2. Mathematical model

compared to thermal desalination technologies. Consequently, the necessity to improve the thermal processes, which are based on the phase change phenomenon of evaporation and condensation, continues to receive a high interest. Condensation on the cooling surfaces is a phenomenon of major significance in the chemical industries, refrigeration, heat exchangers

The mechanism of condensation can be classified by various ways: geometric configurations like tube, channel, internal, external, horizontal or vertical; species of fluid such as steam, refrigerant or mixture with the presence of non-condensable gas; condensing phenomena as filmwise, dropwise or fog; and flow regime like laminar and turbulent. Since the first analysis of Nusselt [1] for film condensation on a vertical plate, a numerous number of studies have been done on improving film condensation modelling and to contribute to the comprehension of this complex phenomenon. Lebedev et al. [2] performed experimentally a combined study of heat and mass transfer from water vapour on a flat plate. They observed an enhancement of the condensation heat transfer with the increase of the inlet relative humidity. Dobran and Thorsen [3] studied the laminar filmwise condensation of a saturated vapour inside a vertical tube. They found that the mechanism of condensation is governed by ratio of vapour to liquid viscosity, Froude number to Reynolds number ratio, subcooling number and Prandtl number of liquid. Siow et al. [4, 5] presented a numerical study of the laminar film condensation with the presence of non-condensable gas in horizontal and then in vertical channels. They analysed the effect of the inlet Reynolds number, the inlet pressure and the inlet-to-wall temperature difference on the condensation mechanism. They studied also the liquid film condensation from steam-air mixtures inside a vertical channel. Results indicate that a higher concentration of non-condensable gas caused substantial reduces in the local Nusselt number, the pressure gradient and the film thickness. Belhadj et al. [6] conducted a numerical analysis to improve the condensation process of water vapour inside a vertical channel. Their results show that the phenomenon of phase change is sensitive to the inlet temperature of liquid film. For different values of the system parameters at the inlet of the tube, Dharma et al. [7] estimated from a numerical study the local and average values of Nusselt number, the pressure drop, the condensate Reynolds number and the gas-liquid interface temperature. Lee and Kim [8] carried out experimental and analytical studies to analyse the effect of the non-condensable gas (nitrogen) on the condensation of water vapour along a vertical tube with a small diameter. The experimental results demonstrate that the heat transfer coefficients become important with a high inlet vapour flow and the reduction of mass fraction of nitrogen. In addition, the authors developed a new correlation to evaluate the heat transfer coefficient regardless the diameter of the condenser tube. Nebuloni and Thome [9] developed a numerical and theoretical model to predict the laminar film condensation inside various channel shapes. They showed that the channel shape strongly affects the overall thermal performance. Chantana and Kumar [10] investigated experimentally and theoretically the heat transfer characteristics of steam-air during condensation inside a vertical tube. They observed that a higher Reynolds number and mass fraction of vapour improve the process of condensation. Dahikar et al. [11] conducted an experimental and CFD studies in the case of the film condensation with downward steam inside a vertical pipe. They found that a larger interfacial shear affects the momentum transfer because of the great velocity gradient especially at the gas-liquid interface.

and desalination units, including thermal desalination.

56 Desalination and Water Treatment

#### 2.1. Physical model and assumptions

The geometry under consideration is a vertical tube with length L and radius R (Figure 1). The tube wall is subjected to a constant temperature. A mixture of water vapour and non-

• Conservation of momentum:

• Conservation of energy:

are written as follows:

• Conservation of mass:

• Conservation of momentum:

• Conservation of energy:

• Species diffusion equation:

• At the tube inlet ð Þ x ¼ 0

∂ ∂x

> ∂ ∂x

2.3. Boundary and interfacial conditions

• At the centre line of the tube ð Þ r ¼ 0

ð Þþ rGuGuG

rGuGCp,GTG <sup>þ</sup>

> ∂ ∂x

∂ ∂x

> ∂ ∂x

ð Þþ rLuLuL

rLuLCp,LTL <sup>þ</sup>

> ∂ ∂x

1 r ∂ ∂r

ð Þþ rGuGw

1 r ∂ ∂r

ð Þ¼� rrLvLuL

1 r ∂ ∂r

ð Þþ rGuG

ð Þ¼� rrGvGuG

1 r ∂ ∂r

1 r ∂ ∂r

The governing equations are subjected to the following boundary conditions:

vG ¼ 0;

