## 4 Spatial Resolution of the Measurements
### 2D Diffusion Influence  
In order to assess the validity of a 1D approach, some calculations using a 2D description of the heater were performed. The general diffusion Equation (1) can be written for a 2D approach as:  
\[\boxed{\rho C_{P}\frac{\partial T}{\partial t}=\lambda(\frac{\partial^{2}T}{ \partial x^{2}}+\frac{\partial^{2}T}{\partial y^{2}})+q^{\prime\prime\prime}} \tag{23}\]  
where \(x\) and \(y\) are the space variables. The boundary conditions are:  
\[\boxed{ \begin{array}{l}\text{Boundary conditions:}\qquad\frac{\partial T}{ \partial x}(0,y,t)=0\\ T(e,y,t)=T_{d,wet}(y,t)\\ \frac{\partial T}{\partial y}(x,0,t)=0\\ \frac{\partial T}{\partial y}(x,l,t)=0\\ \end{array} } \tag{24}\]  
The temperature on the wet side of the plate \(T_{wet}\) is given by:  
\[\boxed{\begin{array}{l}\text{for }t<t_{g}\text{ and }y=y_{0}\\ \text{for }t<t_{g}<t_{g}+t_{w}\text{ and }y=y_{0}\\ \text{\The initial condition is:  
\[T(x,y,0)=T_{d,initial}(x,y) \tag{26}\]  
where \(T_{d,initial}(x,y)\) is the solution of the steady-state problem where the boundary conditions are (\(T_{H}\) is the same as defined in Figure 6):  
\[ \begin{array}{ll}\boxed{\text{Problem equation:}}&\lambda(\frac{\partial^{2}T}{ \partial x^{2}}+\frac{\partial^{2}T}{\partial y^{2}})=-q^{\prime\prime\prime} \\ \text{Boundary conditions:}&\frac{\partial T}{\partial x}(0,y)=0\\ \frac{T}{T}(e,y)=T_{H}\\ \frac{\partial T}{\partial y}(x,0)=0\\ \frac{\partial T}{\partial y}(x,l)=0\end{array}  \tag{27}\]  
The 2D steady-state temperature distribution is presented in Figure 17.  
Equations (23)-(27) are solved by finite differences using a second-order central scheme for the Laplacian and an implicit scheme for time dependence. The temperature \(T(0,t)\) is then calculated. This temperature is noticed \(T_{dry}\). Figure 18 displays, for different thicknesses, the influence of the 2D effects on the temperature beneath the nucleation site at \(y=y_{0}\). This figure aims to point out the diffusion effect between 1D and 2D models. The yellow curves are the 1D direct result of the conduction within the plate, whereas the blue curve depicts the 2D direct calculations of this conduction problem. It shows that the gap between 1D and 2D simulations is decreasing with the thickness: from a maximum gap of 0.21 \({}^{\circ}\)C at 100 \(\upmu\)m to 0.07 \({}^{\circ}\)C at 50 \(\upmu\)m. This gap will not affect the 1D conclusion over the limit thickness. The 1D modeling is then sufficient to raise a first assessment of the plate geometry, as long as the bubble size is not smaller than the thickness of the plate. Figure 16: Diagram of the 2D model with a bubble growing on \(y=y_{0}\). wall (left side), the temperature variation studied in 1D appears on a zone, defined here as being the bubble diameter. The temperature variation on the dry side then appears to be attenuated in amplitude, and the influence area is expanded because of 2D diffusion effects. It is important to check that this temperature variation is measurable in two steps: (i) check that the amplitude \(A\) of this variation is measurable and, therefore, larger than the resolution of the camera (its ability to detect a temperature variation), (ii) check that the size of the spot \(d\) is larger than a camera pixel, to be able to detect it. Let us recall that the aim of these measurements is to detect the timescales of the bubble growth on the plate from low to high-pressure conditions. Unal's correlation [3] gives an order of magnitude of the bubble diameter as a function of the thermal-hydraulic conditions. For our prototypical conditions, the bubble diameter ranges from 40 \(\upmu\)m to 800 \(\upmu\)m for pressure from 110 to 1 bar, respectively. For each of these conditions (pressure, bubble size), the maximum thickness over which the bubble cannot be detected by IR measurements has to be determined. The IR camera generally presents a theoretical temperature resolution of 50 mK (a good calibration must be carried out beforehand).