This work investigates the steam condensation phenomena in an air-cooled condenser. The considered horizontal flattened tube has a 30 mm hydraulic diameter, and its length is a function of the steam quality with a limit value between 0.95 and 0.05. The mass flow rate ranges from 4 to 40 kg/m^{2}.s with a saturated temperature spanning an interval from 40°C to 80°C. A special approach has been implemented using the Engineering Equation Solver (EES) to solve a series of equations for the two-phase flow pattern and the related heat transfer coefficients. A wavy-stratified structure of the two-phase flow has been found when the mass rate was between 4 and 24 kg/m^{2}.s. In contrast, an initially annular flow is gradually converted into a wavy stratified flow (due to the condensation process taking place inside the flattened tube) when the considered range ranges from 32 to 40 kg/m^{2}.s.

Condensation phenomenon is deployed in many industrial applications of equipment such as air-cooled power generation plants, desalination, air-conditioning refrigerant, concentrated solar power, chemical processes and nuclear power plants [

In a horizontal tube, usually, the steam condenses along the inner wall of the tube when the steam forward movement on the way to liquid phase across a varies of qualities. Therefore to understand the two-phase flow development in a horizontal tube, it is important to recognize the flow models along the pipe. Because of the significance of orientation, interaction between gravity, buoyancy, surface tension and inertia forces need to be considered in computational predictions [

The two-phase flow has many different flow patterns, which can be obtained based on the variation of the void fraction [

Both the shear force of a vapor and surface tension force are dominating in the tubes when the tube diameter is less than 3 mm [

Chato [

Baker [_{ε}) method for the determination of vapor void fractions within a range from low pressures to pressures close to the critical point. Indeed, the validity of the new (LM_{ε}) method has been verified by using the thermal condensation model for the annular flow; also, the modified map gave good results after comparing it to several recent flow patterns. Shah [^{2}.s. Mahdi et al. [^{2}.s. Sereda et al. [^{2}.s and the quality of vapor from 0.95 to 0.23. The particular devices were used to measure the heat flux at circumferential of the condensation tube and the features of the heat transfer during the streamlet flow of the phases. The authors presented a CFD simulation through the heat transfer of condensing vapor to the system of cooling water by a cylinder that has a thick wall. The CFD model was evaluated using a practical experiment, which revealed that the results agreed with an error range of 7% to 20%. Under stratified flow conditions, this approach could generalize the experimental results on the condensation of refrigerants R22, R134a, R123, R125, R32, R410a, propane, isobutane, propylene, dimethyl ether, carbon dioxide, and methane with accuracy of 30%. Abdulkareem et al. [

Indeed, many studies investigated the condensation process within different considerations, but there is a lack of information about the mathematical model that can be used to predict the rate of heat transfer which is related to the certain structure of flow pattern during the steam condensation in the flattened tube at low mass flow rate. However, the current work is to present a mathematical model that can be used to predict the mechanism of the heat transfer in the certain structure of the two-phase flow pattern in the steam condensation process inside a flattened horizontal tube which enhanced our previous work presented in [

The Engineering Equation Solver (EES) (Klein) platform Professional V9.478 (2013) was used to investigate the two-phase flow patterns by solving sets of equations that represent the heat transfer coefficient for the condensation phenomena inside the flattened tube of an air-cooled steam condenser.

Certain boundary conditions were used in EES, the range of mass velocity rate was between 4 and 40 kg/m^{2}.s, steam saturated temperature ranged from 40°C to 80°C and steam quality from 0.95 to 0.05.

A horizontal flattened tube as in

The heat transfer model was controlled by the predominant flow structure to analyze the model of heat transfer during the steam condensation process. The flow structures were classified into two categories: shear-dominated and gravity-dominated flows. In the shear force dominated flow model, forced-convection was controlling the condensation heat transfer model. This structure was defined by heat transfer coefficients that were significantly dependent on the steam quality and mass flux. While in the gravity force dominated flow model, a laminar film condensation at the upper part of the tube was controlling the heat transfer model in which the heat transfer coefficients are depended on the temperature difference between the wall and the fluid but were nearly independent of mass velocity. The criteria of dominant flow inside the air-cooled steam condenser tubes are changeable as a result of changing operational situations, and that is why it is obligatory to define the dominant criterion to subsequently move to the analysis of the case. Rosson [

Martinelli’s parameter:

Dimensionless speed:

After the implementation of ^{,}’s parameter for different types of two-phase flow pattern.

