Review of the Wall Temperature Prediction Capability of Available Correlations for Heat Transfer at Supercritical Conditions of Water

Table 1

List of correlations used for comparing supercritical water heat transfer in vertical pipe.

Equation no.	Name of correlation	Correlation	Conditions
Equation (B.1)	Dittus and Boelter (1930)	Nu𝑏=0.023 Re𝑏^0.8Pr𝑏^0.4; [8, 13, 19].	Valid for fully developed turbulent flow in smooth tubes for fluid with Prandtl numbers ranging from about 0.6 to 100 and with moderate temperature difference between wall and fluid conditions.
Equation (B.2)	Mc Adams (1942)	Nu𝑏=0.0243 Re𝑏^0.8Pr𝑏^0.4; [5, 7, 13, 15].	Modified version of (B.1) used for supercritical condition.
Equation (B.3)	Bringer and Smith (1957)	Nu𝑥=0.0266 Re𝑥^0.77Pr𝑤^0.55; [5, 7].	Nu𝑥 and Re𝑥 are evaluated at 𝑡𝑥. Temperature 𝑡𝑥 is defined as 𝑡𝑏 if (𝑡pc−𝑡𝑏)/(𝑡𝑤−𝑡𝑏) < 0, as 𝑡pc if 0 ≤ (𝑡pc−𝑡𝑏)/(𝑡𝑤−𝑡𝑏) ≤ 1, and as 𝑡𝑤 if (𝑡pc−𝑡𝑏)/(𝑡𝑤−𝑡𝑏) > 1 for supercritical water up to 𝑝=34.5 MPa.
Equation (B.4)	Shitsman (1959, 1974)	Nu𝑏=0.023 Re𝑏^0.8Pr^0.8min; [5, 7, 19].	“min” means minimum Pr value, that is, either the Pr value is evaluated at the bulk fluid temperature or the Pr value is evaluated at the wall temperature, whichever is less. Assumption: thermal conductivity is a smoothly decreasing function of temperature near the critical and pseudo-critical points.
Equation (B.5)	Bishop et al. (1964)	Nu𝑏=0.0069 Re𝑏^0.9 (Pr)^0.66av (𝜌𝑤/𝜌𝑏)^0.43  (1+2.4 (𝐷/𝑥)); [5, 7, 8, 11, 13, 15].	𝑥 is the axial location along the heated length.; Pressure = 22.8–27.6 MPa, and bulk fluid temperature = 282–527°C.; Mass flux = 651–3662 kg/m^2s, and heat flux = 0.31–3.46 MW/m^2.
Equation (B.6)	Swenson et al.; [1]	Nu𝑤=0.00459 Re𝑤^0.923 (Pr𝑤)^0.613av (𝜌𝑤/𝜌𝑏)^0.231; hD/𝑘𝑤=0.00459 (GD/𝜇𝑤)^0.923 [((𝐻𝑤−𝐻𝑏) (𝜇𝑤))/((𝑇𝑤−𝑇𝑏) (𝑘𝑤))]^0.613 (𝜌𝑤/𝜌𝑏)^0.231 [1, 5–8, 10, 15, 21].	Pressure = 22.8–41.4 MPa, and bulk fluid temperature = 75–576°C.; Mass flux = 542–2150 kg/m^2s.; Assumption: thermal conductivity is a smoothly decreasing function of temperature near the critical and pseudo-critical points.
