Effect of the temperature gradient on heat transfer and friction in laminar liquid film.

Sinkunas, S. ; Kiela, A.

Nomenclature

a--thermal diffusivity, [m.sup.2]/s; [a.sub.b]--wavy thermal diffusivity, [m.sup.2]/s; [a.sub.t]-turbulent thermal diffusivity, [m.sup.2]/s; c-specific heat, J/(kg-K); d--hydraulic diameter of the film, m; g-acceleration of gravity, m/[s.sup.2]; [Nu.sub.d]-Nusselt number, [alpha]d/[lambda]; [Nu.sub.M]-modified Nusselt number, [([alpha]/[lambda])([v.sup.2]/g).sup.1/3]; Pr--Prandtl number, v/a; q--heat flux density, W/[m.sup.2]; R--cross curvature of the wetted surface (tube external radius), m;Re--Reynolds number, 4[GAMMA]/([rho]v); T--temperature, K; [v.sup.*]--dynamic velocity, [([[tau].sub.w]/g).sup.1/2]; w--film velocity, m/s; y--distance from wetted surface, m; [alpha]-heat transfer coefficient, W/([m.sup.2]-K); [GAMMA]--wetting density, kg/(m-s); [delta]--liquid film thickness, m; [phi]--dimensionless film velocity, w/[v.sup.*] ; [eta]-dimensionless distance from wetted surface, [v.sup.*]y/v; [[eta].sub.[delta]]--dimensionless film thickness, [v.sup.*][delta]/v; [[epsilon].sub.R]--relative cross curvature of the film, [delta]/R; [lambda]--thermal conductivity, W/(m-K); v--kinematic viscosity, [m.sup.2]/s; [??]--temperature field; [rho]--liquid density, kg/[m.sup.3]; [tau]--shear stress, Pa;

Subscripts: b--wavy; f--film flow; g--gas or vapour; s--film surface; t--turbulent; w--wetted surface.

1. Introduction

In recent years the research in the field of twophase flow heat transfer has been constantly increasing due to the rapid growth of the technology applications that require the transfer of high heat rates in a relatively small space and volume. Such applications vary from compact heat exchanger to cooling systems for computer. Twophase flow heat transfer was the subject of numerous researchers [1-3] for the last decade. Unsteady transfer processes taking place in two-phase flow, which is obtained by spraying water into low potential flue gas were researched numerically [4]. It was shown that the change of two-phase system can be classified according to the peculiarities of transfer processes, picking out unsteady and equilibrium modes of the state change. Thin liquid films falling under the influence of gravity are widely encountered in a variety of industrial applications that involve gas-liquid two-phase flow [5, 6]. Flow in nuclear reactor cores, steam condenser, water tube boilers and vertical tube evaporators are some of the practical examples. In order to design these systems with greater efficiency and lower cost, a basic understanding of heat and momentum transfer processes occurring in falling films is needed.

Theoretical model [7] was derived to present the temperature distribution of falling liquid films flowing over a vertical heated/cooled plate with constant temperature. The temperature gradients for different flow rates and different liquid films were also discussed. The temperature distributions for liquid films of water, ethanol aqueous solutions and glycerol aqueous solutions were experimentally investigated with a sensitive thermal imaging system. It was found that the surface temperature of a film flowing over a vertical solid plate has a characteristic relationship with the film flow distance. A lower flow rate of the film or a higher temperature of the wall generally leads to a higher surface temperature in the film inception.

The paper [8] describes an experimental investigation of the hydrodynamics of an evaporating wetting film meniscus in a capillary tube where a temperature gradient is applied along the wall. The results showed that the ability of the evaporating meniscus to wet the capillary tube is degraded by the temperature gradient along the wall.

It has been shown in study [9] that the heat transfer coefficients obtained from using the 1D transient liquid crystal scheme are higher than those obtained from employing the 3D scheme when surface heat transfer is highly nonuniform such on a hot surface subject to jet impingement cooling. This is due to the fact that 1D method does not include the lateral heat flows induced by local temperature gradients.

The paper [10] has investigated the heat and momentum transfer of a water film falling over a tilted plate with solar radiant heating and water evaporation. The results revealed that the gradients of temperature and the mass fraction of water vapor in the gas layer, and the wind velocity played a key role in the heat and momentum transfer along the gas-water interface. The water film Reynolds number related to the film thickness markedly exerted an influence on the eddy viscosity and the turbulent Prandtl number of the water film.

Study [11] reviews experimental local heat transfer data for laminar and turbulent film heat transfer of downward condensing films under the influence of interfacial waviness and shear stress effects. The results demonstrate that the dimensionless film thickness, incorporating shear stress, provides a more appropriate length scale to estimate laminar-wavy film heat transfer as well as transition to turbulence.

Local reflux condensation heat transfer coefficients have been determined inside a vertical tube within water, ethanol and isopropanol as the test fluids in [12]. The heat transfer has found to be impeded by shear stress only in cases of a very thin film, i.e. in the smooth laminar range and it can well be correlated by a simple analytical model. In the laminar-wavy range, including developing turbulence, the heat transfer coefficients are found to increase with the shear stress, an effect which proved to be enhanced with rising Pr numbers.

Noninvasive measuring method based on luminescence indicators to determine the temperature distribution and the local film thickness simultaneously was developed [13]. Results are presented for the temperature distribution measurements in a laminar-wavy water film with Reynolds number of 125. The measured temperature distributions were used to calculate the local heat transfer coefficients and heat flux perpendicular to the wall for different points in the development of a solitary wave.

