_{1}

The paper presents a theoretical study about turbulent film boiling on a horizontal tube with external flowing liquid. The high velocity flowing liquid is determined by potential flow theory. By using Colburn analogy, the present paper successfully addresses a new model to predict the vapor-liquid interfacial shear, applies the interfacial shear into the forced balance equation and then combines the forced balance equation, the energy equation and thermal energy balance equation. At last, both the film thickness and Nusselt number can be obtained. Besides, the present analysis also includes radiation effects, temperature ratio and eddy diffusivity. Finally, a comparison between the results of the present study and those reported in previous theoretical and experimental studies is provided.

The research of film boiling on a horizontal tube was conducted by the pioneering investigator, Bromley [

Laminar film boiling on horizontal tubes has been widely discussed in published literature, and there is also some development on the researches of turbulent film boiling. For example, Sarma et al. [

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

Consider a horizontal tube immersed in flowing liquid with the high velocity _{s}. The wall temperature

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 radius 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:

Under the high velocity liquid, the equation, which describes heat transfer in the flow around the tube, can be expressed as following equation:

where C is a constant in flow configuration. The constant value C is listed in

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

The mean friction coefficient in the stream wise direction may then be written 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 friction coefficient is defined as:

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

Combining Equation (8-10), the local shear stress can be expressed as:

Incorporating the interfacial vapor shear stress

Re_{l} | C | n |
---|---|---|

0.4 - 4 | 0.989 | 0.33 |

4 - 40 | 0.911 | 0.385 |

40 - 4000 | 0.683 | 0.466 |

4000 - 40,000 | 0.193 | 0.618 |

40,000 - 400,000 | 0.0266^{*} | 0.805^{*} |

The forced balance equation Equation (12) 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 (15) are:

where the absolute viscosity equation

Besides, the thermal energy balance equation Equation (4) can be rewritten in dimensionless form as follows:

where the absolute conductivity equation

Furthermore, the dimensionless thermal energy balance equation Equation (18) 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 (13), (15)-(22) and (24)-(25) subject to the relevant boundary conditions given can be used to estimate^{++}:

1) Suitable dimensionless parameters, such as

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

3) At the bottom of the tube, ^{+} is also zero

4) Guess an initial value of

value of

5) Substitute

6) The criterion for the accuracy of

following form:

Furthermore, the error of the numerical calculation is less than

good enough. It can be expressed as the following unequal equation:

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.

9) The flow chart for calculating the vapor thickness is expressed in

^{+}. Because assuming the stagnation point is at the bottom of the tube, the figure shows a linear temperature distribution at the bottom of the tube. Besides, the present paper considers the interfacial

shear with high velocity liquid under the effects of turbulence; thus, when the angular position increases, the non-linear temperature distribution of the profile is becoming more and more significant.

Besides, the figure also compares the present results with the experimental results [

low and mid Ra number. Besides, the increase of the Ra values leads to an increase of the mean Nusselt number. The increase in the Froude number will also bring out an increase in the mean Nusselt number.

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 high velocity of liquid.

2) The increase in the Grash of number will lead to an increase in the mean Nusselt number. Besides, turbulent film boiling under the high velocity of external flowing liquid, the increase in the Froude number will bring out an increase in the mean Nusselt number.

3) It shows a good agreement between the present result which is under the condition of Fr = 0 and the previous studies with free convection.

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-PingHu, (2015) Influences of Interfacial Shear in Turbulent Film Boiling on a Horizontal Tube with External Flowing Liquid. Engineering,07,754-764. doi: 10.4236/eng.2015.711066

a acceleration due to graviton force (m/sec^{2})

D diameter of tube, 2R

Fr Froude number,

g acceleration due to gravity (m/sec^{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,

R Radius of tube (m)

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)

x peripheral coordinate (m)

y coordinate measured distance normal to tube surface (m)

y^{+} dimensionless distance,

^{2}/s)

^{3})

^{2})

l liquid

s vapor at saturation temperature

v vapor

w tube wall

x x-direction