Range of J_{g} & X_{tt} |
Structure of two-phase |
---|---|

_{g} > 1.5; _{tt} < 1 |
Annular |

_{g} ≤ 1.5; _{tt} < 1 |
Stratified-wavy |

_{g} ≤ 1.5; _{tt} ≥ 1 |
Intermittent |

_{g} > 1.5; _{tt} ≥ 1 |
Bubbled |

The structure flow model was determined according to the abovementioned criteria, however, the model proposed by Soliman [

At the medium and low velocity of steam, stratified wavy flow is formulated. In this flow pattern, gravity forces control the flow. The condensation flow structure includes two types of heat transfer models: a film-wise condensation had occurred in the upper part of the tube with overlap effects due to shear of vapor and forced convective heat transfer in the lower part of the tube. Dobson et al. [^{2}.s, steam velocity was higher than 0.5 m/s and Soliman modified Froude number (_{So}) was less than

The Galileo number and Reynold’s number assuming liquid phase flowing alone can be determined as follows:

The average of heat transfer coefficient in this cross-section for this type of flow structure can be represented by:

The film-wise condensation heat transfer coefficient for the upper part of the tubes:

For the liquid part, the heat transfer coefficient as:

The C and D are constants that can be represented as in

Spanning | _{l} ≤ 0.7 |
_{l} > 0.7 |
---|---|---|

C | 7.242 | |

D | 1.773 − 0.169_{l} |
1.655 |

To determine the wavy-stratified angle between the steam portion and the condensate portion, as shown in _{strat,} G_{wavy}, θ_{strat.} And the void fraction

The global term of local heat transfer coefficient h_{TP} at any cross-section of the tube is:

An annular flow structure is formed at a high velocity of steam when the effect of shear force for the steam is pronounced more than the gravity force, while the condensate film covers the circumference of the inner diameter of the tube. According to Dobson et al. [

The Engineering Equation Solver (EES) has been used to solve a set of methodical equations and empirical equations, the mathematical model of the current study, has been validated with experimental results from the literature. The experimental results and operating conditions of the two-phase flow development has been presented by Mahmood et al. [

This section discusses the results that are obtained from the mathematical model using Engineering Equation Solver (EES), after the formation of special codes for the equations of the two-phase flow pattern and heat transfer coefficients.

Both ^{2}.s, at steam saturated temperature of 45°C by using Engineering Equation Solver to detect the type of flow pattern.

G = 16 kg/m^{2}.s |
G = 32 kg/m^{2}.s |
G = 40 kg/m^{2}.s |
||||
---|---|---|---|---|---|---|

Steam quality | Martinelli parameter (Xtt) | Dimensionless speed J_{g} |
Martinelli parameter (Xtt) | Dimensionless speed J_{g} |
Martinelli parameter (Xtt) | Dimensionless speed J_{g} |

