國際學術會議文章
THERMODYNAMIC SECOND LAW ANALYSIS OF FORCED-CONVECTION FILM CONDENSATION ON A HORIZONTAL TUBE WITH UNIFORM WALL HEAT FLUX
Yung-Sian Chen1, Yi-Lung Kang1, Ren-Yi Hung1 and Sheng-An Yang1
1Department of Mold and Die
National Kaohsiung University of Applied Sciences, Taiwan, R.O.C.
ABSTRACT
This paper aims to perform the thermodynamic second law analysis of forced-convection film condensation on a horizontal tube with uniform wall heat flux. We adopted Shekriladze and Gomelauri approach to deal with the interfacial vapor shear stress, and Nusselt film condensation model to investigate laminar film condensation heat transfer. Further, owing to the effect of pressure gradient, the separation angles of the condensate film layerθsare obtained for various dimensionless pressure gradient parametersP∗ and their corresponding dimensionless Grashof’s parameters, Gr∗ via the fourth-order Runge-Kutta numerical method.
Next, based on Bejan’s entropy generation minimization technique, both the local and mean entropy generation rate are also determined to understand which irreversibility factor, like film flow friction, or finite-temperature difference heat transfer dominates the entropy generation rate in terms of pressure gradient parameter, Grashof’s parameter, Brinkman number, and Reynolds number. Note the forced-convection film dominated condensation when modified Grashof’s parameter is smaller than 1, the case without taking account of pressure gradient, i.e. =0P∗ applies to uniform surface heat flux condition.
Finally, the entropy rate and irreversibility factor increases as the Brinkman number increases. Besides, the dimensionless entropy generation numbers are found to increase with Reynolds number and the dimensionless heat transfer coefficient too. The analysis of entropy generation for the case of uniform wall heat flux can also be a good future reference for designing heat pipe.
INTRODUCTION
Numerous thermal engineering systems utilize the circular tubes with constant flux widely in design of condensers for power plants, air-conditioning equipments, chemical industrial process equipments etc. For those in phase-change heat transfer associated with film condensation problems, the entropy generation destroys available energy and also reduces the efficiency, therefore, using the second law analysis to estimates the minimum entropy generation, a optimum energy saved can be achieved is concerned.
Nusselt (1916), the pioneering investigator of laminar film condensation, he obtained a dimensionless parameter of condensation heat transfer, Nusselt number Nu, to evaluate the heat transfer at a boundary within convective and conductive. Bromely (1952) gave an first-order correction closed form considering the effect of the heat capacity of the condensate. Rohsenow (1956) neglected condensate inertia and unrestricted the energy convection in condensate film, accounted for the resulting nonlinear temperature distribution across the condensate by correcting the latent heat of condensation. Sparrow and Gregg (1959) applied
boundary layer equations to the condensate layer both including convective energy and condensate inertia, improved correlation of film condensation data based on or more rigorous application of similarity parameters. Shekriladze and Gomelauri (1966) utilized the asymptotic shear stress, including gravity and velocity of vapor, but neglecting the pressure gradient term around the circular tube, the condensate film boundary layer separation is assumed to be absent in their investigation.
Churchill (1966) provided a closed form solution to account for the secondary effect. Although numerous relative follow researches are presented, the investigation of entropy generation due to convection heat transfer few addressed the film condensation.
Bejan (1979) first proposed the second law analysis of thermodynamics via the minimization of entropy generation for the single phase convection heat transfer. The function of the entropy generation mechanism of film condensation on tube remains an important issue. Further research on the aspects of the entropy generation mechanism, particularly its influence on the heat transfer characteristics, is required. Lin et al. (2001) first performed the second-law analysis on saturated vapor flowing through and condensed inside horizontal cooling tubes.
They found an optimum Reynolds number exists at which the entropy generates at a minimum rate and there was an optimal cooling temperature that generated a minimum of entropy for a given duty.
Saouli and Aiboud-Saouli (2004) showed second law analysis of laminar liquid falling film along an inclined heated plate and found that fluid friction irreversibility dominates over heat transfer irreversibility. Jani (2004) provide optimization of falling film LiBr solution on a horizontal single tube based on minimization of entropy generation that irreversibility of non-isothermal heat transfer dominants in comparison with the fluid flow friction and mass transfer. Li and Yang (2006) conducted the thermodynamic analysis of saturated vapor flowing slowly onto and condensed on an elliptical cylinder, it is the first approach to investigate how the geometric parameter elliptic and surface tension affect local entropy-generation rate during film-wise condensation heat transfer process.
