# Scientific Approaches to Solving the Problem of Joint Processes of Bubble Boiling of Refrigerant and Its Movement in a Heat Pump Heat Exchanger

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Accepted Assumptions

#### 2.1.1. Vapor-Liquid Mixture

#### 2.1.2. Boiling

#### 2.1.3. Flow Mode

#### 2.2. Theoretical Calculation

#### 2.2.1. The Calculation with Assumptions Is Divided into Two Stages

_{H}under pressure p at point 4 in the diagram according to Figure 2. The physical properties of saturated liquid refrigerant are searched for at the obtained temperature according to reference data [14]. The Reynolds number is calculated at a given flow velocity ω (8):

#### 2.2.2. Comparison of Data from Theoretical Calculation and Analytical Calculation for Temperature

#### 2.3. Differential Equations When Solving by Numerical Method

#### 2.3.1. Energy Equations

#### 2.3.2. The Transfer Equation

#### 2.4. Modeling in a Software Environment ANSYS Fluent

- From the experience of modeling the emulsion flow, the condition of a two-phase flow at the inlet is set (previously, only one liquid phase was set at the inlet, and not a mixture);
- The extreme pressures in the cycle were determined (3.5 and 19 bar, respectively);
- The saturation temperature is set from the theoretical calculation;
- Based on points 2 and 3, a certain degree of dryness of the vapor-liquid mixture at the inlet to the evaporator is set.

- −
- Element size a: 133.16 × 133.16 × 3 mm, diagonal: 205 mm;
- −
- Tube inner diameter: 3 mm, tube outer diameter: 5 mm;
- −
- Temperature on the front surface of the element: 60 °C;
- −
- Ambient air temperature: 30 °C.

## 3. Results

#### Comparison of the Received Data

## 4. Discussion

#### 4.1. Limitations of the Study

#### 4.2. Prospects for the Use of a Heat Pump in an Energy Complex for Seawater Desalination

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

i_{1} | enthalpy of condensate, kJ/kg; |

i_{2} | enthalpy of vaporization, kJ/kg; |

q_{boil.} | boiling heat flux density, kW/m^{2}; |

λ | thermal conductivity of liquid Freon, kW/(m·K); |

ΔT_{s} | temperature difference between wall and liquid, K; |

h_{g} | enthalpy of gas, kJ/kg; |

h_{l} | enthalpy of liquid, kJ/kg; |

B | correction factor of free convection heat transfer coefficient, non-dimensional; |

ν | kinematic viscosity, m^{2}/s; |

σ | surface tension coefficient, N/m; |

R_{i} | gas constant for freon vapor, kJ/(kg·K); |

ρ_{g} | gas density, kg/m^{3}; |

q_{conv.} | heat flux density at convective heat transfer kW/m^{2}; |

Nu | Nusselt number, non-dimensional; |

Re | Reynolds number, non-dimensional; |

Pr | Prandtl number, non-dimensional; |

ξ | correction factor that takes into account the features of the flow in a given system, non-dimensional; |

t_{ev. wall} | evaporator wall temperature, °C; |

t_{PEC wall} | PVP surface temperature, °C; |

q | heat flow through PVP, kW/m^{2}; |

R | thermal resistance of layers (protective glass, silicon wafers, etc.), (m^{2}·K)/kW; |

α | heat transfer coefficient, kW/(m^{2}·K); |

Ρ″ | saturated steam density, kg/m^{3}; |

ρ | density of a saturated liquid, kg/m^{3}; |

T_{s} | saturation temperature, K; |

p | pressure, Pa; |

μ | dynamic viscosity, Pa·s; |

ω | flow velocity, m/s; |

d | tube inner diameter, m; |

ε | correction factor taking into account the heating of the liquid from the wall, non-dimensional; |