Equation (B.7)	Krasnoshchekov et al. (1967)	Nu = Nu0 (𝜌𝑤/𝜌𝑏)^0.3 [(Cp)av/Cp𝑏]^𝑛,; where, according to Petukhov and Kirillov (1958), Nu0 = [(𝜉/8) Re𝑏 (Pr)av]/[12.7 Sqrt (𝜉/8) ((Pr)^(2/3)av−1)+1.07] and 𝜉=1/(1.82 log10Re𝑏−1.64)^2.; Later, Krasnoshchekov et al. (1971) added a correction factor to the above equation for the tube entrance region in the form of 𝑓(𝑥/𝐷)=0.95+0.95(𝑥/𝐷)^0.8.; Also, this correction factor can be used for a heated tube abrupt inlet within 2≤𝑥/𝐷≤15 [10].	Exponent 𝑛=0.4 at 𝑇𝑤/𝑇pc≤1 or 𝑇𝑏/𝑇pc>1.2≤1,; 𝑛=𝑛1=0.22+0.18 𝑇𝑤/𝑇pc at 1≤𝑇𝑤/𝑇pc≤2.5, and 𝑛=𝑛1+5⋅𝑛1−2×(1−𝑇𝑏/𝑇pc) at 1≤𝑇𝑏/𝑇pc≤1.2.; Valid within the following range: 8×10^4<Re𝑏<5×10^5, 0.85<(Pr𝑏)av<65, 0.90<(𝜌𝑤/𝜌𝑏)<1.0, 0.02<(Cp)av/Cp𝑏<4.0, 0.9<𝑇𝑤/𝑇pc<2.5, 4.6×10^4<𝑞<2.6×10^6, where 𝑞 is in W/m^2 and 𝑥/𝐷≥15 [5–8].
Equation (B.8)	Kondrat’ev [20]	Nu𝑏=0.020 Re𝑏^0.8; [5, 7, 20].	Valid within the range of 10^4<Re<4×10^5 and 𝑡𝑏=130–600°C.; This equation is not valid within the pseudo-critical region.
Equation (B.9)	Ornatsky et al. (1970)	Nu𝑏=0.023 Re𝑏^0.8 Pr^0.8min (𝜌𝑤/𝜌𝑏)^0.3[5, 7].	Prmin is in the minimum value of Pr𝑤 or Pr𝑏.
Equation (B.10)	Yamagata et al. [3]	Nu𝑏=0.0135 Re𝑏^0.85 Pr𝑏^0.8 𝐹𝑐; [3, 5, 7, 8].	𝐹𝑐 = 1 for 𝐸>1, 𝐹𝑐=0.67 Pr^−0.05pc ((Cp)av/Cp𝑏)^𝑛1 for 0≤𝐸≤1,; 𝐹𝑐 = ((Cp)av/Cp𝑏)^𝑛2 for 𝐸<0,; 𝐸 = (𝑇pc-𝑇𝑏)/(𝑇𝑤-𝑇𝑏),; 𝑛1 = −0.77 (1 + (1/Prpc)) + 1.49,; 𝑛2 = −1.44 (1 + (1/Prpc)) − 0.53.; Pressure = 226–294 bar, and bulk fluid temperature = 230–540°C.; Mass flux = 310–1830 kg/m^2s, heat flux = 116–930 kW/m^2.
Equation (B.11)	Watts and Chou et al. (1982)	For (Gr𝑏)av/(Re𝑏^2.7 (Pr𝑏)^0.5av)≤10^−4,; Nu/Nuvarp = [1–3000(Gr𝑏)av/(Re𝑏^2.7 (Pr𝑏)^0.5av)]^0.295.; For (Gr𝑏)av/(Re𝑏^2.7 (Pr𝑏)^0.5av)≥10^−4,; Nu/Nuvarp = [7000(Gr𝑏)av/(Re𝑏^2.7 (Pr𝑏)^0.5av)]^0.295; [4, 11].	Nu = αD/𝜆𝑏,; Nuvarp = 0.021 Re𝑏^0.8 (Pr𝑏)^0.55av (𝜌𝑤/𝜌𝑏)^0.35,; (Gr𝑏)av =[𝜌𝑏(𝜌𝑏− 𝜌av)gD^3]/𝜇^2𝑏,; Re𝑏 = GD/μ,; (Pr𝑏)av = Cpav 𝜇𝑏/𝜆𝑏,; Cpav = (𝐻𝑤−𝐻𝑏)/(𝑇𝑤−𝑇𝑏).; 𝜌av = [Integral (ρdT)] with limits 𝑇𝑤 and 𝑇𝑏/[𝑇𝑤−𝑇𝑏]
Equation (B.12)	Gorban et al. (1990)	Nu𝑏 = 0.0059 Re𝑏^0.90 Pr𝑏^−0.12; [5, 7].
