## 3 Thermal Study of the Heater
### Inverse Problem
#### 3.3.3 Results  
The inverse problem was solved with finite differences (similar to the direct problem). Figure 14 shows the steps to solve the inverse problem. The direct problem was solved by imposing a temperature variation over the wet side of the plate (internal face) \(T_{d,wet}\). The solution is the temperature variation on the external face of the plate \(T_{d,dry}\). The inverse problem is then solved by imposing on the external face the temperature as \(T_{d,dry}\), resulting from the direct problem calculation. The solution to the inverse problem is the temperature on the internal face \(T_{i,wet}\), which can be compared with the one imposed first \(T_{d,wet}\).  
Figure 14 shows that the inverse solution for the wet temperature is valid: the exact same wet temperature imposed at the beginning for any parameter is calculated (blue and green curves).  
As with all inverse problems, it is sensitive to boundary condition uncertainties. A random uncertainty (white noise) of \(\pm 0.5\)\({}^{\circ}\)C is then applied to the measurement on the outside (dry temperature).  
Imposing this uncertainty to the dry-side temperature \(T_{d}(0,t)\), one can observe (as a consequence) amplification of the noise for the wet temperature \(T_{i,wet}\).  
Figure 15 shows the influence of the plate thickness on the accuracy of the calculated wet temperature. The blue and red triangles are the initial results for the dry and wet temperatures (without any uncertainties). The yellow crosses correspond to the dry temperature taking into account the uncertainty. The blue crosses are the results of the inverse problem with the noisy dry calculated temperature. For a thickness of 100 \(\upmu\)m, the amplification leads to the impossibility of finding the time scales that are to be measured,  
Figure 14: Steps to solve the inverse problem, (a) solving the direct problem by imposing the wet temperature (blue curve, 1) to get the dry temperature (2), (b) solving the inverse problem by imposing the dry temperature (red curve, 3), calculated before (2) to get the wet temperature (4).  
especially the bubble time detachment. Decreasing the thickness leads to decreasing the noise on the reconstructed wet temperature.  
A 70 um thickness enables detecting the change of slope and, therefore, the time of detachment of the bubble, which makes it possible to measure the frequency of bubble detachment from the wall. However, the apparent noise shows an oscillation of 5 \({}^{\circ}\)C around the actual temperature value. For the study of boiling that is envisaged on the future device, a more accurate temperature measurement may be of interest. Indeed, the classical wall-superheat models indicate that a few degrees of variation are expected for the given set of conditions, and the noise of the measurements restricted to 1 or 2 \({}^{\circ}\)C will make it easier to identify the temperature to be compared with the models.  
The 50 um seems to be a good thickness for the plate. The noise is reduced, the identification of the quantity of interest (in time and temperature) is a more accurate (2 \({}^{\circ}\)C) variation around the actual temperature value). As a consequence, 50 um is then the new upper limit for the plate thickness.  
Figure 15: Illustration of the noise amplification solving the inverse problem for various thicknesses. This problem–solution was solved with the following values: \(N=1000\), \(n=50\), \(t_{g}=0.005\)s, \(t_{w}=0.01\)s, \(\Delta t=0.00031\)s, \(P=1\) bar, \(I=400\) A, \(\Delta T=5\)\({}^{\circ}\)C, \(C_{P}=502\)\(\,\)I/K\(\cdot\)kg, \(\lambda=15\) W/m\(\cdot\)K, \(\rho=7960\) kg/m\({}^{3}\).