α_{conv.} | convective heat transfer coefficient, kW/(m^{2}·K); |

α_{boil.} | heat transfer coefficient for freon boiling, kW/(m^{2}·K); |

λ_{wall} | thermal conductivity of the wall, kW/(m·K); |

δ | wall thickness, m; |

Q_{boil.} | heat of saturated steam, kW; |

x | degree of dryness, non-dimensional; |

r | heat of vaporization, kJ/kg; |

l_{ev.} | evaporation section length, m; |

l_{s.h} | refrigerant superheater section, m; |

l_{t} | total length of the PVP section, m; |

t_{f.out} | freon outlet temperature, °C; |

c_{p}″ | freon steam heat capacity, kJ/(kg·K); |

$\frac{\partial (\mathsf{\rho}E)}{\partial t}$ | partial differential of unstable heat transfer, kJ; |

$\nabla \left[V(\mathsf{\rho}E+\mathsf{\rho})\right]$ | a term that takes into account thermal conductivity, kJ; |

${k}_{ef}\Delta T$ | a term that takes into account the diffusion of mixtures, kJ; |

$\sum _{j}{h}_{j}{J}_{j}$ | sum taking into account viscous dissipation, kJ; |

${S}_{h}$ | a term that takes into account the enthalpy of inflows/drains, kJ; |

$\tilde{u}$ | current velocity, m/s; |

ρ_{0} | liquid density at some equilibrium temperature T_{0,} kg/m^{3}; |

Θ | deviation of temperature from equilibrium state (Θ = T − T_{0}), K; |

χ | coefficient of thermal diffusivity, m^{2}/s; |

$\overrightarrow{g}$ | acceleration of gravity, m/s^{2}; |

β | volume expansion coefficient, K^{−1}. |

## References

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**Figure 1.**Designed experimental stand: (

**a**) side view (overall dimensions of the stand: height—1706 mm, width—610 mm, depth—642 mm, angle of inclination of the photovoltaic panel—55 degrees, thickness of the photovoltaic panel redesigned for a thickness of 5 mm); (

**b**) isometry; (

**c**) basic diagram of the energy technological complex: Designations are as follows: 1—low-temperature heat source, 2—evaporator, 3—photovoltaic panel, 4—additional (backup) heater, 5—compressor, 6—condenser, 7—expander (capillary tube), 8—heated water, 9—valve, 10—main desalination tank, 11—steam compressor, 12—distillate heat exchanger-condenser, 13—desalinated water outlet, 14—salt water outlet.

**Figure 2.**Heat pump cycle in the P-i diagram. Symbols and units of measurement: red lines—temperature, °C, blue lines—entropy, kJ/(kg·K), green lines—specific volume, m

^{3}/kg, horizontally—enthalpy in kJ/kg, vertically—pressure in bar.

**Figure 3.**Phase transitions: (

**a**) Emulsion flow mode of the vapor-liquid mixture; (

**b**) Proportion of the liquid phase in the tube cross section, non-dimensional; (

**c**) Ring flow mode (steam inside the liquid); (

**d**) Dispersed flow mode (drops inside the steam). (

**a**,

**b**,

**d**) are borrowed from the literature [13], (

**c**) is the author’s numerical simulation (not described in the literature).

**Figure 4.**Block diagram of the calculation by the Gauss method. Designations of physical quantities are given in the nomenclature.

**Figure 5.**Dependence of solar panel wall temperature on ambient temperature and wind speed: the figure was developed by the authors based on the results of theoretical calculation in a specialized dynamic simulation program Math Lab Simulink.

**Figure 6.**Model building steps: (

**a**) Geometry of the evaporator model; (

**b**) Mesh structure of the evaporator tube.

**Figure 7.**Post-processing: (

**a**) Fluctuations of the discrepancy calculations of the dispersion-ring mode, non-dimensional; (

**b**) Convergence of the degree of dryness at the evaporator output port, non-dimensional; (

**c**) Mixing of the vapor-liquid mixture (Transition k-kl-ω), non-dimensional; (

**d**) Temperature field (Transition k-kl-ω), °C; (

**e**) Share of vapor in the volume (Transition k-kl-ω), non-dimensional.

Title | Designation | Calculation Formula or Method of Determination | Units of Measurement | Result |
---|---|---|---|---|