∂uG <sup>∂</sup><sup>r</sup> <sup>¼</sup> <sup>∂</sup>TG

dp dx þ 1 r ∂ ∂r rμ<sup>L</sup> ∂uL ∂r 

Computational Study of Liquid Film Condensation with the Presence of Non-Condensable Gas in a Vertical Tube

rrLvLCp,LTL <sup>¼</sup> <sup>1</sup>

Similarly, the mass conservation, momentum, energy and diffusion equations for the gas phase

1 r ∂ ∂r

> dp dx þ 1 r ∂ ∂r

rrGvGCp,GTG <sup>¼</sup> <sup>1</sup>

ð Þ¼ rrGvGw

1 r ∂ ∂r

<sup>∂</sup><sup>r</sup> <sup>¼</sup> <sup>∂</sup><sup>w</sup>

r ∂ ∂r

> rμ<sup>G</sup> ∂uG ∂r

r ∂ ∂r

uG ¼ u0; TG ¼ T0; PG ¼ P0; w<sup>G</sup> ¼ w<sup>0</sup> (8)

<sup>r</sup>rGD <sup>∂</sup><sup>w</sup> ∂r

rλ<sup>G</sup> ∂TG ∂r

rλ<sup>L</sup> ∂TL ∂r

ð Þ¼ rrGvG 0 (4)

<sup>þ</sup> <sup>r</sup>Lg (2)

59

<sup>þ</sup> <sup>r</sup>Gg (5)

(6)

(7)

<sup>∂</sup><sup>r</sup> <sup>¼</sup> <sup>0</sup> (9)

(3)

http://dx.doi.org/10.5772/intechopen.76753

Figure 1. Geometry of the problem.

condensable gas enters the tube with a uniform velocity u0, vapour mass fraction w0, temperature T0 and pressure P0. The vapour condenses and forms a liquid film thickness as the mixture flowing downwards.

For the mathematical formulation of the problem, it has been assumed that the gas flow is laminar, incompressible and two-dimensional. The vapour and liquid phases are in thermodynamic equilibrium at the interface. In addition, viscous dissipation and other secondary effects are negligible, and the humid air is assumed to be a perfect gas.

#### 2.2. Mathematical formulation

With respect to the mentioned assumptions, the governing equations for the conservation of mass, momentum and energy, respectively, in the liquid region are written as

• Conservation of mass:

$$\frac{\partial}{\partial \mathbf{x}} (\rho\_L u\_L) + \frac{1}{r} \frac{\partial}{\partial r} (r \rho\_L v\_L) = 0 \tag{1}$$

Computational Study of Liquid Film Condensation with the Presence of Non-Condensable Gas in a Vertical Tube http://dx.doi.org/10.5772/intechopen.76753 59

• Conservation of momentum:

$$\frac{\partial}{\partial \mathbf{x}} (\rho\_L u\_L u\_L) + \frac{1}{r} \frac{\partial}{\partial r} (r \rho\_L v\_L u\_L) = -\frac{dp}{d\mathbf{x}} + \frac{1}{r} \frac{\partial}{\partial r} \left( r \mu\_L \frac{\partial u\_L}{\partial r} \right) + \rho\_L \mathbf{g} \tag{2}$$

• Conservation of energy:

$$\frac{\partial}{\partial x} \left( \rho\_L u\_L \mathbf{C}\_{p,L} T\_L \right) + \frac{1}{r} \frac{\partial}{\partial r} \left( r \rho\_L v\_L \mathbf{C}\_{p,L} T\_L \right) = \frac{1}{r} \frac{\partial}{\partial r} \left( r \lambda\_L \frac{\partial T\_L}{\partial r} \right) \tag{3}$$

Similarly, the mass conservation, momentum, energy and diffusion equations for the gas phase are written as follows:

• Conservation of mass:

$$\frac{\partial}{\partial \mathbf{x}} (\rho\_G u\_G) + \frac{1}{r} \frac{\partial}{\partial r} (r \rho\_G v\_G) = 0 \tag{4}$$

