Guan-Cyun Li and Sheng-An Yang*
Department of Mold and Die Engineering, National Kaohsiung University of Applied Sciences , Kaohsiung , Taiwan , ROC
*E-mail: [email protected]
*Tel.: +886-7-3814526 ext. 5412 *Fax: +888-7-3835015 Address: 415 Chien-Kung Road, Kaohsiung 80778, Taiwan, R.O.C.
Abstract
This paper aims to perform thermodynamic analysis of saturated vapor flowing slowly onto and condensing on an elliptical cylinder. This analysis provides us how the geometric parameter-ellipticity affects entropy generation during film-wise condensation heat transfer process. The results observe that local condensate film thickness decreases with an increase in ellipticity of a cylinder. From the first law point of view, the local heat transfer coefficient enhances as ellipticity increases. Meanwhile, from the second law point of view, entropy generation increases with increasing the value of ellipticity. We derive an expression for entropy generation, which accounts for the combined action of the specified irreversibilities. The result demonstrates that thermal irreversibility dominates over film flow friction irreversibility. Finally, an expression of minimizing entropy generation in laminar film condensation heat transfer is obtained.
elliptical cylinder
1. Introduction
Conservation of useful energy depends on the design of efficient thermodynamic heat-transfer processes, i.e. minimization of entropy generation due to heat transfer and viscous dissipation. Entropy generation in thermal engineering systems destroys system available energy and reduces its efficiency.
As for enhancement of condensation heat transfer, several researches, such as Yang and Hsu [1] and Yang and Chen [2], Ali and McDonald [3], Karimi [4], and Memory et al.
[5] confirmed that cylinders, fins, or extended surfaces of elliptical profiles with major axes aligned with gravity are superior to those of circular profiles. It is a fact that heat transfer enhancement is achieved, however the increase in heat transfer rate is known to augment friction factor due to pumping power.
Bejan [6] pioneered the method of entropy generation minimization in heat and mass transfer analysis. He found thermodynamics optimums of the ratio of film coefficient to pumping power and the dimensionless temperature difference with constant mass flow rate and heat transfer rate per unit length. He devised concrete methods for minimizing entropy generation in engineering equipment for heat transfer. He conducted EGM analysis on ducts with constant heat flux for flat plates; cylinders in cross flow. Sahin [7]
investigated the effect of temperature-dependent viscosity on the entropy generation rate as well as the ratio of pumping power to heat transfer.
Adeyinka and Naterer [8] investigated the physical significance of entropy generation in plate film condensation. Lin et al. [9] first performed the second-law analysis on saturated vapor flowing through and condensing inside horizontal cooling tubes. They noted that in a tube case, an optimum Reynolds number exists at which the entropy generates at a minimum rate. Dung and Yang [10] presented the entropy generation minimization method to optimize a saturated vapor flowing slowly onto and condensing on an isothermal horizontal tube. Their results for the optimizing entropy generation and plate size are expressed in terms of a duty parameter. In addition, they observed that entropy generation provides a useful parameter in the optimization of a two-phase system.
Entropy generation is associated with thermodynamic irreversibility which is common in all types of heat transfer processes. Film condensation belongs to phase-change heat transfer, but little literature regarding its second-law analysis is investigated. The second law analysis of the film condensation outside cylinders still remains an unsettled question so far. More recently, we first conducted a study [11] on the local entropy generation rate of laminar free convection film condensation on an elliptical cylinder. That paper investigated how the geometric parameter-ellipticity affects local entropy-generation rate during film-wise condensation heat transfer process.
Currently, the present study will focus on the minimization of total entropy generation number to give an idea of optimal design on free convection film condensation outside an elliptical cylinder. We derive an expression for the entropy generation number, which accounts for the combined action of finite-temperature difference heat transfer irreversibility and film flow friction irreversibility. Basically, this study makes good engineering sense to focus on the irreversibility of film condensation heat transfer and try
entropy generation minimization will thus help us achieve the complete thermodynamic analysis, including first and second law, on laminar film-wise condensation outside an elliptical cylinder.