Heat flow | Q_{sol} | Given | W/m^{2} | 700 |

Thermal conductivity of silicon | λ_{sil} | Given | W/(m^{2}·K) | 148 |

Absorption capacity of silicon | A_{sil} | Given | - | 0.2 |

Degree of silicon blackness | ε_{sil} | Given | - | 0.71 |

Silicon thickness | δ_{sil} | Given | m | 0.001 |

Stefan-Boltzmann coefficient | σ | Given | W·m^{−2}·K^{−4} | 5.67 × 10^{−8} |

Thermal conductivity of glass | λ_{gl} | Given | W/(m^{2}·K) | 0.96 |

Absorption capacity of glass | A_{gl} | Given | - | 0.1 |

The degree of blackness of the glass | ε_{gl} | Given | - | 0.93 |

Glass thickness | δ_{gl} | Given | m | 0.002 |

Length and width of the PVP | l | Given | m | 0.15675 |

PVP Square | F | ${l}^{2}$ | m^{2} | 0.025 |

Reflectivity of silicon | R_{sil} | Given | - | 0.33 |

Heat absorbed by the PVP | Q | ${Q}_{sol}\cdot F\left(1-{R}_{sil}\right)$ | W | 11.524 |

Heat absorbed by silicon | Q_{abs.sil} | ${Q}_{sol}\cdot F\cdot {A}_{sil}$ | W | 3.44 |

Heat passed through silicon | Q_{past sil.} | ${Q}_{sol}\cdot F\left(1-{R}_{sil}-{A}_{sil}\right)$ | W | 8.084 |

Heat absorbed by glass | Q_{abs.gl.} | ${Q}_{sol}\cdot F\cdot {A}_{gl}$ | W | 1.72 |

Heat passed through the glass | Q_{past gl.} | ${Q}_{sol}\cdot F\left(1-{R}_{sil}-{A}_{sil}-{A}_{gl}\right)$ | W | 6.364 |

Air velocity | ω | Given | m^{2}/s | 1 |

Thermal conductivity of air | λ | Given | W/(m^{2}·K) | 2.4 × 10^{−2} |

Air viscosity | ν_{air} | Given | m^{2}/s | 14.61 × 10^{−6} |

Prandtl number for air | Pr | Given | - | 0.67 |

Ambient air temperature | t_{air} | Selected | °C | 20 |

Ambient air temperature | T_{air} | ${t}_{air}+273$ | K | 293 |

Outside temperature of the PVP | t_{sur}_{1} | Selected | °C | 29.12 |

Outside temperature of the PVP | T_{sur}_{1} | ${t}_{\mathrm{sur}1}+273$ | K | 302.12 |

Reynolds number | Re_{1}, Re_{2} | $\frac{\omega \cdot 1}{{\mathsf{\nu}}_{air}}$ | - | 1.073 × 10^{4} |

Nusselt number | Nu_{1}, Nu_{2} | $0.67R{e}^{0.5}P{r}^{1/3}$ | - | 60.727 |

Heat transfer coefficient | α_{1}, α_{2} | $Nu\frac{\lambda}{1}$ | W/(m^{2}·K) | 9.298 |

Temperature of the inner surface of the PVP | t_{sur}_{2} | ${t}_{\mathrm{sur}1}-\left[{Q}_{sol}\cdot F\left(1-{R}_{sil}-{A}_{gl}-{A}_{gl}\right)\right]$$\times \left(\frac{{\delta}_{sil}}{{\lambda}_{sil}}+\frac{{\delta}_{gl}}{{\lambda}_{gl}}\right)$ | °C | 29.1 |

Losses from above | q_{above} | ${\alpha}_{1}\cdot F\left({T}_{\mathrm{sur}1}-{T}_{air}\right)$ | W | 2.083 |

Silicon radiation losses | q_{sil.rad} | ${\epsilon}_{sil}\cdot \sigma \cdot F\left({T}_{\mathrm{sur}1}^{4}-{T}_{air}^{4}\right)$ | W | 0.951 |