Dung and Yang (2006) investigated the entropy generation of free convection film condensation on horizontal tube, and found that heat transfer coefficients and the entropy generation number have direct relationships that generation number can be reduced by decreasingBr/φ.
The major purpose of this study is to analyze the mechanism of entropy generation onto a horizontal tube encountered in forced convection film condensation with constant heat flux. We derive expressions for entropy
2 generation, which account for the combined action of specified irreversibility.
ANALYSIS
As shown in Figure 1, a horizontal tube immersed in downward flowing vapor along gravity direction that is at uniform flow velocity U∞ and saturated temperature Tsat.. The temperature varies with radial y-direction caused by the constant heat flux kept at the saturation temperature. Thus, a condensation occurs on the wall and a continuous liquid film runs downward over the tube.
Figure 1. Physical model and coordinate system for condensate film flow on a tube.
Based on the model, equations of the conservation of mass, momentum, and energy for steady laminar flow read as follows:
0 condensation corrected for condensate sub-cooling suggested by Rohsenow (1956).
velocity boundary layer
Considering the viscosity in liquid-vapor boundary, the boundary conditions of velocity and temperature are conservation. Shekriladze and Gomelauri suggested it has a well simulation as
m ue
τδ = ′′
(6)
Since the condensation film thickness is much smaller than the curvature of condensation surface, one can express the surface tension and pressure gradient into the Bernoulli’s equation as
e e
dp du
dx = −ρνu dx (7)
Next, by applying potential flow theory to a uniform vapor flow pass over the tube with velocity U∞, the velocity at the edge of boundary describes as
2 sin
ue= U∞ θ (8)
Inserting Eqs. (8) and dx=( / 2)D dθ into Eq. (7), and then substituting Eq. (7) into Eq. (2) yields
( )
Taking the first integration of the Eq. (9) with boundary conditions (4) and (5), and inserting Eqs. (3) and (8) into Eq. (6) to derive the 1st order differential equation, again we secondly integrate to obtain the velocity in condensate film of forced convection
( )
thickness layer distribution
Inserting Eqs. (3) and (10) into (1), one has the condensate film thickness and separation angle over the tube wall as follows
2 8 3
3
Next, considering the following thickness boundary condition
0
dδ∗ dθ= at θ = , 0 (13)
then substituting Eq. (13) into Eq. (11) yields
2 3 3
0 0 0
3 6− δ ∗ −8πGr∗δ ∗ −64π δP∗ ∗ = 0 (14)
From the above equation, as given dimensionless parameters Gr∗and P∗, the initial values of δ0*can be solved. Thus, Eq. (11) can apply fourth-order Runge-Kutta method to obtain the film thickness δ∗ versus separation angle θs. Meanwhile, the local dimensionless Nusselt number can be shown as
dimensionless local entropy generate rate
Based on the second law approach in Bejan (1979), the local entropy generation rate for convection heat transfer can be stated as
2 2
Integrating the energy equation (3) with the temperature boundary conditions of Eqs. (4) and (5) yields the temperature profile of condensate film:
.
For the velocity and temperature gradient along y-direction, Eqs. (10) and (18) represent as follows:
( )
temperature difference.
Substituting Eqs. (19) and (20) into Eq. (17) and assuming η=y δ yields
Defining the characteristic velocity parameter of tube wall
( ) 2
0 1
u =2 ρ ρ− ν gD µ (22)
and substituting Eqs. (12) and (22) into (21) gives local entropy generation rate
2 1/2 1/2
To discuss conveniently, we may define the dimensionless local entropy generate rate as
0
Finally, substituting Eq. (23) into (24), and then equating from dimensionless temperature difference, Brinkman number and uniform heat flux yields the dimensionless local entropy generation as follow:
2
4
it illustrates the dimensionless local entropy generation induced by the finite temperature difference, while the another term indicates the fluid friction irreversibility applying onto the tube wall,describing as follow:
{
To understand which of the condensate flow friction irreversibility N′′F or heat transfer irreversibility N′′ H dominates, one may define the local irreversibility distribution ratio as follows:
local F
total entropy generate rate
Integrating Eq. (26) with respect to η from 0 to 1,
To estimate the dimensionless total entropy generation of condensate film, one may integrate Eq.
(31) over the entire streamline length from top of tube wall till the separation angleθs, as follows:
2 2 2
Similarly, for evaluating which of the irreversibility dominates, a criterion known as the total irreversibility distribution ratio, also defines as follows:
total F H
N
Ψ =N (33)