# Operational Temperature Effect on Positioning Accuracy of a Single-Axial Moving Carrier
## 3 Results and Discussions
### Theoretical Models and Numerical Results of Flow and Temperature Fields
By solving the closed turbulence problem through the \(k\)-\(\varepsilon\) model (kinetic turbulent energy equation and turbulent dissipation equation), the turbulent stress and thermal flux can be acquired. The commercial software, ANSYS Fluent (15.0, ANSYS. Inc, Cecil, PA, USA, 2014), was adopted in the simulation of the flow field and temperature field within the processing area. In this study, the flow fields of the internal and external environments were filled with an ordinary air flow, with its physical parameters listed in Table 3. The carrier and linear guideway components used in this study were made of cast iron and stainless steel, respectively, with their relevant physical parameters listed in Table 4. The heat generating rates from the heat sources of the linear guideway and linear motor were 7575 and 1000 W/m\({}^{3}\), respectively, and were estimated from the simulation results and measurement data. To simulate the movement of the carrier on the guideway, dynamic grids were adopted in the numerical simulation. From the above experimental results, it was found that an operational temperature variation may cause a great increase in positioning accuracy, especially in the operational condition of a temperature \begin{table}
\begin{tabular}{c c c} \hline \hline
**Physical Properties** & **Cast Iron** & **Stainless Steel** \\ \hline Density (kg/m\({}^{3}\)) & 7850 & 7900 \\ Specific heat (J/(kg\(\cdot\)K)) & 440 & 500 \\ Heat conductivity (W/(m\(\cdot\)K)) & 46 & 16 \\ Thermal diffusivity (m\({}^{2}\)/s) & 0.0000133 & 0.0000041 \\ Thermal Expansion (\(\mu\)m/(m\(\cdot\)K)) & 10.5 & 17 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Material parameters of stainless steel and cast iron. \begin{table}
\begin{tabular}{c c} \hline \hline
**Density** & **1.1824 (kg/m\({}^{3}\))** \\ \hline Specific heat & 1005 (J/(kg\(\cdot\)K)) \\ Heat conductivity & 0.0261 (W/(m\(\cdot\)K)) \\ Viscosity & 1.834 \(\times\) 10\({}^{-5}\) (kg/(m s)) \\ Coefficient of heat transfer & 0.00326 (W/(m\({}^{2}\cdot\)K)) \\ Prandtl number & 0.708 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Air flow physical parameters. decline. To maintain a stable operational environment for a moving machine, the moving machine can be operated within an environmental chamber in which the temperature distributions can be controlled by an air conditioning system. Figure 7 illustrates the numerical average temperature distributions on a two-dimensional plane above the moving carrier along the moving path of the periodically-moving carrier. In Figure 7, each average value corresponding to the individual position above the moving carrier was calculated by averaging temperatures along the lateral direction (\(Z\) axis) above the carrier during a moving period. From the average temperature distributions, the temperature deviations in the cases of different inlet/outlet vent locations can be presented. As shown in the figure, the air velocity at the inlets in cases A, B, C, and D was set to 2.36, 1.18, 2.36, and 1.18 m/s, respectively. In order to maintain a stable and stratified flow field in the region near the moving machine, a laminar flow should be ensured in the corresponding region. For fluid dynamics in the internal flow, the non-dimensional parameter, the Reynolds number, should be below 2300. The definition of a Reynolds number is \(Re=\rho VD/\mu\), where \(\rho\) in the density, \(\mu\) is the viscosity of the fluid, \(V\) is the characteristic velocity and \(D\) is the characteristic dimension. In this study, the setting inlet velocities of 1.18 and 2.36 m/s were adopted, and the characteristic velocities around the moving machine were about 0.08 and 0.12 m/s, respectively. The corresponding Reynolds numbers for both cases were 1289 and 1934, respectively, which are both less than the critical value of 2300.