Turbulent Convection Heat Transfer Analysis of Supercritical Pressure CO₂ Flow in a Vertical Tube Based on the Field Synergy Principle

ABSTRACT

Turbulent convection heat transfer of fluids at supercritical pressures can experience abnormal heat transfer phenomena as the fluid temperature approaches the pseudo-critical temperature. A better understanding of these abnormal heat transfer characteristics is essential to develop ways to assure the security and reliability of engineering designs. In this study, we analyze the turbulent convection heat transfer characteristics of supercritical pressure fluid flow in a vertical heated tube using the field synergy principle. A turbulent Prandtl number model that is a function of the flow conditions as well as the physical properties is used in a low Reynolds number turbulence model with validations against experimental data. The validity of the field synergy principle is extended here to analyze convection heat transfer of supercritical pressure fluids where the heat transfer rate is a function of the synergy between the velocity and temperature gradients. Numerical simulation results show that when heat transfer deterioration occurs, the synergy angle between the velocity and temperature gradient vectors increase. The field synergy number could be used as an indicator to quantitatively identify the degree of heat transfer deterioration. The results presented here provide a basis for further research on enhancing turbulent convection of supercritical pressure fluid.

Nomenclature

AKN	=	Abe–Kondoh–Nagano (low Reynolds number turbulence model)
A+	=	effective viscous sublayer thickness
B	=	constant in Eq. (9(9) $$\begin{array}{c}\hfill h_1=1-\exp \left(-\frac{y^{+}}{A^{+}}\right),h_2=0.5\left[1+\tanh \left(\frac{B-y^{+}}{10}\right)\right]\\ \end{array}$$(9) )
Bo*	=	non-dimensional buoyancy parameter (=$\frac{G{r}^{*}}{R{e}^{3.425}P{r}^{0.8}}$)
Cμ, Cϵ1, Cϵ2, cθ	=	constant value in the turbulence model
cp	=	specific heat at constant pressure [J/(kg·K)]
D	=	tube diameter [m]
fμ, f1, f2	=	Functions in the turbulence model
Fc	=	field synergy number
G	=	mass flow rate [kg/h]
Gr*	=	Grashof number (=$\frac{g\beta q_w{D}^4}{\lambda {\vartheta }^2}$)
g	=	gravitational acceleration [m2/s]
h	=	enthalpy [J/kg]
h1, h2	=	Functions in the turbulence Prandtl model
KvT	=	non-dimensional flow acceleration parameter (=$\frac{4q_{w}D{\alpha }_p}{R{e}^2{\mu }_b{c}_p}$)
KvP	=	non-dimensional flow acceleration parameter (=$-\frac{D}{Re}{\beta }_T\frac{dp}{dx}$)
k	=	turbulence kinetic energy [m2/s2]
L	=	tube length [m]
l	=	Mixing length [m]
Nu	=	Nusselt number ($=\frac{q_{w}D}{\lambda \left(T_w-T_b\right)})$
Pr	=	Prandtl number
Prt	=	turbulent Prandtl number
Prt, 0	=	turbulent Prandtl number
p	=	pressure [MPa]
q	=	heat flux [kW/m2]
R	=	tube radius [m]
Re	=	Reynolds number ($=\frac{\rho uD}{\mu })$
r	=	radial coordinate [m]
T	=	temperature [°C]
U	=	velocity [m/s]
$\bar{U}$	=	dimensionless velocity
u	=	velocity components in x direction [m/s2]
v	=	velocity components in r direction [m/s2]
x	=	axial coordinate [m]
y+, y*	=	dimensionless distances from the wall

Greek symbols

αp	=	thermal expansion coefficient [1/K]
β	=	thermal expansion coefficient [1/K]
βT	=	isothermal compression coefficient [1/Pa]
δt	=	thermal boundary [m]
δu, δρ, δcp and δT	=	The increments of u, ρ, cp and T
ϵ	=	dissipation rate of k [m2/s3]
θ	=	field synergy angle [degree,°]
λ	=	thermal conductivity [W/(m·K)]
σk, σϵ	=	constant value in the turbulence model
σt	=	constant value in the turbulence Prandtl model
μ	=	molecular viscosity [kg/(m·s)]
μt	=	turbulent viscosity [kg/(m·s)]
ν	=	kinematic viscosity (=$\frac{\mu }{\rho }$) [m2/s]
ρ	=	fluid density [kg/m3]
τ	=	shear stress [N/m2]
ω	=	specific dissipation rate [1/s]
∇T	=	temperature gradient [K]
$\overline{\nabla \mathrm{T}}$	=	dimensionless temperature gradient

Subscripts

b	=	bulk
f	=	fluid
h	=	heated section
i	=	inlet or local value
m	=	average
o	=	outlet
pc	=	pseudo critical
t	=	turbulent
w	=	wall
x	=	based on length x
∞	=	value at great distance from a body

Additional information

Funding

This project was supported by National Natural Science Foundation of China (No. 51536004), the Science Fund for Creative Research Groups of China (No. 51621062).

Notes on contributors

Zhen-Chuan Wang

Zhen-Chuan Wang is a Ph.D. student in the Department of Energy and Power Engineering, Tsinghua University, China. He received his M.Sc. degree in Engineering Thermophysics from Tianjin University in 2014. His main research interests are convection heat transfer at supercritical pressures and convection heat transfer in porous media.

Pei-Xue Jiang

Pei-Xue Jiang is a Professor in the Department of Energy and Power Engineering, Tsinghua University, China. He received his Ph.D. in the Department of ThermoPower Engineering of Moscow Power Engineering Institute in 1991. He then joined the faculty of Tsinghua University and took the professor post in 1997. His main research interests include convection heat transfer in porous media and enhanced heat transfer, convection heat transfer at supercritical pressures, transpiration cooling and film cooling, thermal transport in nanoscale structures, trans-critical CO₂ air conditioning systems, and heat pumps.

Rui-Na Xu

Rui-Na Xu is an Associate Professor in the Department of Energy and Power Engineering, Tsinghua University, China. She received her Ph.D. in Engineering Thermophysics from the Tsinghua University in 2007. Her main research interests are convection heat transfer and fluid flow in porous media, convection heat transfer at supercritical pressures, and thermal transport in microscale and nanoscale structures.