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A novel passive containment cooling system (PCCS) is proposed to be installed in the advanced nuclear reactor to cope with LOCA and MSLB accident. The internal heat exchanger is located inside the containment and the condensation heat transfer characteristic outside the tube determines the performance of the system. An improved model based on heat and mass transfer is presented to predict the heat and mass transfer accompanying with condensation. Different with Dehibi’s model, the liquid film conduction is considered and the interface temperature is solved by iteration. The results show the effect of different parameter on heat transfer coefficient. And the correlation can well predict the experiment data.

A novel passive containment cooling system (PCCS) is investigated to extend the passive cooling capacity called Time-Unlimited passive containment cooling system (TUPAC). In this system, the internal heat exchangers are adopted inside the containment. During a LOCA or a MSLB, non-condensable gases (hydrogen, air, etc.) mixed with the steam will be injected into the containment, which severely decrease the performance and efficiency of the steam condensation heat transfer. So, it is important to investigate the steam condensation with non-condensable gas on the vertical tube.

Nusselt [

The investigation on the steam condensation of a flat plate or a horizontal tube was conducted by many researchers. And they proposed correlations based on the empirical fits of the data. The correlation would be invalid out of the experimental ranges of conditions. And the correlations fitted to plate may not suitable for tubes. Although Dehbi [

In this paper, experiments were conducted about wall subcooling, total gases pressure and air mass friction over a incline tube under “time free system” condition to investigate the influences of various parameters (subcooling/total gases pressure/air mass friction) on the steam condensation heat transfer process.

In this investigation, a correlation based on heat and mass transfer analogy is presented. The film thickness (δ_{f}) and the h_{f} are considered following Kim’s method [

The non-condensable gas will diffuse to the condensation interface and accumulated on the interface of the gas-liquid, which result in a noncondensable gas boundary [

Colburn [_{cd}), convection heat transfer to the interface (h_{cv}), and heat conduction through the liquid film (hf).

According to Dehib’s model, the total heat transfer coefficient is defined as

h t = q ″ / ( T b − T w ) (1)

where q ″ includes the latent heat and the sensible heat, T_{b} is the bulk temperature, and T_{w} is the wall temperature. The total HTC can also be written as

h t = 1 1 h f + 1 h c d + h c v (2)

According to the energy balance equation:

h f ( T i − T w ) = ( h c d + h c v ) ( T b − T i ) (3)

The condensation rate can be expressed by the mass transfer process, e.g. rate of diffusion of vapor towards the cold tube. Hence, the mass flux of the steam and non-condensable gas in the boundary layer can be expressed as follows:

m n c ' ' = ρ ω n c ν − ρ D ∂ ω n c ∂ n (4)

m s ' ' = ρ ω s ν − ρ D ∂ ω s ∂ n (5)

where D represents the mass diffusion coefficient, w the mass fraction, v the mixture velocity, and n the normal direction to the wall (liquid film). The total mass flux at the liquid-vapor interface can be written as:

m i ' ' = m n c , i ' ' + m s , i ' ' = ( ρ ν ) i (6)

The vapor condensing mass flux at the wall is:

m s , i ' ' = ( ρ ν ) i = − ρ D ∂ ω s , i ∂ n 1 − ω s , i (7)

The vapor condensing mass flux can also be expressed as:

m s , i ' ' = − ρ D ∂ ω s , i ∂ n 1 − ω s , i = h m ( ω s , b − ω s , i ) (8)

By introducing the S h o number,

S h o = h m L D (9)

The mass flux of vapor condensing can be further expressed as:

m s , i ' ' = − ρ D ∂ ω s , i ∂ n 1 − ω s , i = h m ω s , b − ω s , i 1 − ω s , i = D S h 0 L ω s , b − ω s , i 1 − ω s , i (10)

The interface parameter is used in Equation (10) instead of wall parameter. By using the heat and mass transfer analogy (HMTA) and assuming the gas mixture flow is turbulent and naturally driven, one can express the Sherwood number according to a form analogous to the McAdams correlation for free convective flows:

S h = 0.13 ( G r • S c ) 1 / 3 (11)

The Sc number and Gr number can be expressed as follow,

G r = ρ g L 3 ( T b − T i ) / μ 2 (12)

S c = μ / ρ D (13)

As the sensible convection heat transfer is ignored, the condensation heat can be written as:

m s , i ' ' h f g = h t ( T b − T i ) = ( T b − T i ) 1 h c d + 1 h f (14)

Then, we can obtain the total HTC as:

h t = 0.13 ( G r • Pr ) 1 / 3 D • h f g L ω s , b − ω s , i 1 − ω s , i 1 T b − T i (15)

The enhancement in the mass transfer rate, which is due to the suction effect at the gas-liquid interface, can be determined by a free parameter f. And the parameter f depends solely on the Bird factor Θ. The f value will be estimated from the experimental data.

Finally, the total HTC can be expressed as:

h c d = ϕ 0.13 ( G r • Pr ) 1 / 3 D • h f g L ω s , b − ω s , i 1 − ω s , i 1 T b − T i (16)

The conduction through the liquid condensate (h_{f}) is calculated by Kim’s model.