0.95 | 0.0008614 | 0.8867 | 0.0008614 | 1.773 | 0.0008614 | 2.216 |

0.9 | 0.001694 | 0.8397 | 0.001694 | 1.679 | 0.001694 | 2.099 |

0.85 | 0.002565 | 0.7931 | 0.002565 | 1.586 | 0.002565 | 1.982 |

0.8 | 0.003507 | 0.7465 | 0.003507 | 1.492 | 0.003507 | 1.866 |

0.75 | 0.004541 | 0.6998 | 0.004541 | 1.399 | 0.004541 | 1.749 |

0.7 | 0.005692 | 0.6532 | 0.005692 | 1.306 | 0.005692 | 1.633 |

0.65 | 0.006988 | 0.6066 | 0.006988 | 1.213 | 0.006988 | 1.516 |

0.6 | 0.008467 | 0.56 | 0.008467 | 1.12 | 0.008467 | 1.399 |

0.55 | 0.01018 | 0.5133 | 0.01018 | 1.026 | 0.01018 | 1.283 |

0.5 | 0.01219 | 0.4667 | 0.01219 | 0.9331 | 0.01219 | 1.166 |

0.45 | 0.01463 | 0.4197 | 0.01463 | 0.839 | 0.01463 | 1.049 |

0.4 | 0.01758 | 0.373 | 0.01758 | 0.7458 | 0.01758 | 0.9323 |

0.35 | 0.02131 | 0.3264 | 0.02131 | 0.6526 | 0.02131 | 0.8158 |

0.3 | 0.02616 | 0.2798 | 0.02616 | 0.5594 | 0.02616 | 0.6993 |

0.25 | 0.0328 | 0.2332 | 0.0328 | 0.4662 | 0.0328 | 0.5827 |

0.2 | 0.04249 | 0.1865 | 0.04249 | 0.373 | 0.04249 | 0.4662 |

0.15 | 0.05813 | 0.1399 | 0.05813 | 0.2797 | 0.05813 | 0.3497 |

0.1 | 0.08812 | 0.09329 | 0.08812 | 0.1865 | 0.08812 | 0.2332 |

0.05 | 0.1726 | 0.04667 | 0.1726 | 0.09331 | 0.1726 | 0.1166 |

From ^{2}.s, which reflects the high volume of vapor along sections of the flattened tube. This indicates that the flow pattern inside the tube is either wavy-stratified flow or annular flow depending on the other factor, which is the dimensionless speed. At G = 16 kg/m^{2}.s, the values of dimensionless speed are less than the limited value of wavy-stratified flow; subsequently, the structure of the flow will be wavy-stratified, which means that the gravity force has been dominated instead of shear stress. In contrast, the structure of the flow begins at annular flow at mass flux G = 32 kg/m^{2}.s, at steam quality equal to 0.95 to 0.85 corresponding to the dimensionless speed, which is higher than the limited value by more than 1.5, and then converts into wavy-stratified flow at steam quality that equals 0.8 to 0.05. When increasing the mass flux to G = 40 kg/m^{2}.s, the annular flow structure will take a wider range relative to the length of the condensation tube because the shear force (momentum) is higher than that at mass flux G = 32 kg/m^{2}.s, and then the flow has been converted to wavy-stratified flow at a steam quality equal to 0.6 due to the limited value of dimensionless speed parameter in

^{2}.s. The results in the figure can be divided into two categories, at mass velocity 20–24 kg/m^{2}.s and correspond to the values of ^{2}.s the flow pattern consists of two regions, the annular and wavy–stratified flow. The position of conversion of flow pattern for each mass velocity occurs at a different steam qualities, which depends on the forces that to be dominated (shear force or gravity force), so at mass velocity G = 28 kg/m^{2}.s, the point of the conversion of the flow structure takes place at a steam quality that equals 0.9, while at mass velocity G = 40 kg/m^{2}.s, the flow pattern started as annular flow and changed to wavy-stratified flow at steam quality equal to 0.6. From the same figure, the heat transfer coefficient increases when the mass velocity increases from 20 to 24 kg/m^{2}.s at the same steam quality with a wavy-stratified flow which is determined according to ^{2}.s the heat transfer coefficient is degraded from the beginning where the steam quality ranges between 1 and 0.9 because the structure of the flow is an annular flow that depends on the limited values of ^{2}.s the flow structure begins an annular type with a spanning of steam quality from 1 to 0.8 for G = 32 kg/m^{2}.s, while the annular flow begins from steam quality that equals 1 to 0.6 for G = 40 kg/m^{2}.s, then transform to the wavy-stratified flow. This structure of flow indicates degradation in the rate of heat transfer since the liquid film acts as a barrier for the transfer of heat from the vapor (steam); after that, the flow pattern turns to stratified wavy flow, which is characterized by a high heat transfer rate compared to the annular flow pattern. Therefore, it is necessary for the researchers and engineers who design these systems to ensure that the convenient flow pattern includes the highest heat transfer rate and the lowest pressure loss.

