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Fig. 14. Spatial variation of temperature inside the enclosure at mid-plane (X/ L=0.5) at t=1.5s for near-critical carbon dioxide (p;= 7.38 MPa, T;= 304.3 K, p' =3.66 x 10-4, 7 =5.65 x 10-4 and Tc = 3.88 x 10-°) and Ra=7.35 x 10° with and without the effect of bulk viscosity.

Figure 14 Spatial variation of temperature inside the enclosure at mid-plane (X/ L=0.5) at t=1.5s for near-critical carbon dioxide (p;= 7.38 MPa, T;= 304.3 K, p' =3.66 x 10-4, 7 =5.65 x 10-4 and Tc = 3.88 x 10-°) and Ra=7.35 x 10° with and without the effect of bulk viscosity.

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In thermodynamics, a critical point specifies the conditions (temperature and pressure) at which a phase boundary ceases to Fig. 2 shows the density and specific heat vs. temperature dia- gram for carbon dioxide with changing pressure. The density at atmospheric pressure is almost constant, while the density in the supercritical-pressure condition widely varies across the phase interface from the liquid or gas phase to the supercritical fluid phase. Hence, even around 10K above the critical temperature Fig. 1. The p-v diagram for carbon dioxide. 2. Thermo-physical properties at supercritical state
In thermodynamics, a critical point specifies the conditions (temperature and pressure) at which a phase boundary ceases to Fig. 2 shows the density and specific heat vs. temperature dia- gram for carbon dioxide with changing pressure. The density at atmospheric pressure is almost constant, while the density in the supercritical-pressure condition widely varies across the phase interface from the liquid or gas phase to the supercritical fluid phase. Hence, even around 10K above the critical temperature Fig. 1. The p-v diagram for carbon dioxide. 2. Thermo-physical properties at supercritical state
Fig. 3. Ratio of bulk viscosity to shear viscosity (1,/{:) as a function of pressure and temperature for near-critical carbon dioxide. Fig. 2. Variation of density () and specific heat (cp) as functions of pressure and temperature for near-critical carbon dioxide.
Fig. 3. Ratio of bulk viscosity to shear viscosity (1,/{:) as a function of pressure and temperature for near-critical carbon dioxide. Fig. 2. Variation of density () and specific heat (cp) as functions of pressure and temperature for near-critical carbon dioxide.
study, the set of equations developed by Onuki [25] was used fo the determination of the bulk viscosity of supercritical carbor dioxide. Fig. 3 shows the ratio of bulk viscosity to shear viscosity ({/~) of supercritical carbon dioxide as a function of pressure and temperature near the critical point. The bulk viscosity tends to infinity when temperature and pressure approach critica values.
study, the set of equations developed by Onuki [25] was used fo the determination of the bulk viscosity of supercritical carbor dioxide. Fig. 3 shows the ratio of bulk viscosity to shear viscosity ({/~) of supercritical carbon dioxide as a function of pressure and temperature near the critical point. The bulk viscosity tends to infinity when temperature and pressure approach critica values.
Fig. 6. Contours of steady state temperature inside the side heated enclosure filled with supercritical carbon dioxide far from the critical point (p;= 12.0 MPa, Tm = 325.0K) with Ra=10°, p’=6.26 x 107-', T=6.86 x 10°? and Th = 8.71 x 10-7; at (a) t=1.0s and (b) steady state. [Tmax = 325.3 K (left wall), Tmin = 324.7 K (right wall), 5T = 60 mK]. Fig. 5. Contours of steady state temperature inside the side heated enclosure filled with supercritical carbon dioxide near the critical point (p; = 7.38 MPa, T,, = 305.0 K) with Ra = 10°, p’ = 3.66 x 10-4, T =2.9 x 10-3 and T¢ = 3-88 x 10° at (a) t=1.0s and (b) steady state. [Tmax = 305.05 K (left wall), Twin = 304.95 K (right wall), oT = 10 mK].
