RESULTS AND DISCUSSION

Firstly, Figs. 2 and 3 demonstrate that the entropy generation number due to heat transfer irreversibility falls down to nil along the streamwise length as the dimensionless local heat transfer coefficient does. This trend is attributed by the diminished heat transfer coefficient due to increasing film thickness. It is also seen in Eq. (25) that N varies as _H the square of Nusselt number. This may account for the higher contribution to

for a cylinder

for a sphere

to the gravity-driven film flow friction irreversibility increases with the values of Br/Ψ.

Since N varies as _F sin^4/3θ for a cylinder and sin^2/3θ for a sphere, a higher value of ψ

/

Br will enhance the film flow friction irreversibility film to dominate over the heat transfer irreversibility in the lower half of streamwise length.

Fig. 2 Entropy-generation rate profile for varying group parameters Br/Ψ (for a cylinder)

Fig. 3 Entropy-generation rate profile for varying group parameters Br/Ψ (for a sphere)

Secondly, as for the effect of Brinkman number on the entropy generation rate, with a specified value of Ra/Ja, Figs. 4 and 5 indicate obviously that entropy generation number increases with an increase in the parameter Br/Ψ. This confirms that the parameter Br/Ψ plays an important role in influencing its irreversibility induced by

condensate flow friction. Thus, one may achieve the minimization of entropy generation by reducing the value of Br/Ψ. Furthermore, entropy-generation number increases with Ra/Ja as dimensionless local heat transfer coefficient does.

Fig. 4 Entropy-generation rate profiles for varying group parameters Br/Ψ and Ra/Ja.

(for a cylinder)

Fig. 5 Entropy-generation rate profiles for varying group parameters Br/Ψ and Ra/Ja.

(for a sphere)

Finally, Fig. 6 indicates that the irreversibility distribution ratio for a cylinder is less than unity, increasing with the circumferential streamline length around the lower half of cylindrical perimeter (θ =0.7π). But, for the higher values ofBr/Ψ, e. g. Br/ψ ≥0.75, the irreversibility distribution ratio rises up to the value greater than unity in the lower half of a cylinder. This means that heat transfer irreversibility dominates over fluid

the contribution of condensate flow friction is becoming significant, as it goes farther downstream. On account of increasing gravity-driven film velocity, the film flow friction irreversibility enhances much enough so that it dominates over heat transfer irreversibility around the rear lower half of cylindrical perimeter for the cases ofBr/ψ ≥0.75. Moreover, the higher value of Br/Ψ results in the higher contribution to film flow friction irreversibility, as compared with heat transfer irreversibility. Hence, irreversibility distribution ratio increases with Brinkman group parameters.

Fig. 6 Irreversibility distribution ratio (for a cylinder)

The irreversibility distribution ratio for a sphere is illustrated in Fig. 7. Obviously, heat transfer irreversibility dominates in the upper half of a sphere. As for the lower half of a sphere, the flow friction irreversibility becomes dominated. As far as heat transfer irreversibility is concerned, the result obviously confirms that the same temperature difference heat transfer at smaller film thickness leads to a higher heat transfer irreversibility. Thus, heat transfer irreversibility declines along with the streamwise length. At the same time, flow friction irreversibility rises along with the streamwise length owing to increasing condensate film velocity. Thus, the irreversibility distribution

ratio is smaller than unity in the beginning and then rises up to the value greater than unity in the lower half of a sphere.

Fig. 7 Irreversibility distribution ratio (for a sphere)

4. CONCLUSION

This is the first approach using thermodynamic second law to investigate free convection film condensation from saturated vapor flowing slowly onto an isothermal horizontal cylinder/sphere. Basically, the local entropy generation rate induced by heat transfer irreversibility is directly proportion to Nusselt number but inversely proportion to film thickness. On the other hand, the local entropy generation rate induced by gravity-driven film flow friction irreversibility is proportion to Brinkman number. In order to achieve minimization of entropy-generation rate, one should reduce the values of Brinkman group parameters without losing condensation heat transfer coefficient.

Notably, as group Rayleigh parameters increase, dimensionless heat transfer coefficient is enhanced, but entropy generation number is augmented too.

Br Brinkman number, μu₀ /k⋅ΔT

C specific heat of condensate at constant pressure p

D diameter of c a cylinder or sphere g gravity

h latent heat of condensate fg

h^'fg latent heat of condensation corrected for condensate subcooling Ja Jakob number, CpΔT/h^'fg

k thermal conductivity

m condensate mass flux per unit area •

Nu local Nusselt number, hD/k

N dimensionless entropy generation number S

Ra Rayleigh number, ρ(ρ−ρ_v)gPrD³/μ² r radius of a cylinder or sphere

S entropy generation rate G

S characteristics entropy generation rate 0

T saturation temperature sat

T wall w temperature

Δ temperature difference between the interface of vapor and wall, TT _sat- T_w u x-velocity

v y-velocity

Greek symbol

δ local thickness of condensate film

δ* dimensionless thickness of condensate film μ dynamic viscosity

ρ density

θ angle measured from top of cylinder or sphere ψ dimensionless temperature difference, ΔT /T_w

ACKNOWLEDGEMENT

Funding for this study is partially provided by the National Science Council of the Republic of China under contracts NSC94-2212-E-151-020

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