^{2} and G = 40 kg/m^{2}.s. ^{2}.s while

Indeed, the outcome of condensation, which is a result of phase change from vapor to liquid influenced the momentum force at a range of steam quality, which can help to provide an effective database for the condensation system enhancement and heat transfer rate.

A mathematical model has been adopted to investigate the condensation phenomena inside a horizontal flattened tube. The parameters of steam quality, mass velocity and saturated temperature are used to predict the structure of the two-phase flow and its effect on the heat transfer rate, so, the following can be concluded:

1- The two-phase flow pattern has a significant impact on the heat transfer rate, at mass velocity range from 28–40 kg/m^{2}.s, there are degradations in the heat rate because the flow pattern begins as an annular flow and then it is converted to wavy stratified flow while at mass velocity 20–24 kg/m^{2}.s it started as wavy-stratified flow according to the limited criteria.

2- The mass velocity has an impact on the forces that control the structure of the two-phase flow. When mass velocity ranges in 20–24 kg/m^{2}.s the gravitation force is the dominant force, but at mass velocity range of 28–40 kg/m^{2}.s, the shear force is the dominant force at the beginning of the flow while converting to the gravitation force as a dominate force during the decrease in the momentum of the flow.

3- The saturated temperature has no significant impact on the structure of the two-phase during the flow of condensation. In contrast, the heat transfer coefficient increases with the increase of the saturated steam temperature along the condensation process inside the flattened horizontal tube.

4- The heat transfer coefficient is degraded at steam quality that ranges from x = 0.2 to x = 0.05 for mass velocity range of 20–40 kg/m^{2}.s whether the structure of the two-phase had begun as a wavy-stratified flow or annular wavy-stratified flow

Tube cross-section area, [m^{2}]

_{l}

Liquid cross-section area, A_{l} = A (1-ε), [m^{2}]

_{ld}

Dimensionless cross-sectional area occupied by liquid, A_{ld} = A_{l}/d^{2}

_{V}

Vapor cross-sectional area, Av = A ε, [m^{2}]

_{vd}

Dimensionless cross-sectional area occupied by vapor, A_{vd} = A_{v}/d^{2}

_{p}

Specific heat, [J/(kg.K)]

_{h}

Hydraulic diameter, [m]

_{l}

Liquid Froude number, G^{2}/(

Acceleration of gravity, [m/s^{2}]

Total mass flow rate (mass velocity) of liquid and vapor [kg/(m^{2}.s)]

_{strat}

Transition mass velocity of stratified flow [kg/(m^{2}.s)]

_{wavy}

Transition mass velocity of wavy flow [kg/(m^{2}.s)]

Heat transfer coefficient [W/(m^{2}.K)]

_{fg}

Condensation latent heat, [J/kg]

Dimensionless velocity [-]

Thermal conductivity, [W/(m.K)]

Nusselt number, the subscript specifies the characteristic length, (hD_{h})/k_{l}

Pressure, [N/m^{2}]

Prandtl number [-]

Inside radius of tube [m]

Temperature [K]

Vapor quality [–]

_{tt}

Martinelli parameter with both phases turbulent [–]

Increment in the condensation tube

The change of the quality of steam along the condensation process

Difference

Vapor void fraction [–]

_{h}

Homogenous void fraction [-]

_{ra}

Rouhani-Axelsson void fraction [-]

Dynamic viscosity [N s/m^{2}]

Angle of the upper portion tube not wetted by stratified liquid [rad]

_{strar}

Stratified angle about the upper perimeter of the tube [rad]

Density [kg/m^{3}]

Surface tension [N/m]

Bottom portion

Liquid condensate

Critical

Film

Vapor

Liquid

Reduced

Saturated

Two-phase

Upper portion

Wall

The authors would like to thank Mustansiriyah University (