# A Heat Exchanger with Water Vapor Condensation on the External Surface of a Vertical Pipe

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Experiment

#### 2.1. Experimental Device

#### 2.2. Analytical Model

#### 2.3. Experimental Setup

- (a)
- 24 mm channel: 10; 12; 15; 20; 25; 30 and 35 kg∙h
^{−1}, which corresponds to the vapor velocity range, in front of the first testing pipe, from 0.67 to 1.96 m∙s^{−1}. - (b)
- 20 mm channel: 10; 12; 15; 20; 25 and 30 kg∙h
^{−1}, which corresponds to the vapor velocity range, in front of the first testing pipe, from 0.81 to 2.32 m∙s^{−1}.

^{−1}. Throughout all the experiments, its average value was 7.51 ± 0.05 L∙min

^{−1}. Thanks to this, the experiments were carried out under approximately equal conditions; the average value of the Reynolds number on the cooling water side, with all the experiments included, was 22,623 ± 3029 and the average value of the Nusselt number was 104.4 ± 6.6, without distinguishing the required temperature of the cooling water at the inlet into the exchanger. For all the required vapor mass flow rates, three temperature levels were tested of the cooling water at the inlet into the exchanger (${T}_{3}$), in particular 30, 40 and 50 °C, with a permissible deviation of up to ±1.0 °C. The resulting average temperatures of the cooling water at the inlet into the exchanger were 30.5 ± 0.2 °C, 40.1 ± 0.2 °C and 50.0 ± 0.1 °C.

## 3. Results and Comparison

#### 3.1. Results of the Experiments

^{−1}did the output of the exchanger start to decrease for lower cooling water temperatures, because almost all the vapor in the exchanger condensed and, therefore, the power transferred was a function of the vapor mass flow rate. It can be assumed, based on this trend, that the influence of the input cooling water temperature started to be negligible below a vapor flow rate of 10 kg·h

^{−1}.

^{−1}to 30 kg·h

^{−1}. Below a water vapor flow rate of 12 kg·h

^{−1}, and for the input cooling water temperature of 30 °C, a slight increase occurs due to a decrease in the output of the exchanger, and thus, less heating of the cooling water occurs. Above 30 kg·h

^{−1}of the water vapor, the logarithmic temperature gradient increases again. This is caused by an increase in the pressure inside the exchanger due to the mass flow rate being too high and thus the saturation vapor temperature increasing.

^{−1}, where the maximum is reached, and with a further increase in the vapor mass flow rate, it starts decreasing.

^{−1}. The graph shows that the highest values for each condition measured are reached in the mode with an input cooling water temperature of 30 °C. With the temperature increasing, the condensation heat transfer coefficient decreases. This trend applies to the presented interval of the vapor mass flow rate ranging from 12.5 kg·h

^{−1}to 35 kg·h

^{−1}. Below 12.5 kg·h

^{−1}of vapor, with the input cooling water temperatures being 30 °C and 40 °C, the condensation heat transfer coefficient starts to decline rapidly; below this flow rate, it holds that with the cooling water temperature increasing, the condensation heat transfer coefficient increases. This is caused by the output of the exchanger decreasing, as shown in Figure 2. When comparing the wider and the narrower channels, one comes to the conclusion that the width of the channel plays a negligible role. Only with the vapor mass flow rate exceeding 25 kg·h

^{−1}does the condensation heat transfer coefficient reach higher values in the narrower channel than in the wider channel. The difference between heat transfer in the narrower channel and in the wider one is around 5%. Below a vapor flow rate of 25 kg·h

^{−1}, the behaviour of the condensation heat transfer coefficients is very similar for both channel widths.

#### 3.2. Steam Flow Identification by Computational Modeling

^{−1}, the ratio between the flow velocity and the real size of the heat exchanging surface was optimal.

#### 3.3. Comparison with Other Studies

#### 3.3.1. Model S1

#### 3.3.2. Model S2

#### 3.3.3. Model S3

#### 3.3.4. Model S4

#### 3.3.5. Comparing the Models with the Experimental Results

_{p}(of a percentage point).

- (a)
- The basic variant when the authors calculate the Reynolds number from the maximum velocity; see Equation (31). The maximum velocity is determined on the basis of the pipe bundle geometry (spacing of the pipes/row), where, in our case, another adjacent pipe is the channel wall. The velocity considered in this variant is that in front of the first pipe.
- (b)
- Another variant is the mean value of the vapor velocity in front of all the three pipes, i.e., the average value of the vapor velocity in front of the first, second and third pipes. This value is then adjusted as in the previous variant, i.e., it is substituted into Equation (32) and then into Equation (31).
- (c)
- Based on Equations (13) and (14), the maximum velocity at the narrowest point around the first pipe is determined. The amount of condensed vapor in front of the narrow section is $x=0.33$. This velocity is directly substituted into the calculation of the maximum Reynolds number in Equation (31).
- (d)
- As in the previous variant, the maximum vapor velocity is calculated at the first, second and third pipes, and these values are used to calculate the average maximum velocity. This velocity is directly substituted into the calculation of the maximum Reynolds number in Equation (31).

