Characteristics of turbulent heat transfer in an annulus at supercritical pressure

Heat transfer to fluids at supercritical pressure is different from heat transfer at lower pressures due to strong variations of the thermophysical properties with the temperature. We present and analyze results of direct numerical simulations of heat transfer to turbulent ${\mathrm{CO}}_2$ at 8 MPa in an annulus. Periodic streamwise conditions are imposed so that mean streamwise acceleration due to variations in the density does not occur. The inner wall of the annulus is kept at a temperature of 323 K, while the outer wall is kept at a temperature of 303 K. The pseudocritical temperature $T_{pc}=307.7$ K, which is the temperature where the thermophysical properties vary the most, can be found close to the inner wall. This work is a continuation of an earlier study, in which turbulence attenuation due to the variable thermophysical properties of a fluid at supercritical pressure was studied. In the current work, the direct effects of variations in the specific heat capacity, thermal diffusivity, density, and the molecular Prandtl number on heat transfer are investigated using different techniques. Variations in the specific heat capacity cause significant differences between the mean nondimensionalized temperature and enthalpy profiles. Compared to the enthalpy fluctuations, temperature fluctuations are enhanced in regions with low specific heat capacity and diminished in regions with a large specific heat capacity. The thermal diffusivity causes local changes to the mean enthalpy gradient, which in turn affects molecular conduction of thermal energy. The turbulent heat flux is directly affected by the density, but it is also affected by the mean molecular Prandtl number and attenuated or enhanced turbulent motions. In general, enthalpy fluctuations are enhanced in regions with a large mean molecular Prandtl number, which enhances the turbulent heat flux. While analyzing the Nusselt numbers under different conditions it is found that heat transfer deterioration or enhancement can occur without streamwise acceleration or mixed convection conditions. Finally, through a combination of a relation between the Nusselt number and the radial heat fluxes, a quadrant analysis of the turbulent heat flux, and conditional averaging of the heat flux quadrants, it is shown that heat transfer from a heated surface depends on the density and the molecular Prandtl number of both hot fluid moving away from a heated surface as well as the thermophysical properties of relatively cold fluid moving towards it.

Figure 1

Properties of ${\mathrm{CO}}_2$ at 8 MPa as a function of the temperature $T$ (K). Shown are the density $\rho$ (kg/${\mathrm{m}}^3$), the specific heat capacity $c_p$ (J/kg K), the molecular Prandtl number $\text{Pr}$ (–), the thermal conductivity $k$ (W/m K), the kinematic viscosity $\nu ({\mathrm{cm}}^2/\mathrm{s})$, and the thermal diffusivity $a({\mathrm{cm}}^2/\mathrm{s})$.

Figure 2

The annular geometry. The inner and outer wall radii are denoted as $R_{\mathrm{in}}$ and $R_{\mathrm{out}}$, respectively. The corresponding wall temperatures are denoted as $T_{\mathrm{hot}}$ and $T_{\mathrm{cold}}$, respectively. The length of the annulus is denoted with $L$.

Figure 3

Instantaneous cross-sectional visualization of thermophysical properties for the supercritical forced convection case. Left: The enthalpy, temperature, and specific heat capacity are shown. Right: The molecular Prandtl number, thermal diffusivity, and density are shown.

Figure 4

Mean profiles of the density, thermal diffusivity, specific heat capacity, and molecular Prandtl number near the inner wall (a) and the outer wall (b) in the forced convection ${\mathrm{sCO}}_2$ case (II). The vertical dotted line indicates the point where the Favre-averaged enthalpy equals the enthalpy at the pseudocritical temperature.

Figure 5

Root mean square values of the density and the thermal diffusivity near the inner wall (a) and the outer wall (b) in the forced convection ${\mathrm{sCO}}_2$ case (II).

Figure 6

Root mean square values of the velocity fluctuations. (a) forced convection ${\mathrm{sCO}}_2$ case, inner wall region, (b) forced convection ${\mathrm{sCO}}_2$ case, outer wall region, (c) mixed convection ${\mathrm{sCO}}_2$ case, inner wall region, and (d) mixed convection ${\mathrm{sCO}}_2$ case, outer wall region. The gray lines indicate rms values of the velocity fluctuations of the reference case.