Fig. 6. Contours of steady state temperature inside the side heated enclosure filled with supercritical carbon dioxide far from the critical point (p;= 12.0 MPa, Tm = 325.0K) with Ra=10°, p’=6.26 x 107-', T=6.86 x 10°? and Th = 8.71 x 10-7; at (a) t=1.0s and (b) steady state. [Tmax = 325.3 K (left wall), Tmin = 324.7 K (right wall), 5T = 60 mK]. Fig. 5. Contours of steady state temperature inside the side heated enclosure filled with supercritical carbon dioxide near the critical point (p; = 7.38 MPa, T,, = 305.0 K) with Ra = 10°, p’ = 3.66 x 10-4, T =2.9 x 10-3 and T¢ = 3-88 x 10° at (a) t=1.0s and (b) steady state. [Tmax = 305.05 K (left wall), Twin = 304.95 K (right wall), oT = 10 mK].
Fig. 7. Velocity vectors (Ra= 10%) inside the side heated enclosure filled with supercritical carbon dioxide (a) near the critical point (p; = 7.38 MPa, T,, = 305.0 K, p' = 3.66 x 10-4, T = 2.9 x 10-? and T;,, = 3.88 x 10-°) and (b) far from the critical point (p; = 12.0 MPa, Tm = 325.0 K, p’ = 6.26 x 10-1, T =6.86 x 10°? and T,, = 8.71 x 10-7) at steady state.
Fig. 7. Velocity vectors (Ra= 10%) inside the side heated enclosure filled with supercritical carbon dioxide (a) near the critical point (p; = 7.38 MPa, T,, = 305.0 K, p' = 3.66 x 10-4, T = 2.9 x 10-? and T;,, = 3.88 x 10-°) and (b) far from the critical point (p; = 12.0 MPa, Tm = 325.0 K, p’ = 6.26 x 10-1, T =6.86 x 10°? and T,, = 8.71 x 10-7) at steady state.
Fig. 8. Variation of steady state spatially averaged Nusselt (Nu) number along the vertical walls as a function of Rayleigh (Ra) number for supercritical carbon dioxide far from the critical point.
Fig. 8. Variation of steady state spatially averaged Nusselt (Nu) number along the vertical walls as a function of Rayleigh (Ra) number for supercritical carbon dioxide far from the critical point.
Fig. 9. Variation of steady state spatially averaged Nusselt number as a function of T= (= 7) for different p’ = (252) for near-critical carbon dioxide with (a) 8 8 ° 8 Ra=5 x10 (b) Ra=1x10 and(c)Ra=5x10.
Fig. 9. Variation of steady state spatially averaged Nusselt number as a function of T= (= 7) for different p’ = (252) for near-critical carbon dioxide with (a) 8 8 ° 8 Ra=5 x10 (b) Ra=1x10 and(c)Ra=5x10.
Fig. 10. Steady state spatially averaged Nusselt number (Nu) surface map as a function of the dimensionless temperature T’ = (;-"*) and pressure p’ = (22) for near-critical carbon dioxide at Ra = 10°.
Fig. 10. Steady state spatially averaged Nusselt number (Nu) surface map as a function of the dimensionless temperature T’ = (;-"*) and pressure p’ = (22) for near-critical carbon dioxide at Ra = 10°.
Fig. 11. Variation of steady state spatially averaged Nusselt (Nu) number as a function of Rayleigh (Ra) number for supercritical carbon dioxide with different values of the parameter ‘f.
Fig. 11. Variation of steady state spatially averaged Nusselt (Nu) number as a function of Rayleigh (Ra) number for supercritical carbon dioxide with different values of the parameter ‘f.
Fig. 12. Contours of temperatures inside a bottom heated enclosure at t= 1.5 s for near critical carbon dioxide (p;=7.38MPa, 1;=304.3K, p’=3.66 x 10-4, T =5.65 x 10-4 and T,, = 3.88 x 10°*) and Ra = 7.35 x 10° (a) with and (b) without the inclusion of bulk viscosity. [Tmax = 304.4 K (bottom wall), Tmin = 304.3 K (top wall), 6T = 1.0 mK].
