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Research activities involving heat transfer at supercritical pressures have attracted attention in recent years because of possibility of increase in thermal output of heat transfer and industrial equipment. Because of high pressure and temperature conditions associated with heat transfer at supercritical pressures, only few experimental heat transfer studies are being carried out at supercritical conditions. The use of numerical tools for heat transfer and other related studies at supercritical pressures is increasing because of the high-pressure-temperature limitation of experimental studies at supercritical conditions. Heat transfer correlations implemented in these numerical tools are used to obtain numerical heat transfer data to complement experimental heat transfer data provided through experimental studies. In order to further broaden the understanding of fluid flow and heat transfer, this review examines the performance of heat transfer correlations adopted at supercritical pressures. It is found from the review that most of the correlations could predict heat transfer quite well in the low enthalpy region and few of the correlations could predict heat transfer in the high enthalpy region near critical and pseudo-critical conditions (heat transfer deteriorated conditions). However, no single heat transfer correlation is able to accurately predict all the experimental results presented in this work.

The knowledge of heat transfer phenomena and prediction methods for heat transfer at supercritical pressure conditions are required for the design and operation of Supercritical Water Cooled Reactor SCWR. Gathering knowledge available on work done so far relating to heat transfer and prediction methods at supercritical pressure conditions is necessary and important for future research needs towards the development of SCWR. Supercritical pressure conditions of water include conditions above critical pressure and temperature of 22.1 MPa and 373.9˚C respectively.

Research to understand and apply heat transfer at supercritical pressures in various industrial applications including power engineering, aerospace engineering, chemical engineering, cryogenics, and refrigeration engineering have been performed since 1950’s [

1) Normal Heat Transfer (NHT) can be characterised in general with wall heat transfer coefficients similar to those of subcritical convective heat transfer far from the critical or pseudocritical regions, when calculated according to the conventional single-phase Dittus-Boelter-type correlations: Nu = 0.0023Re^{0.8}Pr^{0.4} (

2) Improved or Enhanced Heat Transfer (IHT or EHT) is characterised with higher values of the wall heat transfer coefficient compared to those at the normal heat transfer; and hence lower values of wall temperature within some part of a test section or within the entire test section (

3) Deteriorated Heat Transfer (DHT) is characterized with lower values of the wall heat transfer coefficient compared to those at the normal heat transfer; and hence has higher values of wall temperature within some part of a test section or within the entire test section (_{w} > T_{pc} > T_{b}), and the heat flux is above a certain value, depending on the flow rate and the pressure [

It is known from the existing literature that buoyancy effect and bulk flow acceleration effect are mechanisms responsible for HTD as a result of the extent of variation of thermophysical properties at supercritical pressures [

variation of thermophysical properties of a fluid has greater influence on heat transfer both at subcritical and at supercritical pressures. In fact, thermophysical properties of a fluid going from subcritical to supercritical state are strongly dependent on temperature, especially in the critical and pseudo-critical temperature range where thermodynamic and transport properties show rapid variations. The buoyancy effects are the result of the decrease in the density near the Pseudo-Critical Point (PCP) whereas acceleration effect results from the increase in the thermal expansion coefficient. The acceleration effect can also be explained using mass conservation principle. When a constant heat is provided to a fluid, density of the fluid decreases near the PCP and to conserve the mass, the decreased density will cause an increase in the velocity, accelerating the flow. The buoyancy effect acts always in the direction of gravitational force, irrespective of the flow direction whereas acceleration effect always acts in the direction of a flow irrespective of the gravitational force. Although these two effects are result of large property variation, researchers investigated these effects extensively because large changes in the density and thermal expansion coefficient affect the heat transfer phenomenon more than other properties do [