• Conservation of momentum:

$$\frac{\partial}{\partial x}(\rho\_G u\_G u\_G) + \frac{1}{r}\frac{\partial}{\partial r}(r\rho\_G v\_G u\_G) = -\frac{dp}{dx} + \frac{1}{r}\frac{\partial}{\partial r}\left(r\mu\_G \frac{\partial u\_G}{\partial r}\right) + \rho\_G g \tag{5}$$

• Conservation of energy:

$$\frac{\partial}{\partial x} \left( \rho\_G u\_G \mathbb{C}\_{p,G} T\_G \right) + \frac{1}{r} \frac{\partial}{\partial r} \left( r \rho\_G v\_G \mathbb{C}\_{p,G} T\_G \right) = \frac{1}{r} \frac{\partial}{\partial r} \left( r \lambda\_G \frac{\partial T\_G}{\partial r} \right) \tag{6}$$

• Species diffusion equation:

condensable gas enters the tube with a uniform velocity u0, vapour mass fraction w0, temperature T0 and pressure P0. The vapour condenses and forms a liquid film thickness as the

For the mathematical formulation of the problem, it has been assumed that the gas flow is laminar, incompressible and two-dimensional. The vapour and liquid phases are in thermodynamic equilibrium at the interface. In addition, viscous dissipation and other secondary effects

With respect to the mentioned assumptions, the governing equations for the conservation of

1 r ∂ ∂r

ð Þ¼ rrLvL 0 (1)

mass, momentum and energy, respectively, in the liquid region are written as

ð Þþ rLuL

∂ ∂x

are negligible, and the humid air is assumed to be a perfect gas.

mixture flowing downwards.

Figure 1. Geometry of the problem.

58 Desalination and Water Treatment

2.2. Mathematical formulation

• Conservation of mass:

$$\frac{\partial}{\partial x}(\rho\_G \mu\_G w) + \frac{1}{r} \frac{\partial}{\partial r}(r \rho\_G v\_G w) = \frac{1}{r} \frac{\partial}{\partial r} \left(r \rho\_G D \frac{\partial w}{\partial r}\right) \tag{7}$$

#### 2.3. Boundary and interfacial conditions

The governing equations are subjected to the following boundary conditions:

• At the tube inlet ð Þ x ¼ 0

$$
\mu\_G = \mu\_0; \ T\_G = T\_0; \ P\_G = P\_0; \ \mathbf{w}\_G = \mathbf{w}\_0 \tag{8}
$$

• At the centre line of the tube ð Þ r ¼ 0

$$
\upsilon\_G = 0; \ \frac{\partial u\_G}{\partial r} = \frac{\partial T\_G}{\partial r} = \frac{\partial w}{\partial r} = 0 \tag{9}
$$

• At the wall of the tube ð Þ r ¼ R

$$
\mu\_L = \upsilon\_L = 0; \ T\_L = T\_W \tag{10}
$$

mcd ¼ 2π

<sup>η</sup> <sup>¼</sup> ð Þ� <sup>R</sup> � <sup>δ</sup><sup>x</sup> <sup>r</sup> ð Þ R � δ<sup>x</sup>

<sup>η</sup> <sup>¼</sup> ð Þ� <sup>R</sup> � <sup>δ</sup><sup>x</sup> <sup>r</sup> δx

transformed into η, X as follows:

available in [20, 21].

3.1. Marching procedure

3. Numerical solution method

ðx 0

A transformation of coordinates was performed to ensure that the computational grid would clearly define the gas-liquid interface at each station along the tube. The r, x coordinates are

Computational Study of Liquid Film Condensation with the Presence of Non-Condensable Gas in a Vertical Tube

<sup>X</sup> <sup>¼</sup> <sup>x</sup>

The pure component data (in previous formulations) is approached by polynomials in terms of mass fraction and temperature. For more information, the thermo-physical properties are

The set of non-linear governing equations are discretized using a finite difference numerical scheme. The radial diffusion and the axial convection terms are approximated by the central and the backward differences, respectively. Hence, we arrange the system of discretized algebraic equations coupled with the boundary conditions into a matrix. Finally, the matrix resolution is carried out using the tri-diagonal matrix algorithm (TDMA) [22]. Besides that, a special care was made to ensure accuracy of the numerical computation, by generating a nonuniform grid in both directions. Accordingly, the grid is refined at the interface. In fact, it is important to note that as the liquid goes to the outlet, the film thickness varies along the tube. For that reason, during the downstream marching at each iteration, our finite difference computational grid deals with the variation of the liquid and gas computational domain.

