4. Distribution of axial velocity, temperature and mass fraction profiles along the vertical tube

to describe the heat and mass transfer. Note that NI is the total grid points in the axial direction, NJ is the total grid points in the radial direction at the gas region, and NL is the total

Figure 3. Comparison with numerical study of Hassaninejadfarahani et al. [17] for (a) dimensionless mass fraction at the

grid points in the radial direction at the liquid region.

tube exit, and (b) dimensionless mixture temperature.

64 Desalination and Water Treatment

This chapter investigates the process of the liquid film condensation from the water vapour and non-condensable gas mixtures inside a vertical tube. The results of this study have been obtained for the case of inlet gas temperature T0=70C, inlet pressure P0=1 atm and inlet Reynolds number is fixed at Re0 = 2000. The range of each parameter for this study is listed in Table 1. At first, the air is used as non-condensable gas.

Figures 5–7 Illustrate the profiles of velocity, temperature and the mass fraction of water vapour at different axial locations of the tube. From the distribution of velocity in Figure 5, it is observed that the variation of the velocity in the gas mixture is higher than that in the liquid region. Moreover, as the gas flow progresses along the tube, the velocity in the mixture decreases, while the velocity in the liquid film slightly rises. This behaviour is due to the mass transfer from the mixture to the liquid film. In fact, when the gas mixture loses the mass, it loses velocity too, however, the liquid film gaining mass as well as acceleration. Figure 6 Presents the evolution of the temperature profiles in both mixture and liquid phases at different tube sections. It can be seen that in the liquid phase, the temperature profiles are close to the temperature of the wall and nearly linear. This indicates that the interface temperature


Figure 6. Distributions of axial temperature profile in both the liquid and vapour phases at different tube sections.

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Figure 7. Distributions of axial vapour mass fraction profile in gas phases at different tube sections.

Figure 5. Distributions of axial velocity profile in both the liquid and vapour phases at different tube sections.

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obtained for the case of inlet gas temperature T0=70C, inlet pressure P0=1 atm and inlet Reynolds number is fixed at Re0 = 2000. The range of each parameter for this study is listed in

Figures 5–7 Illustrate the profiles of velocity, temperature and the mass fraction of water vapour at different axial locations of the tube. From the distribution of velocity in Figure 5, it is observed that the variation of the velocity in the gas mixture is higher than that in the liquid region. Moreover, as the gas flow progresses along the tube, the velocity in the mixture decreases, while the velocity in the liquid film slightly rises. This behaviour is due to the mass transfer from the mixture to the liquid film. In fact, when the gas mixture loses the mass, it loses velocity too, however, the liquid film gaining mass as well as acceleration. Figure 6 Presents the evolution of the temperature profiles in both mixture and liquid phases at different tube sections. It can be seen that in the liquid phase, the temperature profiles are close to the temperature of the wall and nearly linear. This indicates that the interface temperature

Tube length (L (m)) 3.0, 4.5, 6.0 Tube radius (R(m)) 0.008, 0.01, 0.012 Inlet vapour mass fraction (w0) 0.05, 0.125, 0.2

Figure 5. Distributions of axial velocity profile in both the liquid and vapour phases at different tube sections.

C)) 5, 20, 35

Non-condensable gas Oxygen, air, nitrogen

Table 1. At first, the air is used as non-condensable gas.

wall temperature (TW(

66 Desalination and Water Treatment

Table 1. The ranges of the physical parameters.

Figure 6. Distributions of axial temperature profile in both the liquid and vapour phases at different tube sections.

Figure 7. Distributions of axial vapour mass fraction profile in gas phases at different tube sections.

decreases from the inlet to the tube exit, resulting in the reduction of the heat transfer across the condensate film. Also, the slopes of the temperature of the mixture are decreasing along the tube owing to the absorbed energy transferred from the gas flow to the liquid film. The distributions of the mass fraction of water vapour in the gas region are illustrated in Figure 7. It is interesting to observe that the vapour mass fraction w0 decreases from the entrance to the tube exit, which implies that the condensation rate is decreasing along the tube. Consequently, w0 is reduced from the Centre line (η = 1) to the liquid-vapour interface (η = 0).

#### 5. Effect of the tube geometry (length L and radius R)

In the desalination units, the tube geometry of which the water vapour condenses (whether it is the length or the radius) contributes positively to the improvement of the condensation process if they are well dimensioned.

To reveal their impacts, we first examined the impact of the length of the tube on the liquid film thickness and the condensing mass flux at the interface along the tube. Figure 8 shows the influence of the tube length on the thickness of the film and condensing mass flux at the interface. It is noted that δ<sup>x</sup> increases with increasing the tube length L. The results also indicate that, for a high tube length, the mechanism of the condensation is important when the distance is less than (X = 0.8), especially for L = 4.5 m and 6.0 m, because for a fixed Reynolds gas number (the inlet velocity is fixed too), the condensed vapour decreases with the increase of tube length. This means that near to the tube exit, the condensation process becomes almost unimportant. It is

also observed that the condensing mass flux reduces from the inlet to the outlet, particularly for the two large values of L approaching zero, because the total amount of the water vapour is

Figure 9. Effect of the tube radius on the variation of the liquid film thickness and the condensing mass flux at the

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The effect of changing the tube radius R is shown in Figure 9. From the curves in this figure, it is interesting to observe that for a fixed value of T0, w0 and Re0, a smaller tube radius corresponds to a thicker liquid film thickness. This trend is true for every position X from the inlet to the outlet of the tube. These results are directly related to the velocity of the gas mixture at the inlet. Obviously, for a fixed Reynolds number, δ<sup>x</sup> increases according to the velocity increases with a weak tube radius. This implies that a higher inlet velocity tends to move the air away from the interface and thus maintains its lower fraction, leading to the increase of the heat transfer coefficient, which improves the condensation process. It is also found that the condensing mass flux increases with the tube radius only near the inlet when X < 4 due to a high interfacial shear

In the majority of thermal desalination units, the water vapour that does not condense at the first effect, with all the non-condensable gases content, is transferred to the second effect, and this produces gas accumulation up to inadmissible concentrations. These gases cause a reduc-

transferred to liquid film and remains only the air at the vapour-liquid interface.

stress. This tendency is reversed as the gas mixture progresses along the tube.

6. Effect of water vapour mass fraction w0

tion in the performance of the system.

interface.

Figure 8. Effect of the tube length on the variation of the liquid film thickness and the condensing mass flux at the interface along the tube.

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Figure 9. Effect of the tube radius on the variation of the liquid film thickness and the condensing mass flux at the interface.

also observed that the condensing mass flux reduces from the inlet to the outlet, particularly for the two large values of L approaching zero, because the total amount of the water vapour is transferred to liquid film and remains only the air at the vapour-liquid interface.

The effect of changing the tube radius R is shown in Figure 9. From the curves in this figure, it is interesting to observe that for a fixed value of T0, w0 and Re0, a smaller tube radius corresponds to a thicker liquid film thickness. This trend is true for every position X from the inlet to the outlet of the tube. These results are directly related to the velocity of the gas mixture at the inlet. Obviously, for a fixed Reynolds number, δ<sup>x</sup> increases according to the velocity increases with a weak tube radius. This implies that a higher inlet velocity tends to move the air away from the interface and thus maintains its lower fraction, leading to the increase of the heat transfer coefficient, which improves the condensation process. It is also found that the condensing mass flux increases with the tube radius only near the inlet when X < 4 due to a high interfacial shear stress. This tendency is reversed as the gas mixture progresses along the tube.