2. Thermal Analysis
Consider a horizontal elliptical cylinder with major axis “2a” in the gravitational direction and minor axis “2b”, situated in a slowly flowing pure vapor which is at its saturated temperatureTsat. Moreover, the wall temperature Tw is considered to be uniform and much lower than the vapor saturation temperatureT . Thus, condensation sat occurs on the wall and a continuous film of the liquid runs downward over the cylinder under the influence of gravity.
Fig.1 illustrates schematically a physical model and coordinate system where the curvilinear coordinates (x, y) are aligned along an elliptical cylinder surface and its normal. The assumptions employed in the formulation of the problem are:
(1) The condensate film flow is laminar and steady.
(2) The inertia effect of the condensate film flow is neglected.
(3) The condensate film thickness is much smaller than the curvature diameter.
(4) Viscous dissipation in the interface is ignored.
(5) Compared with the transversal conduction within the condensate film, the axial conduction is negligible.
According to above assumptions, the condensate film governed equations are written
respectively.
subject to the following boundary conditions:
=0
On account of varying radius of surface curvature, the surface tension forces can be derived here, as expressed in Yang and Chen [2]:
) sin(2 )
Integrating Eqs. (2) and (3) directly with the boundary conditions gives the following formula of the film velocity “ u ” and temperature “T ” profile, respectively.
] By assuming a reference velocity,
μ
Since the temperature distribution in the condensate layer may be assumed linear in the Nusselt-Rohsenow condensation theory, one has
h'fgδ
Using the transformation method from x toφ, as introduced in Yang and Chen [2], we can derive dimensionless local condensate liquid film thickness as.
14
As in Nusselt [12] theory, interpreting the result of model is straightforward by employing the usual idea of a local heat transfer coefficient as follows:
= ∗
According to Bejan [13], together with the fifth item of above-mentioned assumptions, the entropy generation rate for convection heat transfer can be written as
2
On the right-hand side of Eq. (16), the first term and the second term represents the entropy generation due to heat transfer and due to film flow friction, respectively.
Substituting Eqs. (8) and (10) into Eq. (16) and assuming y Tw T <<
Next, integrating Eq. (18) over the entire streamline length, fromφ =0 to π gives
w Tw
k , T and w μ denote thermal conductivity, wall temperature, and dynamic viscosity, respectively. Entropy generation number (N ) is the ratio of the volumetric entropy S generation rate (Sgen) to a characteristics transfer rate (S ). o
N =S
the entropy generation number can be expressed as:
( )
r( )
d H FTo understand which of the condensate flow friction irreversibility (N ), or heat transfer F irreversibility (NH) dominates, we introduce a criterion known as the irreversibility distribution ratio in the following equation:
NS , we find the following optimum that minimizes value of NS
( )
Inserting Eq. (27) into Eq. (20) gives an expression of minimizing entropy generation as follows:The ratio of the actual entropy generation to the minimized entropy generation representsN , which is determined to be S∗
( )
3. Results and Discussion
Fig. 2 indicates the variation of dimensionless entropy generation numbers N S with Ra/Ja under the surface tension effects for various ellipticities. Firstly, from the first law point of view, we also confirm that the mean heat transfer coefficients enhance with value of ellipticity as stated in Yang and Chen [2] study. Secondly, from the second law
generation numbers increases with an increase in the ellipticity and Ra/Ja. Thirdly, entropy generation number is nearly unaffected by surface tension forces at small ellipticity like e≤0.7, but somewhat influenced at large ellipticity for whole perimeters.
Accordingly, the effect of surface tension on the entropy generation number is significant at a larger ellipticity.
Fig. 3 shows that the total dimensionless entropy generation numbers NS is almost induced by the heat transfer irreversibility, NH i.e. the irreversibility due to heat transfer across the finite film temperature difference is much more than that due to the film flow friction. Eq. (27) reads that heat transfer generation number NH varies as the square root of Ra/Ja, while the film flow friction generation number N varies as the inverse F square root of Ra/Ja. Apparently from Fig. 3, heat transfer generation number is much more than the film flow friction generation number when Ra/Ja> 5.