A set of non-linear algebraic equations is realized for uL, vL, TL, uG, vG, TG, w and the two

2. Solve the finite difference forms of Eqs. (2)–(3) and (5)–(7) simultaneously for uL, TL, uG, TG, w.

scalars dp=dx and δx. The computational solution is advanced as follows: 1. For any axial position x, guess an arbitrary values of dp=dx and δx.

3. Numerically, integrate the continuities of Eqs. (1) and (5) to find vL and vG.

rGvIð Þ R � δ<sup>x</sup> dx (17)

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61

0 ≤ r ≤ ð Þ R � δ<sup>x</sup> (18)

ð Þ R � δ<sup>x</sup> ≤ r ≤ R (19)

<sup>L</sup> (20)

• At the interface vapour-liquid ð Þ r ¼ R � δ<sup>x</sup>

Continuities of velocity and temperature:

$$
\mu\_I(\mathbf{x}) = \mu\_{\mathbf{G},I} = \mu\_{L,I};\ T\_I(\mathbf{x}) = T\_{\mathbf{G},I} = T\_{L,I} \tag{11}
$$

Continuity of shear stress:

$$
\pi\_I = \left[ \mu \frac{\partial u}{\partial r} \right]\_{L,I} = \left[ \mu \frac{\partial u}{\partial r} \right]\_{G,I} \tag{12}
$$

Heat balance at the interface:

$$
\lambda\_L \frac{\partial T\_L}{\partial r} = \lambda\_G \frac{\partial T\_G}{\partial r} - \tilde{J} \,\tilde{h}\_{\text{fg}} \tag{13}
$$

where hfg is the latent heat of condensation and J " is the mass flux at the interface J " <sup>¼</sup> <sup>r</sup>GvI � �.

The radial velocity of water vapour-air mixture is calculated by considering that the interface is semipermeable [19] and that the solubility of air in the liquid film is negligibly small, which implies that the air velocity in the radial direction is zero at the interface. The velocity of the steam-air mixture at the interface can be written as

$$w\_{l} = -\frac{\sum\_{i=1}^{2} D\_{\text{G,in}} \frac{\partial w\_{\text{G}}}{\partial r}}{\left(1 - \sum\_{i=1}^{2} w\_{\text{Gi}}\right)} \tag{14}$$

The governing Eqs. (1)–(7) with interfacial conditions (8)–(13) are used to determine the field of variables uL, vL, TL, uG, vG, TG, w. To complete the mathematical model, two equations are used. At every axial location, the overall mass balance in the liquid phase and the gas flow should be satisfied:

$$\frac{m\_{0L}}{2\pi} = \int\_{R-\delta\_{\mathbf{x}}}^{R} (r\rho u dr)\_{L} - \int\_{0}^{\mathbf{x}} \rho\_{G} \upsilon\_{l} (R - \delta\_{\mathbf{x}}) d\mathbf{x} \tag{15}$$

$$\frac{\left(\boldsymbol{R} - \delta\_0\right)^2}{2}\rho\_0\boldsymbol{u}\_0 = \int\_0^{\mathbb{R}-\delta\_x} \left(r\rho udr\right)\_G + \int\_0^x \rho\_G \boldsymbol{v}\_l (\boldsymbol{R} - \delta\_x) d\mathbf{x} \tag{16}$$

A dimensionless accumulated condensation is introduced to estimate the mass transfer along the tube:

Computational Study of Liquid Film Condensation with the Presence of Non-Condensable Gas in a Vertical Tube http://dx.doi.org/10.5772/intechopen.76753 61

$$m\_{cd} = 2\pi \int\_0^\mathbf{x} \rho\_G \upsilon\_l (R - \delta\_\mathbf{x}) d\mathbf{x} \tag{17}$$