Fig. 12. Contours of temperatures inside a bottom heated enclosure at t= 1.5 s for near critical carbon dioxide (p;=7.38MPa, 1;=304.3K, p’=3.66 x 10-4, T =5.65 x 10-4 and T,, = 3.88 x 10°*) and Ra = 7.35 x 10° (a) with and (b) without the inclusion of bulk viscosity. [Tmax = 304.4 K (bottom wall), Tmin = 304.3 K (top wall), 6T = 1.0 mK].
Fig. 13. Velocity vectors inside a bottom heated enclosure at t=1.5s for near- critical carbon dioxide (p; = 7.38 MPa, T; = 304.3 K, p’ = 3.66 x 10~4, T’ =5.65 x 10-4 and T,,, = 3.88 x 10~°) and Ra = 7.35 x 10° (a) with and (b) without the inclusion of bulk viscosity.
Fig. 13. Velocity vectors inside a bottom heated enclosure at t=1.5s for near- critical carbon dioxide (p; = 7.38 MPa, T; = 304.3 K, p’ = 3.66 x 10~4, T’ =5.65 x 10-4 and T,,, = 3.88 x 10~°) and Ra = 7.35 x 10° (a) with and (b) without the inclusion of bulk viscosity.
Fig. 15. Temporal variation of spatially averaged Nusselt number (Nu) at the left wall (X/L=0.0) of the enclosure at t=1.5s for near-critical carbon dioxide (p; = 7.38 MPa, T; = 304.3 K, p' = 3.66 x 10-4, T’ =5.65 x 10-4 and T,, = 3.88 x 10-°) and Ra = 7.35 x 10° with and without the effect of bulk viscosity.
Fig. 15. Temporal variation of spatially averaged Nusselt number (Nu) at the left wall (X/L=0.0) of the enclosure at t=1.5s for near-critical carbon dioxide (p; = 7.38 MPa, T; = 304.3 K, p' = 3.66 x 10-4, T’ =5.65 x 10-4 and T,, = 3.88 x 10-°) and Ra = 7.35 x 10° with and without the effect of bulk viscosity.
Fig. 16. Contours of viscosity ratio (j1,/44) inside a bottom heated enclosure at t=1.5s for near-critical carbon dioxide (p; = 7.38 MPa, T; = 304.3 K, p! = 3.66 x 1074, T =5.65 x 10-4 and T,,¢ = 3.88 x 10-5) and Ra=7.35 x 108 [(jtp//)max = 1200 (top wall), ({p//0)min = 50 (bottom wall), 5({tp/p0) = 50]. The temporal variation of spatially averaged Nusselt number at the left (heated) wall of the enclosure is shown in Fig. 15. At the beginning, the temporal heat transfer rate at the left wall is very high due to the heating. The value decreases as the system ap- proaches thermal equilibrium. It is observed that the case includ- ing the bulk viscosity effects exhibits a relatively smaller Nusselt number compared to that without the effect of bulk viscosity. This is mainly due to the larger effective viscosity in the system. For the case without the inclusion of bulk viscosity, the rapid increase and decrease in temporal Nusselt number around 1.0 ~ 1.3 s indicates the collapse of the bottom boundary layer and release of thermal
Fig. 16. Contours of viscosity ratio (j1,/44) inside a bottom heated enclosure at t=1.5s for near-critical carbon dioxide (p; = 7.38 MPa, T; = 304.3 K, p! = 3.66 x 1074, T =5.65 x 10-4 and T,,¢ = 3.88 x 10-5) and Ra=7.35 x 108 [(jtp//)max = 1200 (top wall), ({p//0)min = 50 (bottom wall), 5({tp/p0) = 50]. The temporal variation of spatially averaged Nusselt number at the left (heated) wall of the enclosure is shown in Fig. 15. At the beginning, the temporal heat transfer rate at the left wall is very high due to the heating. The value decreases as the system ap- proaches thermal equilibrium. It is observed that the case includ- ing the bulk viscosity effects exhibits a relatively smaller Nusselt number compared to that without the effect of bulk viscosity. This is mainly due to the larger effective viscosity in the system. For the case without the inclusion of bulk viscosity, the rapid increase and decrease in temporal Nusselt number around 1.0 ~ 1.3 s indicates the collapse of the bottom boundary layer and release of thermal
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EngineeringNatural ConvectionBulk Viscosity
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