Unlike buoyancy-influenced deterioration, deterioration due to acceleration occurs only with supercritical fluids. The reason for this is the need for large variation in properties to achieve large differences in inlet and exit densities. Heat transfer deterioration HTD due to flow acceleration is present for both upward and downward flows. But, because of negligible effect of buoyancy in downward flows, it is generally believed that downward flow shows a better heat transfer behaviour than that in upward flow conditions and heat transfer enhancement is more significant in downward flow conditions compared to that of upward flows [

HTD is shown by graphical representation of experimental data showing broad wall temperature peaks at higher heat fluxes for horizontal flows. For a vertical upward flow, sharp temperature peaks are observed (

q (W/cm^{2}) | m ˙ / A (gm/(s/cm^{2})) | d (cm) | Flow direction | ||
---|---|---|---|---|---|

a | Shitsman | 34 | 43 | 0.8 | vertical upward |

b | Shitsman | 28.5 | 43 | 0.8 | vertical upward |

c | Shitsman | 28.0 | 43 | 0.8 | vertical upward |

d | Domin | 72.5 | 68.6 | 0.2 | horizontal |

e | Domin | 72.5 | 72.4 | 0.2 | horizontal |

Jackson [

Licht et al. [

Zhang et al. [_{2} in square and triangular array tube bundle using CFD code FLUENT and compared it to that of H_{2}O. It was established that the heat transfer performance of CO_{2} was better than that of H_{2}O. Liao and Zhao [

4) Experimental and numerical studies in heat transfer

Experimental research to obtain data on heat transfer at supercritical pressures is important for the development of SCWR in the near future. Experimental data describe the actual behaviour of fluid flow and heat transfer, and as such are used for design and also used to validate analytical and numerical tools. Several experimental studies have been performed at supercritical conditions in trying to understand fluid flow and heat transfer at these supercritical conditions.

Some of these experimental studies include studies carried out by Yamagata et al. [

The recent increase in computing power and computational methods has made it possible for CFD codes to be used to reproduce experimental data and obtain data that cannot be obtained through experiments. Thus, CFD codes can be used for making predictions and providing greater detail than is offered by experiments at supercritical pressure conditions. Some of the numerical studies performed using these numerical tools include studies carried out by Zhu [

The major findings obtained from these experimental and numerical studies include:

1) The Normal Heat Transfer (NHT), Improved Heat Transfer (IHT) and Deteriorated Heat Transfer (DHT) are three main heat transfer regimes observed at supercritical pressures.

2) Buoyancy and bulk flow acceleration are two main mechanisms causing heat transfer deterioration at supercritical pressures as a result of significant variation of thermo-physical properties of fluids. Impaired force convection and Impaired mixed convection are two other different mechanisms reported recently to cause HTD at supercritical pressures.

3) HTD caused by flow acceleration can occur both in upward and downward flows whereas HTD caused by buoyancy can occur only in upward flows.

4) HTD caused by bulk flow acceleration can occur only at supercritical pressures whereas HTD caused by buoyancy can occur both at subcritical and supercritical pressures.

5) It was observed that a decrease in the hydraulic diameter of test geometry improves heat transfer at supercritical pressures.

6) At a high mass flux, HTD is caused by bulk flow acceleration mechanism, and at a low mass flux, HTD is caused by buoyancy mechanism.

7) Heat transfer performance of CO_{2} is better than that of H_{2}O at supercritical pressures due to its significant lower critical parameters compared to that of H_{2}O.

8) Flow instability could cause HTD and might have significant effect on supercritical pressure heat transfer. Therefore more experimental and theoretical studies at supercritical pressures are needed to help further understand effects of flow instability on heat transfer.

9) There is possibility of cyclic occurrence of heat transfer deterioration HTD and restoration during flow oscillations in heated channels and HTD could occur before the occurrence of unstable behaviour at supercritical pressures.

10) There is the need for developing and installing new turbulent models in commercial CFD codes to enable CFD codes accurately predict heat transfer at supercritical pressures. It has been established that CFD modeling could help understand heat transfer phenomenon in supercritical pressure fluid.

