# Enhancing Small Heat Source Performance through Gravitational Loop Heat Pipes

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## Abstract

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## 1. Introduction

## 2. Device for Small Heat Source Efficiency Improvement

#### 2.1. Container Materials

#### 2.2. Working Medium

#### 2.3. Design of the Proposed Device

## 3. Mathematical Model

#### 3.1. Evaporator

_{c}is the total heat transfer by convection, and S

_{he}is the surface area of the heat exchanger. Convection facilitates efficient heat transfer from the flue gases to the evaporator walls. The heat transfer rate q is calculated using Fourier’s law of thermal conduction (2).

_{ss}represents stainless-steel conductivity. This coefficient characterizes the efficiency of heat transfer and is crucial for analyzing thermal conductivity in a system.

_{δ}(5). This temperature plays a significant role in determining the thermal conductivity value.

_{L}denotes the Reynolds number based on the length L. This coefficient characterizes the average surface losses within the flow and is derived from scientific principles. It provides valuable insights into the heat transfer behavior in the system.

_{L}, is a dimensionless parameter used to characterize the flow regime. For laminar flow conditions (7), w represents the velocity of the fluid, L is the characteristic length, and v denotes the kinematic viscosity of the fluid. This criterion indicates that the flow is considered laminar when the Reynolds number falls below the specified threshold value of 3 × 10

^{5}.

_{L}is less than 10

^{5}and the Prandtl number Pr falls within the range of 0.1 to 1000, the Nusselt number Nu can be determined using the correlation Equation (9).

_{s}) accounts for the ratio of fluid viscosities between the working fluid and the reference fluid.

_{1}and the small heat source A

_{2}. The external radiation heat transfer, denoted as Q

_{r,ext}, is determined by the emissivity of the surfaces ε, which affects their ability to emit and absorb thermal radiation. The equation for Q

_{r,ext}is given by Equation (12), where the temperatures of the evaporator and the fireplace insert are denoted as T

_{1}and T

_{2}, respectively, and C

_{o}is a constant.

_{12}, is determined by the individual emissivities of the surfaces ε

_{1}and ε

_{2}and their respective areas A

_{1}and A

_{2}. Radiant flux between surfaces A

_{1}and A

_{2}forming a closed system is shown on Figure 4.

_{12}(13) considers the interaction of the two surfaces in the radiation heat transfer process.

_{B}, Prandtl number Pr

_{k}, and pressure p. In the case of bubble boiling, there are two criterial equations that describe the Nusselt number Nu

_{B}for different ranges of Reynolds number. Equation (14) applies to ${10}^{-5}\le R{e}_{B}\le {10}^{-2}$,while Equation (15) applies to ${10}^{-2}\le R{e}_{B}\le {10}^{4}$.

_{B}, Re

_{B}, and Pr

_{k}, allowing for the estimation of heat transfer performance in boiling systems within the specified range. It is important to note that the Prandtl number should be between 0.86 and 7.6, and the pressure should range from 4500 to 17.5 × 10

^{6}Pa. These criterial equations serve as valuable tools in the design and optimization of heat transfer systems involving boiling, enhancing our understanding and control of heat transfer processes in such applications.

_{B}is a key parameter that quantifies the convective heat transfer. It is defined by Equation (16), where h represents the convective heat transfer coefficient, B is a characteristic length, and λ

_{k}is the thermal conductivity of the fluid.

_{B}characterizes the fluid flow behavior in boiling. It is determined using Equation (17), where q is the heat flux, B is a characteristic length, l

_{b}is the latent heat, ρ

_{s}is the density of the fluid, and ν

_{c}is the kinematic viscosity.

_{k}describes the relative importance of momentum diffusion to thermal diffusion in boiling. It is determined by Equation (18), where ν

_{c}is the kinematic viscosity and α

_{c}is the thermal diffusivity of the fluid. The Prandtl number helps in understanding the relative rate of momentum and heat transfer during boiling.

_{s}is the specific heat capacity, ρ

_{c}is the density of the solid, σ is the Stefan–Boltzmann constant, T

_{s}is the surface temperature, l

_{b}is the latent heat, and ρ

_{c}is the density of the fluid. The characteristic length B accounts for the specific thermodynamic properties of the system and enables a better understanding of the heat transfer phenomena during boiling processes.

^{5}Pa, the heat transfer coefficient h can be calculated using Equations (20) and (21), where ∆T is the temperature difference and q is the heat flux.

#### 3.2. Condenser

_{α}, the Reynolds number Re, and the Prandtl number Pr.

_{1}, n

_{r}, l

_{1}, ϕ, S

_{r}, S

_{t}, and η

_{r}represent geometric parameters associated with the arrangement of the tube bundle.