Losses from below | q_{und.} | ${\alpha}_{2}\cdot F\left({T}_{\mathrm{sur}2}-{T}_{air}\right)$ | W | 2.083 |

Glass radiation losses | q_{gl.rad} | ${\epsilon}_{gl}\cdot \sigma \cdot F\left[{\left({T}_{\mathrm{sur}2}^{}+273\right)}^{4}-{T}_{air}^{4}\right]$$+{Q}_{sol}\cdot F\cdot {A}_{sil}+{Q}_{co\u043b}\cdot F\cdot {A}_{gl}$ | W | 6.399 |

Heat losses in the system | Q_{losses} | ${q}_{above}+{q}_{sil.rad}+{q}_{und.}+{q}_{gl.rad}$ | W | 11.516 |

Error rate | ΔQ | $\frac{Q-{Q}_{losses}}{Q}\cdot 100$ | % | 0.067 |

Wind Speed, m/s | 1 | 5 | 10 |
---|---|---|---|

The temperature of the outer wall according to the theoretical calculation, °C | 40.74 | 37.20 | 34.30 |

External wall temperature according to analytical calculation, °C | 40.57 | 37.06 | 34.21 |

Ф | P | D | Γ_{ϕ} | |
---|---|---|---|---|

Kinetic energy | k | ${\mathsf{\tau}}_{y}\frac{\partial {\overline{u}}_{i}}{\partial {x}_{j}}$ | $p\mathsf{\epsilon}$ or ${\mathsf{\beta}}^{\ast}pk\mathsf{\omega}$ | $\frac{{\mathsf{\mu}}_{t}}{{\mathsf{\sigma}}_{k}}$ |

Kinetic energy dissipation rate | ε | ${c}_{\mathsf{\epsilon}1}\frac{\mathsf{\epsilon}}{k}{\mathsf{\tau}}_{ij}\frac{\partial {\overline{u}}_{i}}{\partial {x}_{j}}$ | ${c}_{\mathsf{\epsilon}2}p\frac{{\mathsf{\epsilon}}^{2}}{k}$ | $\frac{{u}_{t}}{{\mathsf{\sigma}}_{e}}$ |

Specific rate of dissipation | ω | $\mathsf{\alpha}\frac{\mathsf{\omega}}{k}{\mathsf{\tau}}_{ij}\frac{\partial {\overline{u}}_{i}}{\partial {x}_{j}}$ | $\mathsf{\beta}p{\mathsf{\omega}}^{2}$ | ${\mathsf{\sigma}}_{\mathsf{\omega}}{u}_{t}$ |

Wind Speed, m/s | 1 | 5 | 10 |
---|---|---|---|

Numerical modeling calculation, °C | 40.23 | 36.94 | 34.10 |

External wall temperature according to analytical calculation, °C | 40.57 | 37.06 | 34.21 |

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## Share and Cite

**MDPI and ACS Style**

Osintsev, K.; Aliukov, S.; Kovalev, A.; Bolkov, Y.; Kuskarbekova, S.; Olinichenko, A.
Scientific Approaches to Solving the Problem of Joint Processes of Bubble Boiling of Refrigerant and Its Movement in a Heat Pump Heat Exchanger. *Energies* **2023**, *16*, 4405.
https://doi.org/10.3390/en16114405

**AMA Style**

Osintsev K, Aliukov S, Kovalev A, Bolkov Y, Kuskarbekova S, Olinichenko A.
Scientific Approaches to Solving the Problem of Joint Processes of Bubble Boiling of Refrigerant and Its Movement in a Heat Pump Heat Exchanger. *Energies*. 2023; 16(11):4405.
https://doi.org/10.3390/en16114405

**Chicago/Turabian Style**

Osintsev, Konstantin, Sergei Aliukov, Anton Kovalev, Yaroslav Bolkov, Sulpan Kuskarbekova, and Alyona Olinichenko.
2023. "Scientific Approaches to Solving the Problem of Joint Processes of Bubble Boiling of Refrigerant and Its Movement in a Heat Pump Heat Exchanger" *Energies* 16, no. 11: 4405.
https://doi.org/10.3390/en16114405