The general purpose of heat transfer research at supercritical pressures is to provide data, heat transfer correlations and tools used for simulation, for the design of heat transfer equipment, licensing heat transfer facilities and for safe operation of these heat transfer facilities. In an attempt to achieve this purpose, several Authors made efforts to develop correlations for predicting heat transfer at supercritical pressures. Figures 6-10 were used to demonstrate the performance of some of the correlations mentioned in this section for predicting heat transfer behavior at supercritical conditions.

The Dittus-Boelter correlation which is widely used at subcritical pressures quite well predicted the experimental data in the low enthalpy region and largely over-predicted the experimental data in the high enthalpy region associated with critical and pseudo-critical conditions (Figures 7-10) [

Dittus-Boelter correlation [

N u D B , f = 0.023 R e f 0.8 P r f 0.4 (1)

Equation (1) is valid for single-phase heat transfer in channels within the range of parameters (Pioro and Duffey, 2007): 0.7 ≤ P r ≤ 160 ; R e ≥ 10 , 1000 and L / D h y ≥ 10.

The Krasnoshchekov-Protopopov correlation generally predicted the experimental data quite well in the low and high enthalpy regions (Figures 7-10) except in

N u b = N u o ( μ b μ w ) 0.11 ( λ b λ w ) 0.3 ( C p ¯ C p b ) 0.35 (2)

Nu_{o} is given by Equation (3).

N u o = ( ξ / 8 ) R e b P r ¯ 12.7 ( ξ / 8 ) ( P r ¯ 2 / 3 − 1 ) + 1.07 (3)

ξ = 1 / ( 1.82 log 10 R e b − 1.64 ) 2 (4)

Equation (2) is valid within the range of parameters [

2 × 10 4 < R e b < 8.6 × 10 5 , 0.85 < P r ¯ b < 65 , 0.90 < μ b / μ w < 3.60 , 1.00 < k b / k w < 6.00 and 0.07 < C ¯ p / C p b < 4.50

Correlation of Bishop et al. developed for annular channels predicted quite well the heat transfer coefficient experimental data for all the enthalpy range except for

Bishop et al. correlation [

N u b = 0.0069 R e b 0.9 P r ¯ b 0.66 ( ρ w ρ b ) x 0.43 ( 1 + 2.4 D x ) (5)

where x is the axial location along the heated length, ρ w is the density of the fluid at the wall temperature, ρ b is the density of the fluid at bulk temperature

and the last term accounts for entrance-region effects. Equation (5) is valid within the following range of flow and operating parameters: 22.8 - 27.6 MPa, 282˚C - 527˚C bulk-fluid temperature, 651 - 3662 kg/m^{2}s mass flux and 0.31 - 3.46 MW/m^{2} heat flux [

Correlation of Dyadyakin and Popov developed for fuel bundles predicted closely experimental heat-transfer coefficients over a narrow range of bulk-fluid enthalpy (

Dyadyakin and Popov correlation [

N u x = 0.0021 R e x 0.8 P r ¯ x 0.7 ( ρ w ρ b ) x 0.45 ( μ b μ i n ) x 0.2 ( ρ b ρ i n ) x 0.1 ( 1 + 2.5 D h y x ) (6)

where x is the axial location along the heated length and D_{hy} is the hydraulic diameter. Equation (6) is valid within the range of the parameters: 24.5 MPa, 90˚C - 570˚C bulk temperature (400 - 3400 kJ/kg bulk enthalpy), <4.7 MW/m^{2} heat flux and 500 - 4000 kg/m^{2}s mass flux [

Ornatsky et al. correlation [

N u b = 0.023 R e b 0.8 P r b 0.8 ( ρ w ρ b ) 0.3 (7)

Equation (7) is valid within the range of parameters: 22.6 - 29.4 MPa, 420 - 1400 kJ/kg inlet enthalpy, 0.28 - 1.2 MW/m^{2} heat flux and 450 - 3000 kg/m^{2}s mass flux [

