_{1}

Film boiling on a horizontal elliptical tube immersed in external flowing nitrogen liquid is investigated in the present paper. The isothermal wall temperature is high enough to induce turbulent film boiling, and then a continuous vapor film runs upward over the surface. The high velocity of the flowing saturated liquid at the boundary layer is determined by potential flow theory. In addition, the present paper addresses a new model to predict the vapor-liquid interfacial shear on an elliptical tube under forced convection turbulent film boiling. In the results, film thickness and Nusselt number can be obtained under different eccentricity and Froude number. And a comparison between the results of the present study and those reported in previous experimental studies is provided. The results show that there is a good agreement between the present paper and the experimental data.

The pioneering investigator, Bromley [

Laminar film boiling had been widely discussed in published literature, and so has turbulent film boiling. For example, Sarma et al. [_{2} on an ellipsoid. However, the study just researched into a simple theoretical model for turbulent film boiling heat transfer on an ellipsoid under a quiescent liquid. Furthermore, Hu [

Even though there were many researches about laminar film boiling and turbulent film boiling, there was little publication about the turbulent film boiling on a horizontal elliptical tube which his high velocity liquid was flowing outside. Predicting interfacial shear in a turbulent film boiling system under high velocity liquid was not easy. However, the present paper successfully predicted the vapor-liquid interfacial shear by using Colburn analogy. The present study applied the interfacial shear into the forced balance equation, and then combined the forced balance equation with the energy equation and thermal energy balance equation. At last, both the film thickness and Nusselt number were obtained. Then, the present analysis also included eddy diffusivity, radiation effects and temperature ratio. Finally, a comparison between the results of the present study and those reported in previous experimental studies was provided. It was found that a good agreement exists between the two sets of results.

Consider a horizontal elliptical tube immersed in an up flowing LN2 of the high velocity _{s}. The wall temperature

For thin film flow of turbulent film boiling under the forced convection, the viscosity component and the buoyancy effect are assumed more significant than the inertia force. Then the force balance equation for the vapor film can be expressed as:

It is assumed the thickness of vapor film is much thinner than the diameter of the tube (

The boundary conditions of energy equation under isothermal condition are as follows:

For a pure substance, the thermal energy balance equation of the vapor film can be expressed as:

The differential arc length for ellipse can be expressed with the following equation:

where

where

Substitute the

In the turbulent region the semi-empirical equation which describes heat transfer in the flow parallel to a moderately curved surface may also be used to describe the heat transfer in the flow parallel to an elliptical surface. Jakob [

where C is a constant in flow configuration, C = 0.034.

According to Colburn analogy, the friction factor can be written as the following equation:

The mean friction coefficient in the streamwise direction may then be calculated as:

Furthermore, the local friction can be obtained as:

The turbulent boundary layer exerts a friction force on the liquid-vapor boundary. The shear stress is estimated by considering the external flowing liquid across the surface of the tube when there is no vapor film on the surface. The local shear stress is defined as:

According to potential flow theory, when the uniform liquid flow of velocity

Combining Equations (12)-(14), the local shear stress can be expressed as:

Incorporating the interfacial vapor shear stress

The forced balance equation Equation (17) yields the following dimensionless equation:

It’s further assuming the pressure across the boundary layer is constant and the density variation across the boundary layer is given by the following equation:

The energy equation Equation (2) yields the following dimensionless energy equation:

The dimensionless boundary conditions of Equation (19) are:

where the absolute viscosity equation

Besides, the thermal energy balance equation Equation (8) can be rewritten in dimensionless form as follows:

where the absolute conductivity equation

Furthermore, the dimensionless thermal energy balance equation Equation (22) requires the velocity profile

The boundary condition is:

The eddy diffusivity distribution presented by Kato et al. [

The heat transfer of turbulent film boiling can be given by the following equation:

Obviously, the local Nusselt number can be expressed as:

The mean Nusselt number for the entire surface of the tube can be written as:

The dimensionless governing Equations (17), (22)-(26) and (28), (29) subject to the relevant boundary conditions given can be used to estimate^{++}:

1) Suitable dimensionless parameters, such as e,

2) The boundary conditions of velocity and temperature are as follows:

3) Since the ^{+} is also zero (

4) Guess an initial value of

5) Substitute

6) The criterion for the accuracy of

If the calculation is a convergence, process the film thickness of next angular position. If the calculation is not a convergence, guess a new thickness and repeat processes (4)-(6).

7) The process above is repeated at the next node position, i.e.

8) The local Nusselt number and mean Nusselt number are then calculated.

^{+}, u^{+} will increase to a maximum value, and then it will slightly decrease. The results also show that the dimensionless velocity increases with an increasing angular position on the elliptical surface.

To validate the present model, a comparison is made between this work and previous studies for different

cases. For this reason, a modified Rayleigh number is introduced as

fects of Ra on the mean Nusselt numbers on an elliptical tube under e = 0 with special case of a tube subject to turbulent film boiling. It shows that the mean Nusselt number of the present study has a good agreement with previous experimental data [_{m} increases with Ra at fixed radiation parameter. The increase in the radiation parameter, conceivably, will bring out an increase in the mean Nusselt number at a given Ra.

The following conclusions can be drawn from the results of the present theoretical study:

1) With the help of Colburn analogy, the present research successfully predicts the shear stress of the vapor- liquid interface in a film boiling system under liquid of high velocity on an elliptical tube.

2) The increase in the eccentricity parameter of the elliptical tube will lead to a decrease in the mean Nusselt number. Besides, turbulent film boiling under the external flowing liquid with high velocity, the increase in both the Froude number and Grashof number will bring out an increase in the mean Nusselt number.

3) The present paper includes interfacial shear, radiation effects, temperature ratio, eddy diffusivity and thermal properties of temperature dependent. It can predict the forced convection turbulent film boiling more exactly. Besides, under the condition of free convection film boiling, it shows a good agreement between the present result and the previous experimental studies.

The authors gratefully acknowledge the support provided to this projects by the Ministry of Science and Technology of Taiwan under Contract Number MOST 104-2221-E-019-052.

Hai-Ping Hu, (2016) Heat Transfer Efficiency of Turbulent Film Boiling on a Horizontal Elliptical Tube with External Flowing Liquid. World Journal of Engineering and Technology,04,206-219. doi: 10.4236/wjet.2016.42020

a, b semimajor, semiminior axis of ellipse

D_{e} equivalent circular diameter of elliptical tube (m)

e eccentricity of ellipse,

Fr Froude number,

g acceleration due to gravity (m/s^{2})^{ }

h heat transfer coefficient, W/(m^{2}∙K)

h_{fg} latent heat (J/kg)

k thermal conductivity (W/m∙K)

Nu_{m} mean Nusselt number

NR radiation parameter,

Pr Prandtl number,

Ra modified Rayleigh number,

S heat capacity parameter,

St Stanton number,

T temperature (K)

T^{+} dimensionless temperature,

u vapor velocity in x-direction (m/s)

v velocity normal to the direction of flow (m/s)

^{2})

x peripheral coordinate (m)

y coordinate measured distance normal to tube surface (m)

y^{+} dimensionless distance,

^{2}/s)

^{3})

^{2})

^{2}∙K^{4}

l liquid

s vapor at saturation temperature

v vapor

w tube wall

x x-direction