A transformation of coordinates was performed to ensure that the computational grid would clearly define the gas-liquid interface at each station along the tube. The r, x coordinates are transformed into η, X as follows:

$$\eta = \frac{(R - \delta\_{\mathbf{x}}) - r}{(R - \delta\_{\mathbf{x}})} \quad 0 \le r \le (R - \delta\_{\mathbf{x}}) \tag{18}$$

$$\eta = \frac{(R - \delta\_{\mathfrak{x}}) - r}{\delta\_{\mathfrak{x}}} \qquad (R - \delta\_{\mathfrak{x}}) \le r \le R \tag{19}$$

$$X = \frac{x}{L} \tag{20}$$

The pure component data (in previous formulations) is approached by polynomials in terms of mass fraction and temperature. For more information, the thermo-physical properties are available in [20, 21].

#### 3. Numerical solution method

• At the wall of the tube ð Þ r ¼ R

60 Desalination and Water Treatment

• At the interface vapour-liquid ð Þ r ¼ R � δ<sup>x</sup>

where hfg is the latent heat of condensation and J

steam-air mixture at the interface can be written as

m0<sup>L</sup> 2π ¼

ð Þ R � δ<sup>0</sup> 2 <sup>2</sup> <sup>r</sup>0u<sup>0</sup> <sup>¼</sup>

ðR

R�δ<sup>x</sup>

τ<sup>I</sup> ¼ μ

λL ∂TL <sup>∂</sup><sup>r</sup> <sup>¼</sup> <sup>λ</sup><sup>G</sup>

vI ¼ �

∂u ∂r � �

L,I ¼ μ ∂u ∂r � �

The radial velocity of water vapour-air mixture is calculated by considering that the interface is semipermeable [19] and that the solubility of air in the liquid film is negligibly small, which implies that the air velocity in the radial direction is zero at the interface. The velocity of the

P<sup>2</sup>

<sup>i</sup>¼<sup>1</sup> DG,im

ðx 0

> ðx 0

ð Þ rrudr <sup>G</sup> þ

A dimensionless accumulated condensation is introduced to estimate the mass transfer along

<sup>1</sup> � <sup>P</sup><sup>2</sup>

The governing Eqs. (1)–(7) with interfacial conditions (8)–(13) are used to determine the field of variables uL, vL, TL, uG, vG, TG, w. To complete the mathematical model, two equations are used. At every axial location, the overall mass balance in the liquid phase and the gas flow should be

ð Þ rrudr <sup>L</sup> �

ð<sup>R</sup>�δ<sup>x</sup> 0

∂wGi ∂r

<sup>i</sup>¼<sup>1</sup> wGi

∂TG <sup>∂</sup><sup>r</sup> � <sup>J</sup> "

Continuities of velocity and temperature:

Continuity of shear stress:

Heat balance at the interface:

satisfied:

the tube:

uL ¼ vL ¼ 0; TL ¼ TW (10)

(12)

" <sup>¼</sup> <sup>r</sup>GvI � �.

uIð Þ¼ x uG,I ¼ uL,I; TIð Þ¼ x TG,I ¼ TL,I (11)

G,I

hfg (13)

" is the mass flux at the interface J

� � (14)

rGvIð Þ R � δ<sup>x</sup> dx (15)

rGvIð Þ R � δ<sup>x</sup> dx (16)

The set of non-linear governing equations are discretized using a finite difference numerical scheme. The radial diffusion and the axial convection terms are approximated by the central and the backward differences, respectively. Hence, we arrange the system of discretized algebraic equations coupled with the boundary conditions into a matrix. Finally, the matrix resolution is carried out using the tri-diagonal matrix algorithm (TDMA) [22]. Besides that, a special care was made to ensure accuracy of the numerical computation, by generating a nonuniform grid in both directions. Accordingly, the grid is refined at the interface. In fact, it is important to note that as the liquid goes to the outlet, the film thickness varies along the tube. For that reason, during the downstream marching at each iteration, our finite difference computational grid deals with the variation of the liquid and gas computational domain.