The Shitsman correlation mostly over-predicted experimental values at high Nusselt numbers as shown in

N u b = 0.023 R e b 0.8 P r min 0.8 (8)

Equation (8) is valid within the range of parameters: 22.6 - 27.4 MPa, 180˚C - 580˚C bulk temperature, 0.28 - 8.4 MW/m^{2} heat flux and 170 - 3000 kg/m^{2}s mass flux [

Griem correlation slightly deviated in predicting the experimental data at low and high heat flux conditions shown in

N u b = 0.0169 R e b 0.8356 P r b 0.432 (9)

Equation (9) is valid within the range of parameters: 23 - 25 MPa, 0.3 - 0.6 MW/m^{2} heat flux, and 500 - 2500 kg/m^{2}s mass flux [

The correlation of Mokry et al. under-predicted experimental heat-transfer coefficient over entire range of bulk-fluid enthalpy as shown in

N u b = 0.0061 R e b 0.904 P r ¯ b 0.684 ( ρ w ρ b ) x 0.564 (10)

Equation (10) is valid within the range of parameters: 24 MPa, 320˚C - 350˚C inlet temperature, ≤1250 kW/m^{2} heat flux and 200 - 1500 kg/m^{2}s mass flux [

The Jackson correlation [

N u b = 0.0183 R e b 0.82 P r b 0.5 ( ρ w ρ b ) 0.3 ( C p ¯ C p b ) n (11)

n = 0.4 for T b < T w < T p c and for 1.2 T p c < T b < T w (12)

n = 0.4 + 0.2 ( T w T p c − 1 ) for T b < T p c < T w (13)

n = 0.4 + 0.2 ( T w T p c − 1 ) [ 1 − 5 T b T p c − 1 ] for T p c < T b < 1.2 T p c (14)

where T_{b}, T_{pc} and T_{w} are in K. Hence it can be expected that the Jackson correlation will follow closely a trend predicted by the Krasnoschekov et al. correlation. Equation (11) is valid within the range of parameters [

8 × 10 4 < R e b < 5.10 5 , 0.85 < P r ¯ b < 65 , 0.90 < ρ w / ρ b < 1.0 , 0.9 < T w / T p c < 2.5 , 4.6 × 10 4 < q < 2.6 × 10 6 ( q in W / m 2 ) , 0.02 < C ¯ p / C p b < 4.0 and x / D ≥ 15

Cheng et al. [

N u = 0.023 R e B 0.8 P r 1 / 3 ⋅ F (15)

F = min ( F 1 , F 2 ) (16)

F 1 = 0.85 + 0.776 ( π A ⋅ 10 3 ) 2.4 (17)

F 2 = 0.48 ( π A , P C ⋅ 10 3 ) 1.55 + 1.21 ⋅ | ( 1 − π A π A , P C ) | (18)

where π A is the acceleration parameter given by

π A = q ⋅ β B G ⋅ C p . B (19)

Equation (15) is valid within the range of parameters (test data for Herkenrath et al. [^{2} heat flux and 700 - 3500 kg/m^{2}s mass flux [

Wang et al. [

N u b = 0.01 R e b 0.88 P r b 0.64 ( ρ m ρ b ) 1.76 ( C p ¯ C p b ) 0.49 (20)

The qualitative temperature for density is the film temperature but not the wall temperature in the correlation of Jackson. Equation (20) is valid within the range of parameters: 23 - 28 MPa, <500˚C bulk temperature, 200 - 1000 kW/m^{2} heat flux and 700 - 3500 kg/m^{2}s mass flux [

The Watt and Chou correlation performed quite well in predicting heat transfer for downward vertical flow as shown in