Fig. 4 shows minimum entropy generation rate versus Br/Θ. From Eq. (30), one may clearly see that the optimal value of Ra/Ja varies as Br/Θ. We thus know relationship of minimum entropy generation rate and Br/Θ in Eq. (31) i.e. (NS)OPT varies as square root of Br/Θ. As indicated in Fig. 4, the effect of surface tension on minimum entropy generation can be ignored.
Whenϕ<1 in Fig. 5, the heat transfer irreversibility dominates over the flow friction irreversibility. This may account to the finite temperature difference heat transfer across the condensate film thickness. Although, there exists a gravity-induced film flow friction irreversibility within the condensate film, this viscous drag contribution to the entropy generation rate declines across the film. Therefore, irreversibility distribution ratio ϕ
decreases with Ra/Ja, but increases with Br/Θ. In general, for the case of free-convection film condensation, the entropy-generation rate due to gravity-induced film flow friction is usually small.
Considering the effect of geometrical parameters-ellipticity on the irreversibility distribution ratio, we may observe that irreversibility distribution ratio decreases with an increase in the ellipticity. This means that the heat transfer irreversibility dominated is more significant for the lower ellipticity.
Next, minimum entropy generation rate versus ellipticities in Fig. 6 demonstrates that total dimensionless entropy generation numbers increase with Br/Θ and ellipticities, similar to the mean heat transfer coefficient trend as described in Yang and Hsu [1].
4. Conclusions
This is the first approach using the entropy generation minimization to investigate free convection film-wise condensation on an elliptical cylinder. The result obtained only applies to the very slow or quiescent vapor condensed outside horizontal elliptical cylinders, and to very long elliptical cylinders, with negligible interfacial vapor shear drag. The foregoing results can be summarized as follows:
1. The entropy generation number was found to be a function of the group Rayleigh, Brinkman numbers and geometrical parameters-ellipticity.
2. The effect of group parameters on entropy generation number and the irreversibility distribution ratio was examined, respectively.
3. We can find that heat transfer generation number N and dimensionless
Ra/Ja, but film flow friction generation numberNF declines to nil as Ra/Ja increases.
4. Because of assuming y Tw T <<
Δ δ , the actual entropy generation rate is less than
that of the present calculated result.
5. The optimal design can be achieved by analyzing entropy generation in film condensation on elliptical cylinders; however, the practical ellipticity is limited to 0.9 owing to manufacturing availability.
References
1. Yang, S. A. and Hsu, C. H., Free and forced convection film condensation from a horizontal elliptical tube with a vertical plate and horizontal tube as special cases, Int. J. Heat and Fluid Flow 18, (1997), 567-574.
2. Yang, S. A. and Chen, C. K., Role of surface tension and ellipticity in laminar film condensation on horizontal elliptical tube, Int. J. Heat and Mass Transfer 36, No.12, (1993), 3135-3141.
3. Ali A. F. M., and McDonald, T. W., Laminar film condensation on horizontal elliptical cylinders: A first approximation for condensation on inclined tubes, ASHRAE Trans. Vol. 83, (1977), 242-249.
4. Karimi, A., Laminar film condensation on helical reflux condensers and related configurations, Int. J. Heat and Mass Transfer 20, (1977), 1137-1144.
5. Memory, S. B. and Rose, J. W., Free convection laminar film condensation on a horizontal tube with variable wall temperature, Int. J. Heat Mass Transfer 34, (1991), 2775-2778.
6. Bejan, A., A study of Entropy generation in fundamental convective heat transfer, Transactions of the ASME, Vol.101, (1979), 718-725.
7. Sahin, A. Z., Thermodynamics of laminar viscous flow through a duct subjected to constant heat flux, Energy, 21~12, (1996), 1179-1187.
8. Adeyinka, O. B. and Naterer, G. F., Optimization correlation for entropy production and energy availability in film condensation, Int. Comm. Heat Mass Transfer, Vol.31, No.4, (2004), 513-524.