#### 3.1. Marching procedure

A set of non-linear algebraic equations is realized for uL, vL, TL, uG, vG, TG, w and the two scalars dp=dx and δx. The computational solution is advanced as follows:


$$
\delta\_{\mathbf{x}}^{it+1} = \delta\_{\mathbf{x}}^{it} - \frac{\delta\_{\mathbf{x}}^{it} + \delta\_{\mathbf{x}}^{it-1}}{E\_L^{it} - E\_L^{it-1}} E\_L^{itt} \tag{21}
$$

resolved by Thomas algorithm. Furthermore, the boundary conditions on u<sup>n</sup>þ<sup>1</sup>

M\_ <sup>n</sup>þ<sup>1</sup> <sup>i</sup> � <sup>M</sup>\_ <sup>n</sup>

3.3. Mesh stability and validation of the numerical model

where the integral is estimated using numerical means. The M\_ <sup>n</sup>þ<sup>1</sup>

<sup>j</sup> .

fied, u<sup>n</sup>þ<sup>1</sup>

p,j <sup>¼</sup> 0. The solution of unþ<sup>1</sup>

the correct values of the velocity u<sup>n</sup>þ<sup>1</sup>

equation then allows to calculate vnþ<sup>1</sup>

Figure 2. Comparison of sensible heat Nusselt number.

of the global mass flow, unþ<sup>1</sup>

ent with the boundary conditions of the velocity. At boundaries, where the velocity is speci-

Computational Study of Liquid Film Condensation with the Presence of Non-Condensable Gas in a Vertical Tube

<sup>i</sup> ¼ 2πΔH

value specified in the initial conditions. The required value of ΔH is given by Eq. (24), whereas

To validate the grid independency of results and to avoid convergence problems due to the use of thin grids, it is helpful to choose an optimum solution between computational time and result precision. Several grid sizes have been examined to ensure that the results are grid independent. Figure 2 shows that in all grid arrangements, the difference in local Nusselt number of sensible heat is always less than 3%. The grid with NI � (NJ + NL) = 131 � (81 + 31) is chosen because it gives results close enough to those of the thin grid and sufficiently accurate

p,j is used to calculate ΔH, noting that to satisfy the constraint

<sup>j</sup> may be determined from Eq. (23). Besides, the continuity

p,j ΔH is the correction in velocity at each point. So, we can write

rrunþ<sup>1</sup>

ð<sup>R</sup>�δ<sup>x</sup> 0

p,j must be coher-

63

p,j dr (24)

http://dx.doi.org/10.5772/intechopen.76753

<sup>i</sup> in Eq. (24) is the known

The convergence criteria used is Eitt <sup>L</sup> <sup>¼</sup> <sup>10</sup>�<sup>5</sup> . Usually, six to seven iterations suffice to get converged solution.


$$E\_{rr} = \frac{\max\left|\mathbf{Y}\_{i,j}^{n} - \mathbf{Y}\_{i,j}^{n-1}\right|}{\max\left|\mathbf{Y}\_{i,j}^{n}\right|} < 10^{-5} \tag{22}$$

The solution for the actual axial position is complete. If not, repeat procedures (1) to (7), where ϒ represents the variables uL, TL, uG, TG, w.

#### 3.2. Velocity and pressure coupling

Owing to satisfy the global mass flow constraint, the pressure correction gradient and axial velocity profile are performed applying a method proposed by Raithby and Schneider [24], described by Anderson et al. [25]. To fully explain, we let H ¼ dp=dx. Due to an initial guesses for �dp ð Þ¼ � <sup>=</sup>dx dp ð Þ <sup>=</sup>dx <sup>∗</sup> , we calculate provisional velocities unþ<sup>1</sup> j <sup>∗</sup> and a mass flow rate of gas <sup>M</sup>\_ <sup>n</sup>þ<sup>1</sup> j <sup>∗</sup> . Because of the linearity of the equation of momentum with frozen coefficients, the correct velocity at each point from an application of Newton's method is as follow:

$$
\mu\_{j}^{n+1} = \left(\mu\_{j}^{n+1}\right)^{\*} + \frac{\partial \mu\_{j}^{n+1}}{\partial H} \Delta H \tag{23}
$$