N u = N u v a r P [ 1 − 3000 G r b ¯ R e b 2.7 ⋅ P r b ¯ 0.5 ] 0.295 for G r b ¯ R e b 2.7 ⋅ P r b ¯ 0.5 < 10 − 4 (21)

N u = N u v a r P [ 7000 G r b ¯ R e b 2.7 ⋅ P r b ¯ 0.5 ] 0.295 for G r b ¯ R e b 2.7 ⋅ P r b ¯ 0.5 > 10 − 4 (22)

where

N u v a r P = 0.021 R e b 0.8 ⋅ P r b ¯ 0.55 ( ρ w ρ b ) 0.35 (23)

G r b ¯ = g ⋅ ρ b ⋅ d 3 ⋅ ( ρ b − ρ ¯ ) μ b 2 (24)

P r b ¯ = C p ¯ ⋅ μ b λ b (25)

C p ¯ = h w − h b T w − T b (26)

ρ ¯ = ρ w + ρ b 2 (27)

Equation (21) is valid within the range of parameters: 25 MPa, 150˚C - 310˚C bulk temperature, 0.175 - 0.44 MW/m^{2} heat flux and 106 - 1060 kg/m^{2}s mass flux [

Jackson [

N u b N u f = [ | 1 ± 1875 B o b F V 1 ( N u b N u f ) − 1.1 | ] 0.46 (28)

where

N u f = 0.023 R e b 0.8 P r b 1 / 3 (29)

B o b = G r b R e b 2.625 P r b 1 / 3 (30)

F V 1 = ( μ ¯ μ b ) ( ρ ¯ ρ b ) − 0.5 (31)

Recently, Jackson introduced new Buoyancy parameter to further improve upon his prediction method for heat transfer of supercritical fluids [

B o = G r B R e B 3.425 P r B 0.8 (32)

where

G r B = g ⋅ ρ B ⋅ d 3 ⋅ ( ρ B − ρ w ) μ B 2 (33)

Zhao et al. [

N u = 0.023 R e b 0.8 P r 1 / 3 ⋅ F (34)

F = min ( F 1 , F 2 ) (35)

F = F 1 = 0.62 + 0.06 × ln ( π B ) (36)

F = F 2 = 11.46 × ( ln π B ) (37)

where π B is the Buoyancy parameter given by Equation (38).

π B = q ⋅ β B ⋅ d λ B (38)

Zhao et al. compared the performance of their newly developed correlation with their experimental data and heat transfer correlations of Watts and Chou, Jackson, Dittus-Boelter, Krasnoschkov et al., and Cheng (b). The correlation of Zhao et al. performed quite well in predicting the experimental data compared to the other correlations. Equation (34) is valid within the range of parameters: 23 - 26 MPa, 280˚C - 410˚C bulk temperature, 0.17 - 1.4 MW/m^{2} heat flux and 450 - 1500 kg/m^{2}s mass flux [

McAdams et al. correlation [

N u b = 0.0214 R e f 0.8 P r f 0.33 ( 1 + 2.3 d h y / l ) (39)

Equation (39) is valid within the range of parameters: 0.8 - 24 MPa, 221˚C - 538˚C bulk temperature, 0.035 - 0.336 MW/m^{2} heat flux and 75 - 224 kg/m^{2}s mass flux [

Gang et al. [

For normal heat transfer condition as shown in

For heat transfer deterioration condition as shown in

Similar studies have been carried out using different assessment methods including statistical method to evaluate heat transfer correlations at supercritical pressures. Results of these studies are presented in the IAEA-TECDOC-1746 (2014) and in papers written by Zahlan et al. [

Heat transfer deterioration HTD phenomenon is an issue as far as fluid flow and heat transfer processes are concerned and there is the need to determine the onset of HTD. Determining the onset of HTD is therefore important for the design and operation of heat transfer equipment at supercritical pressures.