## 4. Conclusions

- (a)
- The width of the channel is negligible, and deviations fall within the statistical error margins.
- (b)
- The heat flow in the exchanger increases with a decreasing input temperature of the cooling water. By decreasing the temperature of the cooling water by 20 °C, the output of the exchanger increases by approx. 25%, on average.
- (c)
- At the moment when not all the vapor condenses in the exchanger, the influence of the vapor velocity at the inlet into the exchanger decreases.
- (d)
- It holds for the particular exchanger geometry that with a higher vapor mass flow rate, and thus, a higher velocity, the condensation heat transfer coefficient also increases up to a flow rate of 25 kg·h
^{−1}, where the maximum is reached; with a further increase in the flow rate, it starts decreasing.

^{−1}, at which the best ratio is achieved between the flow velocity and the real size of the heat exchanging surface, which is smaller than the theoretical area size given by the pipe shell.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

c_{p} | specific heat capacity, J∙kg^{−1}∙K^{−1} |

d | tube diameter, m |

g | gravity acceleration, m∙s^{−2} |

Gr | Grashof number, - |

Δh_{c} | latent heat of condensation, J∙kg^{−1} |

i | Enthalpy, J∙kg^{−1} |

Ja | Jacob number, - |

k_{c} | overall HTC, W∙m^{−1}∙K^{−1} |

L | characteristic length, m |

$\dot{M}$ | Mass flow rate, kg∙s^{−1} |

Nu | Nusselt number, - |

p | absolute pressure, Pa |

Pr | Prandtl number |

$\dot{Q}$ | heat transferred, W |

Re | Reynolds number, - |

t | temperature, °C |

T | temperature, K |

$\dot{V}$ | Volume flow rate, m^{3}∙s^{−1} |

w | speed, m∙s^{−1} |

Greek letters | |

α | HTC, W∙m^{−2}∙K^{−1} |

η | dynamic viscosity, Pa∙s |

λ | thermal conductivity, W∙m^{−1}∙K^{−1} |

ν | kinematic viscosity, m^{2}∙s^{−1} |

ρ | density, kg∙m^{−3} |

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**Figure 3.**Developments of the logarithmic temperature gradients in the exchanger for different input cooling water temperatures.

**Figure 4.**Developments of the condensation heat transfer coefficient (HTC) in the exchanger for different input cooling water temperatures.

**Figure 6.**Vapor flow velocities at a distance of one tenth of the pipe diameter (i.e., 1.4 mm from the pipe wall), depending on the input vapor mass flow rate. (

**a**) Velocity field around pipe 1, (

**b**) positions of points (1–16) around the pipe perimeter, (

**c**) velocity field around pipe 2, (

**d**) velocity field around pipe 3.

**Figure 7.**Relative error of the maximum velocity at the narrowest point around the pipe determined using the CFD model and the analytical model.

Model | Average Error [%] | Standard Deviation [%_{p}] |
---|---|---|

S1 | −131.3 | ±28.5 |

S2 | −121.9 | ±27.3 |

S3 | −14.7 | ±14.4 |

S4—a | 23.4 | ±19.3 |

S4—b | 34.1 | ±19.5 |

S4—c | 45.3 | ±13.2 |

S4—d | 53.6 | ±13.1 |

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**MDPI and ACS Style**

Kracík, P.; Toman, F.; Pospíšil, J.; Kraml, S.
A Heat Exchanger with Water Vapor Condensation on the External Surface of a Vertical Pipe. *Energies* **2022**, *15*, 5636.
https://doi.org/10.3390/en15155636

**AMA Style**

Kracík P, Toman F, Pospíšil J, Kraml S.
A Heat Exchanger with Water Vapor Condensation on the External Surface of a Vertical Pipe. *Energies*. 2022; 15(15):5636.
https://doi.org/10.3390/en15155636

**Chicago/Turabian Style**

Kracík, Petr, Filip Toman, Jiří Pospíšil, and Stanislav Kraml.
2022. "A Heat Exchanger with Water Vapor Condensation on the External Surface of a Vertical Pipe" *Energies* 15, no. 15: 5636.
https://doi.org/10.3390/en15155636