9. Lin, W. W., Lee, D. J., and Peng, X. F., Second-law analysis of vapor condensation of FC-22 in film flows within horizontal tubes, J. Chin. Inst. Chem. Engrs., Vol. 32, (2001), 89-94.
10. Dung, S. C. and Yang, S. A., Second law based optimization of free convection film-wise condensation on a horizontal tube, Accepted in Int. Comm. Heat Mass Transfer. (to appear in 2006).
11. Li, G. C. and Yang, S. A., Thermodynamic analysis of free convection film condensation on an elliptical cylinder, Accepted in J. of The Chinese Institute Engineers. (to appear in 2006)
12. Nusselt, W., Die oberflächen Kondensation des Wasserdampfes, Zeitschrift des Vereines Deutscher Ingenieure, VDI, Vol.60, No.4, (1916), 541 -546; 569-575.
13. Bejan, A., Entropy generation minimization, CRC Press, New York, (1996), 71-90.
Fig. 1 Physical model and coordinate system for condensate film flow on an elliptical surface
0 20 40 60 80 100
0 1 2 3 4 5 6 7 8
Br/Θ =1
e=0.5 e=0.7 e=0.9
NS
Ra/Ja
Bo=100 Bo=0
Fig. 2 The variation of dimensionless entropy generation numbers N with surface S tension effect and ellipticities versus Ra/Ja
0 20 40 60 80 100 0
2 4 6 8 10 12 14 16 18 20
Br/Θ =1 Bo=100
e=0.5
e=0.7 e=0.9
Ra/Ja
NH NF NS
Fig. 3 Dimensionless entropy generation number versus Ra/Ja for every kind of ellipticities
0.01 0.1 1
0.1 1 10
e=0.5 e=0.9
(NS)OPT
Br/Θ
Bo=100 Bo=0
Fig. 4 Minimum entropy generation rate versus Br/Θ
0 20 40 60 80 100 0.0
0.2 0.4 0.6 0.8
Bo=100
e=0.5
e=0.9
φ
Ra/Ja
Br/Θ =0.5
Fig. 5 The irreversibility distribution ratio versus Ra/Ja
0.0 0.2 0.4 0.6 0.8 1.0
0 1 2 3 4 5 6 7 8
Ra/Ja=50, Bo=100
NS
e
Br/Θ =1 Br/Θ =0.7 Br/Θ =0.5
Fig. 6 Minimum entropy generation rate versus ellipticities
Nomenclature
a semi-major axis of ellipse b semi-minor axis of ellipse Bo Bond number, (ρ−ρv)ga2/σ Br Brinkman number, μu02/kΔT
Cp specific heat capacity of condensate De equivalent diameter of elliptical cylinder
e ellipticity of ellipse g acceleration due to gravity
h condensing heat transfer coefficient at angle φ
h′fg latent heat of condensation corrected for condensate subcooling Ja Jakob number, Cp(Tsat −Tw)/h'fg
k thermal conductivity of condensate
m• condensate mass flow rate per unit length of elliptical cylinder
NF film flow friction irreversibility NH heat transfer irreversibility
NS the entropy generation number Nu local Nusselt number, hDe/k
'''
So characteristic transfer rate
Ra Rayleigh number, (ρ−ρv)ρgPrDe3/μ2
Tsat saturation temperature of vapor Tw wall temperature
u velocity component in x direction v velocity component in y direction
Greek symbols
δ thickness of condensate film
δ∗ dimensionless thickness of condensate film, defined in equation (14)
θ angle measured from top of the cylinder μ absolute viscosity of condensate
ρ density of condensate
ρv density of vapor
σ surface tension coefficient in the film
φ angle between the tangent to cylinder surface and the normal to direction of gravity
φ the irreversibility distribution ratio, defined in equation (26) Θ dimensionless temperature difference, ΔT/T
Subscripts
opt optimal sat saturation
v vapor
w cylinder wall
Superscripts
* indicates dimensionless