes a much smaller difference. Therefore, it can be concluded that cp instead of the flow structure is the dominating factor contributing to the heat transfer difference at various pressures. As the specific heat is highly dependent on pressure, the pressure has an obvious influence on heat transfer in a weak buoyancy-influenced heat transfer case.

For the strong buoyancy-influenced case of G1000-q100 shown on the right column of Fig. 19, the pressure effect is not as strong as that in the case of G1000-q30 with the heat transfer coefficient peak value experiencing a mild increase as the pressure declines. Even though cp at 1.02 pc is significantly higher than that at 1.2 pc and 1.4 pc in the core flow region, the advantage of cp at a lower pressure within the layer of y+ < 30 is not as strong as that of the lower q/G case. At the pressure of 1.02 pc, ktop > kbottom appears, implying that new vortexes formed on the top, as analyzed in section 3.1.1. This can be proven by the vortex shown in Fig. 20. As the pressure increases, the property variations weakens; thus, the buoyancy effect caused by density gradient weakens, and no newly developed vortexes appear at higher pressures. In this case, the buoyancy effect is the dominating factor for the heat transfer, which is closely related with the density variation on the cross section. The density variation with pressure is not as strong as that of the specific heat, resulting in a smaller difference in the heat

1.02pc

10

10

1.02pc

HTC(kW/m K)

1.2pc

8

1.4pc

2

1.4pc

8

2

HTC(kW/m K)

1.2pc

6 4 2

6 4 2

0

0

320

360

400

440

300

hb(kJ/kg)

350

400

450

500

1

10

hb(kJ/kg)

(a) 80

1.02pc

1.02pc

60

1.2pc

60

1.4pc

cp(kJ/kg)

cp(kJ/kg)

1.2pc 40 +

y =30 20

1.4pc

40

+

y =30 20

0

0 1E-3

0.01

0.1

1

10

R-r(mm)

1E-3

0.01

0.1

R-r(mm)

(b)

0.020 1.02pc

0.05

1.2pc

2

k(m /s )

0.04

1.4pc

0.03

2

0.010

2

2

k(m /s )

0.015

0.005

1.02pc

0.01

1.2pc 0.000

1.4pc 1E-3

0.01

0.1

R-r(mm)

0.02

0.00

1

10

1E-3

(c)

0.01

0.1

1

10

R-r(mm)

Fig. 19. Parameter distributions under various operating pressures, (a) heat transfer coefficient, (b) specific heat, and (c) turbulent kinetic energy (left column: G = 1000 kg/m2 s, q = 30 kW/m2; right column: G = 1000 kg/m2 s, q = 100 kW/m2; solid line: bottom, dashed line: top).

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R. Tian et al. / International Journal of Heat and Mass Transfer 144 (2019) 118607

1.02pc

1.2 pc

1.4 pc

Y X Fig. 20. Density distribution and secondary flow at various pressures, G = 1000 kg/m2 s, q = 100 kW/m2.

transfer coefficient at various pressures compared with the lower q/G case.

51236004) and the Science Fund for Creative Research Groups (No. 51621062).

4. Conclusions

References

Numerical studies were performed for mixed convection heat transfer with supercritical pressure R134a in horizontal tubes to further understand the heat transfer mechanisms. The effects of buoyancy and property variations on heat transfer were analyzed, then the diameter and pressure effects on the heat transfer were subsequently investigated. The primary conclusions are summarized as follows:

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1) In weak buoyancy-affecting heat transfer cases of horizontal flow, the heat transfer coefficient peak was caused by the distribution of high cp in the boundary layer, especially within y+ < 30. The difference in the turbulent kinetic energy with ktop < kbottom was the dominating factor resulting in the non-uniform heat transfer. 2) In the strongly buoyancy-affected mixed convection, the flow process could be divided into three regions according to the turbulence kinetic energy distribution. In region I (Tw increasing section) and region III (gas-like section), ktop < kbottom because of the greater velocity gradient at the bottom; in region II, where Tw,top reaches a peak and subsequently decreases, ktop > kbottom because the newly developed vortexes near the top intensifies the turbulence. 3) Stronger buoyancy effect was developed in tubes of larger inner diameters. No new vortex formed in the 2-mm tube. While, multiple vortexes were developed in the upper region of the 16-mm and 26-mm tubes, providing stronger turbulence for the heat transfer to recover. 4) In lower q/G cases, the heat transfer coefficient increases obviously with decreasing pressure because cp is the dominating factor and is sensitive to the pressure variation. While in heavy-buoyancy cases, buoyancy dominates the heat transfer and the density variation with pressure is moderate, so the pressure has less effect on heat transfer.

Declaration of Competing Interest The authors declared that there is no conflict of interest. Acknowledgements This work was supported by the Project funded by China Postdoctoral Science Foundation (No. 2018M640078), National Natural Science Foundation of China (No. 51806117), the State Key Program of the National Natural Science Foundation of China (No.

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