ΔH is the change in the gradient of the pressure required to satisfy the global mass flow constraint. In addition, we specify u<sup>n</sup>þ<sup>1</sup> p,j <sup>¼</sup> <sup>∂</sup>unþ<sup>1</sup> j <sup>∂</sup><sup>H</sup> . The difference equations are indeed differentiated with respect to the pressure gradient (H) to have difference equations for unþ<sup>1</sup> p,j , which have a tridiagonal form. The coefficients for the unknowns in these equations will be the same as for the original implicit difference equations. The system of algebraic equations for unþ<sup>1</sup> p,j is resolved by Thomas algorithm. Furthermore, the boundary conditions on u<sup>n</sup>þ<sup>1</sup> p,j must be coherent with the boundary conditions of the velocity. At boundaries, where the velocity is specified, u<sup>n</sup>þ<sup>1</sup> p,j <sup>¼</sup> 0. The solution of unþ<sup>1</sup> p,j is used to calculate ΔH, noting that to satisfy the constraint of the global mass flow, unþ<sup>1</sup> p,j ΔH is the correction in velocity at each point. So, we can write

$$\dot{M}\_i^{n+1} - \dot{M}\_i^n = 2\pi\Delta H \int\_0^{R-\delta\_x} r\rho u\_{p,j}^{n+1} dr\tag{24}$$

where the integral is estimated using numerical means. The M\_ <sup>n</sup>þ<sup>1</sup> <sup>i</sup> in Eq. (24) is the known value specified in the initial conditions. The required value of ΔH is given by Eq. (24), whereas the correct values of the velocity u<sup>n</sup>þ<sup>1</sup> <sup>j</sup> may be determined from Eq. (23). Besides, the continuity equation then allows to calculate vnþ<sup>1</sup> <sup>j</sup> .

#### 3.3. Mesh stability and validation of the numerical model

4. The interfacial conditions of velocity, temperature, shear stress and heat balance are

6. The best approximation to the thickness of the liquid film is then obtained using the secant

<sup>x</sup> � <sup>δ</sup>itt

<sup>L</sup> <sup>¼</sup> <sup>10</sup>�<sup>5</sup>

max ϒ<sup>n</sup>

 

max ϒ<sup>n</sup> i,j 

Eitt <sup>L</sup> � Eitt�<sup>1</sup> L

8. Check the satisfaction of the convergence of velocity, temperature and species concentrations. If the relative error between two consecutive iterations is small enough, that is

> i,j � <sup>ϒ</sup><sup>n</sup>�<sup>1</sup> i,j

The solution for the actual axial position is complete. If not, repeat procedures (1) to (7),

Owing to satisfy the global mass flow constraint, the pressure correction gradient and axial velocity profile are performed applying a method proposed by Raithby and Schneider [24], described by Anderson et al. [25]. To fully explain, we let H ¼ dp=dx. Due to an initial guesses for

. Because of the linearity of the equation of momentum with frozen coefficients, the

∂unþ<sup>1</sup> j

þ

ΔH is the change in the gradient of the pressure required to satisfy the global mass flow

have a tridiagonal form. The coefficients for the unknowns in these equations will be the same as for the original implicit difference equations. The system of algebraic equations for unþ<sup>1</sup>

, we calculate provisional velocities unþ<sup>1</sup>

correct velocity at each point from an application of Newton's method is as follow:

<sup>j</sup> <sup>¼</sup> unþ<sup>1</sup> j <sup>∗</sup>

> p,j <sup>¼</sup> <sup>∂</sup>unþ<sup>1</sup> j

ated with respect to the pressure gradient (H) to have difference equations for unþ<sup>1</sup>

u<sup>n</sup>þ<sup>1</sup>

 

<sup>x</sup> <sup>þ</sup> <sup>δ</sup>itt�<sup>1</sup> x

Eitt

<sup>G</sup> using Eq. (16).

 

> j <sup>∗</sup>

<sup>L</sup> using Eq. (15).

<sup>L</sup> (21)

. Usually, six to seven iterations suffice to get

< 10�<sup>5</sup> (22)

and a mass flow rate of gas

p,j , which

p,j is

<sup>∂</sup><sup>H</sup> <sup>Δ</sup><sup>H</sup> (23)

<sup>∂</sup><sup>H</sup> . The difference equations are indeed differenti-

obtained from Eqs. (11)–(13).

The convergence criteria used is Eitt

7. Calculate the error in the gas flow balance Eitt

method [23]. Thus

62 Desalination and Water Treatment

converged solution.