Reference | Correlation |
---|---|

Vikhrev et al. [ | q ˙ D H T = 0.4 ⋅ G |

Styrikovich et al. [ | q ˙ D H T = 0.58 ⋅ G |

Yamagata et al. [ | q ˙ D H T = 0.2 ⋅ G 1.2 |

Mokry et al. [ | q ˙ D H T = 58.97 + 0.745 ⋅ G |

Cheng et al. [ | q ˙ D H T = G ⋅ 1.354 ⋅ 10 − 3 ⋅ ( C p , p c β p c ) |

Li et al. [ | q ˙ D H T = d i ⋅ ( 0.36 ⋅ ( G d i ) − 1.1 ) 1.21 |

Schatte et al. [ | q ˙ D H T = 1.942 ⋅ 10 − 6 ⋅ G 0.795 ⋅ ( 30 − d i ) 0.339 ⋅ ( C p , p c β p c ) 2.065 |

Kondrat’ev et al. [ | q ˙ D H T = 5.815 ⋅ 10 − 7 ⋅ R e 1.7 ⋅ ( p 1.01325 ) 4.5 |

Protopopov et al. [ | q ˙ D H T = G ⋅ 1.3 ( T p c − T b ) ⋅ C p , b ⋅ ( ξ 8 ) ⋅ ( ν w ν p c ) 1.3 |

Petukhov et al. [ | q ˙ D H T = 0.1875 ⋅ ξ ⋅ G ⋅ ( C p , p c β p c ) |

the correlations under-predicted the experimental at low mass fluxes and over-predicted the experimental data at high mass fluxes. Correlations of Styrikovich et al. [

In fluid flow and heat transfer systems where the fluid flow is affected by buoyancy, the existing heat transfer correlations are unable to capture heat transfer characteristics including the onset of HTD. Researchers have therefore made effort to develop criterion to predict the occurrence of buoyancy in fluid flow and heat transfer systems.

The onset of HTD due to buoyancy could be determined using the criterion of Jackson and Hall, or the criterion of Seo et al. [

The most accurate Jackson and Hall buoyancy criterion is given by Equation (40).

G r ¯ b R e b 2.7 < 10 − 5 (40)

Jackson and Hall criterion limiting value of less than 10^{−5} is needed for negligible buoyancy effects.

The recent buoyancy criterion of Seo et al. is given by Equation (41).

1 F r ≈ 1 ρ b ρ b − ρ s T s − T b Q ″ D H k s g D H 3 v s 2 1 R e s 2.8 N u < 0.3 (41)

In Equation (41), the s subscript denotes properties similar to a film temperature and is defined as an average of the inlet and pseudo-critical temperature. The Nusselt number Nu is based on the Dittus-Boelter Nusselt correlation, and an inverse Froude number of less than 0.3 is needed for negligible buoyancy effects.

The various relations used for estimating the total pressure drop or the total hydraulic resistance are in terms of Design variables and Control variables or Operating parameters. The value of the total pressure drop associated with the designed system is used to check whether the Design and Control Variables are within the safety limits of operation. Small pressure drop indicates good design and the range of control variables associated with small pressure drop are within the safety limit of operation. Large pressure drop indicates poor system design and the associated design and control variables are not within the safety limit of operation.

The total pressure drop for forced convection flow inside a closed-loop system can be determined using the expression [

Δ P = Σ Δ P f r + Σ Δ P l + Σ Δ P a c + Σ Δ P g (42)

where Δ P f r , Δ P l , Δ P a c and Δ P g are the pressure drop due to frictional resistance, local flow obstruction, acceleration of flow, and gravity respectively.

The pressure drop due to frictional resistance can be calculated using

Δ P f r = ξ f r L ρ u 2 2 D = ξ f r L G 2 2 D ρ (43)

where ξ f r is the frictional resistance coefficient or simply called friction factor. The friction factor ξ f r is not to be confused with the friction coefficient, sometimes called the Fanning friction factor. Denoting the Fanning friction factor by f, it follows that

ξ f r = 4 f (44)

The thermophysical properties in the equations should be evaluated based on the arithmetic average of inlet and outlet values.