A physical model of the film condensation outside a vertical tube is described.

The thickness of the liquid film is:

δ ( x ) = δ P 1 + 1 6 δ P R (17)

where the δ P is the thickness of film condensed on the cold plate.

Then, the local condensation HTC can be expressed as:

h x ≅ ( k δ P ) 1 1 − 2 3 ( δ P R ) = h P 1 − 2 3 ( δ P R ) (18)

And the average condensation HTC of the tube is:

h ¯ f ≅ h ¯ P , L 1 − 1 2 ( δ P , L R ) (19)

where the h ¯ P , L is the average condensation HTC on the vertical plate, which is deduced by Nusselt.

Then, the theoretical expressions are provided for the average heat transfer coefficient of vertical cylinder [

Generally, the liquid film thickness is smaller than 1 mm. So, the tube radius is much larger than that of film thickness. According to the Equation (19), the effect of the tube radius on the variation of the condensation heat transfer coefficient can be ignored, as long as the radius of condenser tubes is much larger than the liquid film thickness.

In order to predict the heat transfer coefficient, the following parameters are needed: the containment pressure, containment temperature, the wall temperature. Different from the Dehbi’s model, the interface temperature should be calculated in the process.

1) Guess an interface temperature Ti.

2) Obtain the corresponding interface noncondensable mass fraction Wi using the Gibbs-Dalton ideal gas mixture relation and the assumption that steam is at saturation conditions.

3) Calculate the vapor condensation HTC at the interface (h_{cd}) by (16).

4) Calculate the liquid condensate conduction resistance at the interface (h_{f}) by (19).

5) Compute the heat flux across the condensate film by (3).

6) Judge the energy balance of the heat flux.

7) If the Ti iteration has not converged then go back to step 1.

8) Stop the calculation if the heat flux is balance between both sides of interface.

As less total HTC data are obtained by the present experiment, Dehbi’s data are used. The model described above was used for parametric studies to determine the impact of conduction through the liquid condensate.

The effect of film conductive thermal resistance is revealed in

shown that the impact of conduction through the liquid condensate is small when the steam mass fraction is less than 70%. With the increasing of the steam mass fraction, the film conduction is more and more important. If it is not considered, the HTC deviation will be larger than 20%, which is not acceptable. So, the conduction through the liquid condensate (h_{f}) is highly recommended to be included in the model especially when the Ws larger than 70%.

The wall subcooling temperature is defined as the difference of wall and bulk temperature. And the interface subcooling temperature is defined as the difference of wall and interface temperature. As shown in

The effect of non-condensable gas mass fraction and subcooling on HTC is shown in

the condensation film is increased, which increase the film conductivity resistance. Then the HTC decreases with the increase of wall subcooling. When the wall subcooling temperature increase from 12˚C to 24.5˚C，the HTC increases by 34%. And with the increasing of non-condensable gas mass fraction, the diffusion layer is accumulated near the tube. The mass transfer process is further inhibited, which results in the decrease of HTC. When the non-condensable gas mass fraction increase from 68% to 78% in 16˚C subcooling，the HTC decrease by 81%.

h t = 0.296 D 2 / 3 ( ρ i + ρ b ) ( t b − t i μ ) 1 / 3 ln ( ω s , b − ω s , i 1 − ω s , i ) h f g t b − t i (20)

The unknown parameter in Equation (14), which is assumed to include the suction effect, is correlated with the data obtained by recent experiment. And the expression is provided in Equation (20). As is shown in

A correlation based upon heat and mass transfer analogy method was improved by this investigation. The interface temperature is introduced and determined by iteration. The experiment is conducted and the data are used to fit for the enhanced factor. The influence of the steam mass fraction, total pressure and wall subcooling, on heat transfer coefficient are analyzed. The main conclusions are as follow:

1) The diffusion layer is modelled to simulate the heat transfer process of steam condensation with non-condensable gas on the vertical tube. Different with Dehibi’s model, the liquid film conduction resistance is introduced. And the interface temperature is solved by iterative. The heat balance across boundary layers between the liquid film and the vapor-gas mixture calculations can be reached by this process. Then the heat transfer correlation is improved.

2) The effect of different parameter on HTC is analyzed. With the increase of wall subcooling and mass fraction of non-condensable gas, the HTC decreases.

3) We now conduct the experiment study on the heat transfer in presence of non-condensable gas with a vertical tube of 2 m in length and 0.38m in diameter. In the experiment, the variation of condensation heat transfer coefficients with pressure, mass fraction of air, wall subcooling are investigated. The enhanced factor φ due to suction effect is correlated.

National Science and Technology Major Project of the Ministry of Science and Technology of China (No. 2015ZX06004004).

Zhang, S.J., Cheng, X. and Shen, F. (2018) Condensation Heat Transfer with Non-Condensable Gas on a Vertical Tube. Energy and Power Engineering, 10, 25-34. https://doi.org/10.4236/epe.2018.104B004