Equation (B.13)	Griem (1996)	Nu𝑏=0.0169 Re𝑏^0.8356 Pr𝑏^0.432; [5, 7, 8].	It covers the entire enthalpy range due to a new method for determining a representative specific heat capacity.; Heat capacities were computed with semiempirical equations at five reference temperatures.
Equation (B.14)	Kitoh et al. (1999)	Nu𝑏=0.015 Re𝑏^0.85 Pr^𝑚𝑏; [5, 7, 10].	𝑚=0.69–81000/𝑞dht + 𝑓𝑐 𝑞.; The heat flux (𝑞dht) is that at which deterioration-rated heat transfer occurs (W/m^2).; The heat flux is calculated according to 𝑞dth = 200 𝐺^1.2.; The coefficient 𝑓𝑐 is calculated according to; 𝑓𝑐 = 29 × 10^−8 + 0.11/𝑞dht for 0 ≤ 𝐻𝑏 ≤ 1500 kJ/kg,; 𝑓𝑐 = −8.7 × 10^−8  − 0.65/𝑞dht for 1500 ≤ 𝐻𝑏 ≤ 3300 kJ/kg,; 𝑓𝑐 = −9.7 × 10^−7  − 1.30/𝑞dht for 3300 ≤ 𝐻𝑏 ≤ 4000 kJ/kg.; Valid for 𝑇𝑏 from 20°C to 550°C (bulk fluid enthalpy from 100 to 3300 kJ/kg), 𝐺 from 100 to 1750 kg/m^2s, and 𝑞 from 0 to 1.8 MW/m^2.
Equation (B.15)	Jackson (2002)	Nu𝑏=0.0183 Re𝑏^0.82 (Pr𝑏)^0.5av (𝜌𝑤/𝜌𝑏)^0. 3    [(Cp)av/Cp𝑏)]^𝑛; [5–7, 11, 15].	Exponent; 𝑛=0.4 for 𝑇𝑏<𝑇𝑤<𝑇pc and for 1.2   𝑇pc<𝑇𝑏<𝑇𝑤,; 𝑛=0.4+0.2 ((𝑇𝑤/𝑇pc)−1) for 𝑇𝑏<𝑇pc<𝑇𝑤,; 𝑛=0.4+0.2 ((𝑇𝑤/𝑇pc)−1) [1–5((𝑇𝑏/𝑇pc)−1) for 𝑇pc<𝑇𝑏<1.2 𝑇pc, and 𝑇𝑏<𝑇𝑤.; 𝑇𝑏, 𝑇pc, and 𝑇𝑤 are in 𝐾.; Valid for forced convection heat transfer in water and carbon dioxide at supercritical pressures.
Equation (B.16)	Kang and Chang et al. [11]	Nu𝑏=0.0244 Re𝑏^0.762 Pr^0.552av (𝜌𝑤/𝜌𝑏)^0.0293; [11].	Fluidfreon, HFC134a.; Pressure: 4.1 to 4.5 MPa, mass flux: 600 to 2000 kg/m^2s, and heat flux: up to 160 kW/m^2.
Equation (B.17)	Zhu et al. [12]	Nu𝑏=0.0068 Re𝑏^0.90 (Pr𝑏)^0.63av (𝜌𝑤/𝜌𝑏)^0.17(𝑘𝑤/𝑘𝑏)^0.29 [12].	Pressure: 90–300 bar, mass flux: 600–1200 kg/m^2s, and heat flux: 200–600 kW/m^2.
Equation (B.18)	Mokry et al. [15]	Nu𝑏=0.0061 Re𝑏^0.904 (Pr𝑏)^0.684av (𝜌𝑤/𝜌𝑏)^0.564; [15].	Pressure: 24 MPa, inlet fluid temperature: 320–350°C, mass flux: 200–1500 kg/m^2s, and heat flux ≤1250 kW/m^2.