Mikheev correlation for frictional resistance coefficient ξ f r of non-isothermal flow of water and other fluids is given by Equation (45).

ξ f r = 1 ( 1.82 log 10 R e b − 1.64 ) 2 ( P r w P r b ) 1 3 (45)

The Equation is valid for smooth tubes within a range of R e > 4000 (turbulent stabilized flow) [

Equation (46) was recommended for calculating frictional resistance coefficient ξ f r for turbulent flows at supercritical pressures [

ξ f r = f o ( ρ w ρ B ) 0.4 (46)

With

f o = ( 1.82 log ( R e / 8 ) ) − 2.0 (47)

The Colebrook equation can be used to estimate friction factor ξ r at supercritical pressures. The equation was derived by combining experimental results of laminar and turbulent flow in pipes. An approximate explicit form of the Colebrook equation was given by Haaland [^{−6} m [

ξ r = [ − 1.8 log ( ( ε 3.7 D ) 1.11 + 6.9 R e ) ] − 2 (48)

The friction factor ξ r can also be determined using the Blasius and McAdams relations for smooth pipe at supercritical pressures (Equation (49)) [

ξ r = { 64 R e R e ≤ 2200 max ( 64 R e , 0.316 R e 0.25 ) 2200 < R e < 3000 0.316 R e 0.25 R e ≥ 3000 (49)

The Blasius friction factor equation (Equation (50)) which is based on the assumption of fully developed turbulent flow in a smooth tube can also be used to estimate friction factor ξ r at supercritical pressures. It is applicable for bulk Reynolds numbers (Re_{b}) up to 10^{5} [

ξ r = 0.316 R e b 1 / 4 (50)

The implicit and iterative Colebrook and White (C-W) equation (Equation (51)) is also used to obtain friction factor at supercritical conditions [

I f C - W 1 / 2 = − 2 log ( ε / D 3.7 + 2.5 I R e b f C - W ) (51)

The frictional resistance coefficient changes at supercritical pressures due to significant property changes near the critical and pseudocritical points. Other correlations for calculating friction factor or frictional resistance coefficient can be found in Pioro and Duffey [

The pressure drop due to local flow obstructions is defined as:

Δ P l = ξ l G 2 2 D (52)

where the local resistance coefficient, ξ l , is determined using appropriate correlations for different flow obstructions. Significant property changes at supercritical flow conditions also affect the local resistance coefficients.

The pressure drop due to acceleration of flow is defined as:

Δ P a c = G 2 ( 1 ρ o u t − 1 ρ i n ) (53)

The pressure due to gravity is defined as:

Δ P g = ± g ( H o u t ρ o u t + H i n ρ i n H o u t + ρ i n ) L sin θ (54)

where θ is the test-section inclination angle to the horizontal plane, the sign “+” is for the upward flow, and the sign “−” is for the downward flow. Equation (42) is applicable for subcritical and supercritical pressures [

Research activities involving heat transfer at supercritical pressures have attracted attention in recent years because of possibility of increase in thermal output of heat transfer and industrial equipment. Because of high pressure and temperature conditions associated with heat transfer at supercritical pressures, only few experimental heat transfer studies are being carried out at supercritical conditions. The use of numerical tools for heat transfer and other related studies at supercritical pressures is increasing because of the high-pressure-temperature limitation of experimental studies at supercritical conditions. Heat transfer correlations implemented in these numerical tools are used to obtain numerical heat transfer data to complement experimental heat transfer data provided through experimental studies. In order to further broaden the understanding of fluid flow and heat transfer, this review examines the performance of heat transfer correlations adopted at supercritical pressures. It is found from the review that most of the correlations could predict heat transfer quite well in the low enthalpy region and few of the correlations could predict heat transfer in the high enthalpy region near critical and pseudo-critical conditions (heat transfer deteriorated conditions). However, no single heat transfer correlation is able to accurately predict all the experimental results presented in this work. The following are other major findings obtained as a result of this review:

1) Some of the heat transfer correlations for predicting heat transfer in upward flows of vertical geometry could also be used for predicting heat transfer in downward flows of vertical geometry. The two different choices of heat transfer correlations could behave closely similar outside the regions of near critical and pseudo-critical points. The correlations of Watts and Chou [

2) Some of the heat transfer correlations predicted quite well experimental data while others less predicted experimental data at different design parameters and operating conditions. This gives indication that heat transfer at supercritical pressures depends strongly on test geometry and operating conditions.

3) There is no single heat transfer correlation available that could predict experimental data very well within the acceptable error limit over all the experimental operating conditions considered in this study. However, correlations of Jackson [

4) Among the existing correlations for predicting the onset of heat transfer deterioration HTD presented in this work, correlation of Schatte et al. [

The readers of this review would be able to adopt the best performing correlations based on the comparison between the heat transfer correlations and experimental data presented in this work for their future activities involving fluid flow and heat transfer. It is recommended that more experiments on fluid flow and heat transfer at supercritical pressures be carried out to enable validation of similar numerical studies.

Shitsi, E., Debrah, S.K., Agbodemegbe, V.Y. and Ampomah-Amoako, E. (2018) Performance of Heat Transfer Correlations Adopted at Supercritical Pressures: A Review. World Journal of Engineering and Technology, 6, 241-267. https://doi.org/10.4236/wjet.2018.62014

The Dimensionless numbers, Greek letters, Symbols and Subscripts adopted in the assessment of heat transfer correlations include:

Dimensionless Numbers

Gr Grashof number, G r B = g ⋅ ρ B ⋅ d 3 ⋅ ( ρ B − ρ w ) μ B 2 ; G r ¯ Averaged Grashof number, G r b ¯ = g ⋅ ρ b ⋅ d 3 ⋅ ( ρ b − ρ ¯ ) μ b 2 ; Nu Nusselt number [ h d / λ ]; Pr Prandtl

number [ μ C p / λ ]; Re Reynolds number [ G D / μ ]; and P r ¯ Average cross-sectional Prandtl number [ μ C ¯ p / λ ].

Greek Letters

λ Thermal conductivity [W/(m∙K)]; μ Dynamic viscosity [kg/(m∙s)]; ν Kinematic viscosity, ν = μ / ρ [m^{2}/s]; ρ Density of a fluid [kg/m^{3}]; ρ ¯ Average density, ( ρ w + ρ b ) / 2 [kg/m^{3}]; ξ Friction factor; β Thermal expansion coefficient [1/K]; π A Dimensionless parameter, representing acceleration

effect, π A = q ⋅ β B G ⋅ C p . B ;and π B Dimensionless parameter, representing buoyancy effect, π B = q ⋅ β B ⋅ d λ B .

Symbols

Bo Dimensionless parameter, representing buoyancy effect; C_{p}_{ }Specific heat

capacity [J/(kg∙K)]; C ¯ p Average specific heat, H w − H b T w − T b , [J/(kg∙K)]; D, d

Diameter [m]; D_{hy}, D_{h} Hydraulic diameter [m]; G Mass flux [kg/(m^{2}s)]; g Gravitational acceleration [m/s^{2}]; H, h Specific enthalpy [J/kg]; HTC, h Heat transfer coefficient [W/(m^{2}/K)]; P Pressure [Pa]; Q, Q'', q Heat flux [W/m^{2}]; T Temperature [K]; t Time [second]; and x, l axial location [m].

Subscripts

ave, v average; B, b Bulk; cr, crit Value at critical point; DB Dittus-Boelter; ex exit; f fluid; in inlet; out outlet; pc Pseudo-critical; w Wall.