# 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  
In order to understand the effect of flow patterns on positioning accuracy within an environmental chamber, flow fields and temperature fields were simulated. To simplify the simulation, we assume that the air is in an environment containing low air velocity and is non-compressible, and the flow field is in an unsteady-state in terms of the computational domain. Navier-Stokes equations within the \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multirow{2}{*}{**Time Point**} & \multirow{2}{*}{\begin{tabular}{c} **Chamber** \\ **Temperature** \\ **(\(\mathrm{\SIUnitSymbolCelsius}\)****** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Time** \\ **(\(\mathrm{\SIUnitSymbolCelsius}\)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Positioning** \\ **(\(\mathrm{\SIUnitSymbolMicro m}\)**)** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Variation of** \\ **Positioning** \\ **Accuracy** (\(\mathrm{\SIUnitSymbolMicro m}\)**)** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Variation Rate of** \\ **Positioning Accuracy** \\ **(\(\mathrm{\SIUnitSymbolMicro m}\)**/ **(\(\mathrm{\SIUnitSymbolMicro m}\)**-\(\mathrm{\SIUnitSymbolCelsius}\)**)** \\ \end{tabular} } \\ \hline \multirow{5}{*}{5} & Summer 0 & 23.5 & 0 & 12.42 & 0 & 0 \\  & Summer 180 & 31.5 & 180 & 11.85 & \(-\)0.57 & \(-\)0.00040 \\  & Summer 213 & 31.8 & 213 & 11.33 & \(-\)0.52 & -0.05253 \\  & Summer 338 & 23.5 & 338 & 15.74 & 4.41 & 0.00425 \\  & Summer 386 & 23.3 & 386 & 11.85 & \(-\)3.89 & \(-\)0.40521 \\ \hline \multirow{5}{*}{6} & Winter 0 & 23.7 & 0 & 13.42 & 0 & 0 \\  & Winter 43 & 23.1 & 43 & 10.45 & \(-\)2.97 & \(-\)0.11512 \\ \cline{1-1}  & Winter 249 & 15.4 & 249 & 24.06 & 13.61 & 0.00858 \\ \cline{1-1}  & Winter 362 & 22.5 & 362 & 27.09 & 3.03 & 0.00378 \\ \cline{1-1}  & Winter 391 & 22.7 & 391 & 26.61 & \(-\)0.48 & \(-\)0.08276 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Corresponding positioning accuracies and corresponding variations of a periodically-moving carrier operating in the summer daytime and winter nighttime modes, respectively. Figure 6: Temperature distributions within the environmental chamber according to time under summer daytime and winter nighttime operational conditions. time average featuring the unsteady-state and three-dimensional non-compressible flow fields can be described as follows:  
\[\frac{\partial U_{j}}{\partial x_{j}}\ =\ 0\,\ j\ =\ 1,\ 2,\ 3 \tag{1}\]  
\[\frac{\partial\big{(}U_{i}U_{j}\big{)}}{\partial x_{j}}\ =\ -\ \frac{1}{\rho}\frac{ \partial P}{\partial x_{j}}\ -\ g_{i}\beta(T-T_{\infty})\ +\ \frac{\partial}{ \partial x_{j}}\Bigg{[}v\frac{\partial U_{i}}{\partial x_{j}}-\overline{u_{i} u_{j}}\Bigg{]} \tag{2}\]  
\[\frac{\partial\big{(}TU_{j}\big{)}}{\partial x_{j}}\ =\ \frac{\partial}{ \partial x_{j}}\Bigg{[}\frac{v}{P_{r}}\frac{\partial T}{\partial x_{j}}\ -\ \overline{T^{\prime}u_{j}}\Bigg{]} \tag{3}\]  
In the equations, \(U_{j}\), \(T\), and \(P\) represent the average velocity, average temperature, and average pressure, respectively. The Boussinesq buoyancy effect was taken into consideration, and \(\beta\) refers to the coefficient of thermal expansion. With a closed turbulence model, the turbulent stress \(-\overline{u_{i}u_{j}}\) and turbulent thermal flux \(-\overline{T^{\prime}u_{j}}\) can be obtained by the following expression:  
\[-\overline{u_{i}u_{j}}\ =\ 2v_{t}S_{ij}\ -\ \frac{2}{3}k\delta_{ij}-\ \overline{T^{ \prime u_{j}}}\ =\ \frac{v_{t}}{Pr_{t}}\Bigg{[}\frac{\partial T}{\partial x_{j}}\Bigg{]} \tag{4}\]  
In the expression, \(v_{t}\) and \(Pr_{t}\) represent the turbulent flow viscosity and the turbulent Prandtl number, respectively. \(K\) refers to the kinetic turbulent energy, while \(k=\overline{u_{x}^{2}}+\overline{u_{y}^{2}}+\overline{u_{z}^{2}}/3\) and \(\delta_{ij}\) indicates the Kronecker delta.