# Experimental and Theoretical Study on Operation Characteristics of an Oscillating Heat Pipe
## 3 Mathematical Analysis
### Extended Thermal Resistance  
As mentioned in the literature, the thermal performance of OHPs has been experimentally studied for different structures, working fluids, and other operating conditions. However, no formulation was proposed to evaluate the thermal performance of OHPs theoretically, which limits the wide application of findings in the literature. It is well known that the liquid film heat conduction and phase transition are the major factors influencing thermal resistance. In this paper, an empirical correlation is proposed to describe the relationship between thermal resistance and heat input based on experimental and mathematical results.  
According to the literature [5; 6; 24], the thermal resistance of the OHP mainly depends on the radial thermal conductivity. The other influencing factors could be corrected using a correlation coefficient. Using this assumption, the relationship between the heat input and liquid film thickness is almost linear, and, hence, the thermal resistance can be determined as follows:  
\[R=\varepsilon\cdot\frac{1}{k_{r}}+\theta\cdot\frac{1}{k_{a}} \tag{16}\]  
\[Q=\frac{1}{\varphi\delta_{lf}} \tag{17}\]  
where the two terms in the right hand of Equation (16) are the radial heat conduction term and the axial heat conduction term, respectively. \(E\) and \(\theta\) are the regression coefficients for thermal resistance, which are measures of the importance of radial and axial heat conductions relative to the thermal resistance. These two coefficients are influenced by the geometric parameters such as thickness and area as well as the OHP characteristics such as filling ratio and working fluid. \(\varphi\) is the regression coefficient for heat inputs. It reflects the relationship between heat input and liquid film thickness. The correlation function of \(\varphi\) leads the value of \(1/(\varphi\cdot Q)\) to approach the magnitude of \(\delta_{lf}\). To simplify the calculation, Equation (13) can be substituted into Equation (16).  
\[R=\varepsilon\Bigg{[}\frac{1}{k_{w}}+\frac{\delta_{lf}}{k_{lf}\delta_{w}} \Bigg{]}+\theta\cdot\frac{1}{k_{a}} \tag{18}\]Substituting Equation (17) in Equation (18), the equivalent thermal resistance formula relative to the heat input can be expressed as follows:  
\[R=\frac{\varepsilon}{\varphi}\cdot\frac{1}{k_{lf}\delta_{w}}\cdot\frac{1}{Q}+ \left(\theta\cdot\frac{1}{k_{a}}+\varepsilon\cdot\frac{1}{k_{w}}\right) \tag{19}\]  
According to Equation (19), the linear correlation is expressed. The slope is described by the liquid film term. The intercept is described as a function of the axial thermal conductivity term and the wall thermal conductivity term. Then, Equation (18) can be expressed as follows:  
\[R=\frac{\varepsilon}{\varphi}\cdot\frac{1}{k_{lf}\delta_{w}}\cdot\frac{1}{Q}+ C(k_{w},k_{a}) \tag{20}\]  
where \(C(k_{w},k_{a})\) is a constant associated with the thermal conductivities \(k_{w}\) and \(k_{a}\) for a given condition. The value of \(\varepsilon/\varphi\) and \(C(k_{w},k_{a})\) can be determined by the experimental data. The validation of this correlation will be discussed in the following section.