Figure 7

Mean profiles of the enthalpy and the temperature in the forced convection ${\mathrm{sCO}}_2$ (a) and the mixed convection ${\mathrm{sCO}}_2$ (b) cases. The gray lines indicate results of the reference case, where $\overline{h}=\overline{T}$. The vertical dotted line indicates the point where the Favre-averaged enthalpy equals the enthalpy at the pseudocritical temperature.

Figure 8

Root mean square values of the enthalpy in the forced convection ${\mathrm{sCO}}_2$ and the mixed convection ${\mathrm{sCO}}_2$ cases near the inner wall (a) and the outer wall (b). The gray lines and crosses indicate rms values of the enthalpy in the reference case (I), where $h_{\mathrm{rms}}=T_{\mathrm{rms}}$, due to the nondimensionalization of $h$ and $T$.

Figure 9

Probability density functions of the enthalpy and the temperature in the forced convection ${\mathrm{sCO}}_2$ case (II). (a) forced convection ${\mathrm{sCO}}_2$, $y^{+}=5$, (b) forced convection ${\mathrm{sCO}}_2$, $y^{+}=10$, (c) mixed convection ${\mathrm{sCO}}_2$, $y^{+}=5$, and (d) mixed convection ${\mathrm{sCO}}_2$, $y^{+}=10$.

Figure 10

Heat fluxes in the forced convection ${\mathrm{sCO}}_2$ case (II) and the reference case (I) near the inner wall region (a) and the outer wall region (b). Gray lines indicate results of the reference case.

Figure 11

The radial turbulent heat flux $\widetilde{u^{″}{h}^{″}}$ (a) and the streamwise turbulent heat flux $\widetilde{w^{″}{h}^{″}}$ (b) near the inner wall in the reference case (I), the forced convection case (II), and the mixed convection case (III).

Figure 12

Budgets of the turbulent heat flux transport equation. (a) forced convection ${\mathrm{sCO}}_2$, radial turbulent heat flux, (b) forced convection ${\mathrm{sCO}}_2$, streamwise turbulent heat flux, (c) mixed convection ${\mathrm{sCO}}_2$, radial turbulent heat flux, and (d) mixed convection ${\mathrm{sCO}}_2$, streamwise turbulent heat flux.

Figure 13

Viscous parts of the diffusion and dissipation terms of the radial turbulent heat flux for reference case (I) and the forced convection ${\mathrm{sCO}}_2$ case. The results of the reference case (I) are in gray.

Figure 14

(a) Typical mushroom structures near the hot wall of the annulus in the forced convection ${\mathrm{sCO}}_2$ case. The black contour lines denote low-speed fluid regions (streaks). (b) A physical interpretation of the flux quadrants using the mushroom structure as an example.

Figure 15

Graphic representation of the hole parameter $H$. The “hole” is indicated by the shaded area.

Figure 16

Contributions of the turbulent heat flux events as a function of the hole size $H$ in the forced convection ${\mathrm{sCO}}_2$ case (II) (black) and the reference case (I) (gray). (a) hot inner wall, (b) cold outer wall.

Figure 17

Contributions of the turbulent heat flux events as a function of the hole size $H$ in the mixed convection ${\mathrm{sCO}}_2$ case (II) (black) and the reference case (I) (gray). (a) hot inner wall, (b) cold outer wall.

Figure 18

Expected value of the wall normal velocity $u^{″}$, the density $\rho$, and the molecular Prandtl number $\text{[}$ denoted as $\text{E}\left(u^{″}\right)$, $\text{E}\left(\rho \right)$, and $\text{E}\left(\text{Pr}\right)$, respectively] conditioned on all four quadrants of the turbulent heat flux in the forced convection ${\mathrm{sCO}}_2$ case (II) and the reference case (I) at $y^{+}=20$ near the inner wall (left column) and the outer wall (right column).