5. Calculate the error of the liquid film mass balance Eitt

δittþ<sup>1</sup> <sup>x</sup> <sup>¼</sup> <sup>δ</sup>itt

Err ¼

where ϒ represents the variables uL, TL, uG, TG, w.

3.2. Velocity and pressure coupling

constraint. In addition, we specify u<sup>n</sup>þ<sup>1</sup>

�dp ð Þ¼ � <sup>=</sup>dx dp ð Þ <sup>=</sup>dx <sup>∗</sup>

<sup>M</sup>\_ <sup>n</sup>þ<sup>1</sup> j <sup>∗</sup> To validate the grid independency of results and to avoid convergence problems due to the use of thin grids, it is helpful to choose an optimum solution between computational time and result precision. Several grid sizes have been examined to ensure that the results are grid independent. Figure 2 shows that in all grid arrangements, the difference in local Nusselt number of sensible heat is always less than 3%. The grid with NI � (NJ + NL) = 131 � (81 + 31) is chosen because it gives results close enough to those of the thin grid and sufficiently accurate

Figure 2. Comparison of sensible heat Nusselt number.

In order to check the accuracy and validity of the numerical method, the obtained results were first compared to those reported by Hassaninejadfarahani et al. [17] in the case of laminar condensation of a steam and non-condensable gas in a vertical tube, in which the tube wall is maintained at a constant temperature. A good agreement was found between the current computational study and the results provided by Hassaninejadfarahani et al. [17] as shown in Figure 3a, b, which illustrates the vapour mass fraction evolution and

Computational Study of Liquid Film Condensation with the Presence of Non-Condensable Gas in a Vertical Tube

http://dx.doi.org/10.5772/intechopen.76753

65

Figure 4. Comparison with experimental data of Lebedev et al. [2] for local condensate heat transfer coefficient.

The computations have been also compared with experimental results of Lebedev et al. [2]. It is important to indicate that Lebedev et al. [2] examined the simultaneous heat and mass transfer during humid air condensation in a vertical duct. So, to obtain the case of Lebedev et al. [2], equivalent hydraulic diameter de is chosen [7, 26]. Figure 4 is a plot of local condensate heat transfer coefficient compared with the study of Lebedev et al. [2] for T0 = 60C, P0 = 1 atm, L = 0.6 m, TW = 5C and de = 0.02 m. A good agreement between our computations and the experiment curves is found with a maximum relative error of 4.7% for both curves (u0 = 1.4 m/s

4. Distribution of axial velocity, temperature and mass fraction profiles

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

mixture temperature, respectively.

and u0 = 0.7 m/s).

along the vertical tube

Figure 3. Comparison with numerical study of Hassaninejadfarahani et al. [17] for (a) dimensionless mass fraction at the tube exit, and (b) dimensionless mixture temperature.

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 grid points in the radial direction at the liquid region.

Computational Study of Liquid Film Condensation with the Presence of Non-Condensable Gas in a Vertical Tube http://dx.doi.org/10.5772/intechopen.76753 65

Figure 4. Comparison with experimental data of Lebedev et al. [2] for local condensate heat transfer coefficient.

In order to check the accuracy and validity of the numerical method, the obtained results were first compared to those reported by Hassaninejadfarahani et al. [17] in the case of laminar condensation of a steam and non-condensable gas in a vertical tube, in which the tube wall is maintained at a constant temperature. A good agreement was found between the current computational study and the results provided by Hassaninejadfarahani et al. [17] as shown in Figure 3a, b, which illustrates the vapour mass fraction evolution and mixture temperature, respectively.

The computations have been also compared with experimental results of Lebedev et al. [2]. It is important to indicate that Lebedev et al. [2] examined the simultaneous heat and mass transfer during humid air condensation in a vertical duct. So, to obtain the case of Lebedev et al. [2], equivalent hydraulic diameter de is chosen [7, 26]. Figure 4 is a plot of local condensate heat transfer coefficient compared with the study of Lebedev et al. [2] for T0 = 60C, P0 = 1 atm, L = 0.6 m, TW = 5C and de = 0.02 m. A good agreement between our computations and the experiment curves is found with a maximum relative error of 4.7% for both curves (u0 = 1.4 m/s and u0 = 0.7 m/s).