_{α}and n are determined on the basis of the specific geometric attributes of the tubes (Table 1). For the specific configuration of tubes arranged in a linear fashion, the value of G

_{α}is unity. This simplification is applicable to the arrangement of the four ribbed tubes within the condenser. By considering these geometric parameters and employing the pertinent equations, the hydraulic and thermal characteristics of the condenser system can be effectively evaluated with enhanced precision.

_{e}represents the equivalent diameter of the ribbing, which is considered a characteristic dimension.

_{h}is employed to estimate d

_{e}and is calculated using relationship (27). In this expression, S

_{t}denotes the total surface area, d

_{t}refers to the tube diameter, S

_{r}represents the rib surface area, and n

_{r}corresponds to the number of ribs per tube.

_{v}is the density of the vapor, g denotes the acceleration due to gravity, ρ

_{p}corresponds to the density of the fluid, h

_{fg}represents the average latent heat of vaporization, D represents the characteristic length or diameter, μ

_{v}is the dynamic viscosity of the vapor, k

_{v}denotes the thermal conductivity of the vapor, T

_{sat}is the saturation temperature, and T

_{s}represents the surface temperature.

_{s}

_{1}, can be calculated using Equation (33)

_{s}

_{2}, can be determined by Equation (34).

_{m}, can be determined by Equation (35).

_{1}and t

_{2}represent the inlet temperatures of the first and second fluids, α

_{1}and α

_{2}are the heat transfer coefficients of the respective fluids, S

_{1}and S

_{2}are the heat transfer surface areas on the respective sides, D

_{1}and D

_{2}are the diameters of the channels, L is the length of the heat exchanger, and λ

_{ekv}represents the equivalent thermal conductivity of the composite wall.

#### 3.3. Results

_{r,int}, which acts upon the inner wall of the evaporator. Remarkably, this radiative heat transfer rate surpasses the convective heat outputs, Q

_{k,ext}and Q

_{k,int}, on the outer and inner surfaces of the evaporator, respectively. Furthermore, the heat outputs on the inner surface of the evaporator demonstrate higher magnitudes compared to those observed on the outer shell of the evaporator.

_{22}also rose. Consequently, the temperature difference between the combustion air and steam increased. This larger temperature difference enhanced the efficiency of heat transfer between the condenser and the combustion air, leading to improved heat exchange performance.

_{c}and the steam mass flow rate ṁ

_{s}, a point was identified with equal temperature and pressure, where the amount of evaporated condensate was equal to the amount of condensed condensate. The cooperation between the evaporator and the condenser is contingent upon maintaining identical temperature and pressure in both devices. The collaboration of the evaporator and the condenser occurs at a steam temperature of 160 °C and a saturated steam pressure of 618.3 kPa.

## 4. Experimental Verification

_{2}) content, carbon dioxide (CO

_{2}) concentration, and carbon monoxide (CO) levels. These measurements helped assess the combustion efficiency and environmental impact of the gas fireplace insert.

## 5. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 6.**Heat output by the evaporator and the temperature of the flue gases at the outlet of the evaporator depending on the boiling temperature.

**Figure 7.**Heat output by the condenser and at the outlet of the evaporator depending on condensation temperature.

**Figure 10.**Course of measured values on a fireplace insert without LHP filling (nomenclature in Figure 9).

**Figure 11.**Course of measured values on a fireplace insert with filling of 0.1 L of water in LHP (nomenclature in Figure 9).

Reynolds Number | K_{α} | n |

$Re\le 1500\frac{{d}_{e}}{{d}_{h}}$ | $1.15{\left(\frac{{d}_{e}}{{d}_{h}}\right)}^{0.22}$ | 0.41 |

$1500\frac{{d}_{e}}{{d}_{h}}<Re\le 70,000$ | 0.23 | 0.63 |

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## Share and Cite

**MDPI and ACS Style**

Martvoňová, L.; Malcho, M.; Jandačka, J.; Ďurčanský, P.; Holubčík, M.; Drozda, J.; Adamička, M.
Enhancing Small Heat Source Performance through Gravitational Loop Heat Pipes. *Machines* **2023**, *11*, 813.
https://doi.org/10.3390/machines11080813

**AMA Style**

Martvoňová L, Malcho M, Jandačka J, Ďurčanský P, Holubčík M, Drozda J, Adamička M.
Enhancing Small Heat Source Performance through Gravitational Loop Heat Pipes. *Machines*. 2023; 11(8):813.
https://doi.org/10.3390/machines11080813

**Chicago/Turabian Style**

Martvoňová, Lucia, Milan Malcho, Jozef Jandačka, Peter Ďurčanský, Michal Holubčík, Július Drozda, and Martin Adamička.
2023. "Enhancing Small Heat Source Performance through Gravitational Loop Heat Pipes" *Machines* 11, no. 8: 813.
https://doi.org/10.3390/machines11080813