# Experimental and Theoretical Study on Operation Characteristics of an Oscillating Heat Pipe
## 3 Mathematical Analysis  
### Equivalent Thermal Conductivity  
To better understand the thermal performance of the OHP, the equivalent thermal conductivity is deduced theoretically. The OHP is simplified as a unit consisting of one liquid plug and two halves of the vapor bubbles. The OHP is assumed to be two-dimensional. Figure 3 shows the simplified model.  
When the heat flux direction is parallel to the axial direction, the thermal resistance can be described as a parallel thermal resistance circuit. The temperature gradient across the OHP is the same, while the heat fluxes are different. The effect of the liquid film region on the axial heat conduction is ignored due to its small magnitude. Therefore, the axial thermal conductivity is determined by the following:  
\[k_{a}=\frac{d_{i}^{2}}{d_{o}^{2}}k_{l}+\left(1-\frac{d_{i}^{2}}{d_{o}^{2}} \right)k_{w} \tag{11}\]  
where \(k_{a}\) is the axial thermal conductivity, W/(m\(\cdot\)K); \(d_{i}\) is the inner diameter, m; \(d_{o}\) is the outer diameter, m; \(k_{l}\) is the thermal conductivity of the liquid, W/(m\(\cdot\)K); and \(k_{w}\) is the thermal conductivity of the copper, W/(m\(\cdot\)K).  
When the heat flux direction is perpendicular to the axial direction, the thermal resistance can be described as a series thermal resistance circuit. The heat flux passed through the wall and the liquid film. Assuming that the wall and the liquid film have the same flow area, the heat fluxes are equal. The radial thermal conductivity is determined as follows:  
\[\frac{1}{k_{r}}=\frac{\delta_{w}}{k_{w}\left(\delta_{w}+\delta_{lf}(x)\right) }+\frac{\delta_{lf}}{k_{lf}\left(\delta_{w}+\delta_{lf}(x)\right)} \tag{12}\]  
where \(k_{r}\) is the radial thermal conductivity, W/(m\(\cdot\)K); and \(\delta_{lf}\) is the liquid film thickness, m.  
Figure 3: A simplified model of a single slug and plug unit.  
According to the literature [23; 24], the magnitude of the liquid film is around \(10^{-8}\sim 10^{-6}\), \(\delta_{w}\gg\delta_{lf}\). It is assumed that the influence of the absorbed region is negligible, Equation (12) can be expressed as follows:  
\[\frac{1}{k_{r}}=\frac{1}{k_{w}}+\frac{\delta_{lf}}{k_{lf}\delta_{w}} \tag{13}\]  
Substituting the thermal conductivity of the copper and liquid film into Equation (13), the value of characteristic thermal conductivity can be calculated. When the thickness of the liquid film becomes smaller, heat is transferred more effectively through the liquid film.  
For copper and water, the thermal conductivity can be calculated using a correlation that is a function of temperature [25]. It is given by the following:  
\[k=A+B\cdot\mathrm{T}+C\cdot T^{2} \tag{14}\]  
For inorganic compounds, the correlation is expressed as follows:  
\[\log k=A+B\bigg{(}1-\frac{T}{C}\bigg{)}^{\frac{2}{7}} \tag{15}\]  
where \(A\), \(B\), and \(C\) are the regression coefficients for a chemical compound, and \(T\) is in Kelvin temperature scale. In this study, the mean temperature (70 \({}^{\circ}\)C) of the evaporation section after heating for twenty minutes is chosen as the reference temperature.