2. Stabilized heat transfer in a liquid film flow

For stabilized turbulent liquid film flow on a vertical surface shear stress can be expressed as

[tau] = [rho](v + [v.sub.t]) dw/dy (1)

and dimensionless form is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2)

By taking into account a variation of liquid physical properties, Eq. (2) can be rewritten as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)

Then, the dimensionless film velocity can be expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4)

In the case of liquid density variation, the following expression can be used

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5)

Then, shear stress in the turbulent film can be defined by the following expression

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6)

In practice, the operation of real film heat ex changers is based on vertical tubes. Therefore, the equation for heat flux calculation across the turbulent film evaluating the surface cross curvature is as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (7)

In the case of plane film flow ([[epsilon].sub.R] = 0), Eq. (7) can be rearranged as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (8)

Let us denote that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (9)

Then, from the Eq. (8), we can obtain the expression defining temperature field in the film

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (10)

The temperature field intensity and heat flux density in the liquid film, when [[epsilon].sub.R] = 0 and [q.sub.s] = 0 can be defined by the energy equation

c[rho]w [partial derivatives]T/[partial derivatives]x + dq/dy = 0 (11)

Twice integrating of Eq. (11) within the limits from 0 to y, [q.sub.w] to q and from 0 to [delta], [q.sub.w] to 0 respectively, allows obtaining the ratio of heat flux densities in the film: when [T.sub.w] = const

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (12)

and when [q.sub.w] = const

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (13)

In accordance with Eq. (5), shear stress on the wall can be defined by the following expression

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (14)

Then, assuming that [t.sub.w] = [[rho].sub.f][v.sup.*2], the dynamic velocity of the film can be determined as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (15)

Dividing of Eq. (10) for temperature gradient [DELTA]T = [T.sub.w]-[T.sub.f] and taking into account that [alpha] = [q.sub.w]/[DELTA]T, leads to the expression of relative temperature field in the film

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (16)

In this case, the following correlation can be obtained

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (17)

It is evident that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (18)

Then, taking into account above-mentioned ratio, we obtain the following expression from Eq. (16)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (19)

In regard to the variation of liquid physical properties, the film Reynolds number can be defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (20)

Heat transfer and shear stress calculations were performed for laminar flows of water, transformer oil, compressor oil, fuel oil and glycerine films respectively. The method of gradual approximation was applied for the calculations. The results of calculations are presented in Figs. 1 and 2.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

As we can see from Figs. 1 and 2, regardless of very different temperature dependent physical properties of used liquids and different boundary conditions on the wall, the calculations can be generalized in a good agreement with the following correlations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (21)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (22)

where index 0 means heat transfer and shear stress under steady physical properties.

It evidently is seen from Figs. 1 and 2, that in the case of film heating m [congruent to] 0.32 and n [congruent to] 0.28 is respectively, but m [congruent to] 0.25 and n [congruent to] 0.23 is respectively, when the film is cooling.

Since the exponent n little depends on the heat flux direction, with sufficient accuracy it can be taken equal 0.25.

Therefore, in accordance with Eq. (21), we can obtain the equation for the dimensionless film thickness (friction) calculation in laminar plane flow

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (22)

In the case of temperature variation, liquid density varies very little. Then, in accordance with Eq. (15) the dynamic velocity of the film can be defined by the following equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (23)

and the dimensionless form respectively

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (24)

Consequently, the following correlation can be obtained

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (25)

Considering that, [Nu.sub.M] = 2.07[Re.sup.-1/3], [Nu.sub.d] = 7.5 for boundary condition [T.sub.w] = const and [Nu.sub.M] = 2.27[Re.sup.-1/3], [Nu.sub.d] = 8.24 for boundarycondition [q.sub.w] = const , we can obtain the following correlations for heat transfer calculations in laminar film: when [T.sub.w] = const

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (26)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (27)

and when [q.sub.w] = const

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (28)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (29)

For both cases, hydraulic diameter of the film in the Nud numbers can be determined like as for the isothermal film

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (29)

3. Conclusions

1. The main difficulties for the wavy and turbulent film flow are determining the thermal diffusivity coefficient a, the turbulent thermal diffusivity coefficient [a.sub.t] and the turbulent kinematic viscosity [v.sub.f] under varying physical properties of the film. Therefore, heat transfer and friction calculations were carried out for laminar flows of water, transformer oil, compressor oil, fuel oil and glycerine films respectively.

2. The dependencies of stabilized heat transfer and friction on temperature gradient in the case of laminar film flow of various liquids with respect to variability of liquid physical properties were estimated analytically.

3. In the case of nonisothermality, transformation of the film thickness first of all is related to the variation of liquid viscosity. However, viscosity variation depends on the film temperature field, which is determined by the liquid thermal properties. Therefore, the influence of nonisothermality on the film thickness is more reasonable to evaluate using the ratio [Pr.sub.f]/[Pr.sub.w].

Received December 01, 2008

Accepted January 15, 2009

References

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[13.] Schagen, A., Modigell, M., Dietze, G., Kneer, R. Simultaneous measurement of local film thickness and temperature distribution in wavy liquid films using a luminescence technique.-Int. J. of Heat and Mass Transfer, 2006, v.49, p.5049-5061.

S. Sinkunas *, A. Kiela **

* Kaunas University of Technology, Donelaicio 20, 44239 Kaunas, Lithuania, E-mail: [email protected]

** Kaunas College, Pramones 22, 50387Kaunas, Lithuania, E-mail: [email protected]