## 3 Results and Discussion
### Expansion Measurements
\[DEF_{X(t)}= 0.0230~{}t^{3}-0.3893~{}t^{2}-1.9397~{}t+222.1223 \tag{4}\] \[DEF_{Y(t)}= 0.0038~{}t^{6}-0.0607~{}t^{5}+0.8899~{}t^{4}-5.7830~{}t^{3}+17.05 51~{}t^{2}-17.9077~{}t~{}+3.1502 \tag{5}\]  
where \(DEF_{X(t)}\) and \(DEF_{Y(t)}\) represent the expansion in the X (width) and Y (height) directions, respectively, as functions of baking time, with \(t\) denoting the baking time in seconds. The regression analysis yielded the following values: for \(DEF_{X(t)}\), \(R^{2}\) value of 0.9987 and a Residual Standard Error of 0.57. For \(DEF_{Y(t)}\), \(R^{2}\) value of 0.9932 and Residual Standard Error = 4.0. This empirical modeling approach, using polynomial regression to describe expansion over time, aligns with methods highlighted by [PERSON] [34; 52] as  
Figure 8: Expansion in the X and Y directions during baking for four tests. effective for food process optimization when full physics-based modeling is impractical or data-limited. For advanced analysis, the mean expansions in X (width) and Y (height) directions and the inside bread temperature are illustrated in Figure 10. To measure the bread core temperature during baking, the thermocouple is inserted inside the dough. This insertion may cause some gases to leak out, which possibly affects the bread expansion. Experimental tests performed on bread without and with sensor insertion using the same initial conditions showed that expansion in the X (width) and Y (height) axes follows the same pattern without remarkable deviation, concluding that sensor insertion had no significant effect on expansion. In Figure 10, the Y (height) expansion and temperature show a similar trend with a small phase delay between the two curves. Inflation begins, and fast temperature increases  
Figure 10: Relationship between expansion in the Y direction and core temperature during baking. Figure 9: Average expansion of the dough in the X and Y directions during baking. are time-shifted by 1 s. The fast temperature increase begins in the 4 th second, starting at a temperature of 60 \({}^{\circ}\)C and reaching a peak of 99 \({}^{\circ}\)C after 10 s. One second later, the Y (height) inflation starts, rapidly increasing towards a maximum of 100 mm after 11 s of baking and stabilizing. Similarly, the temperature and Y (height) expansion stabilize with the same time shift of one second. On the other hand, the X (width) expansion shows a slight contraction as the Y (height) expansion rapidly increases, indicating a redistribution of dough volume during baking. This behavior aligns temporally with the sharp rise in bread core temperature, suggesting a correlation between thermal expansion and structural transformation. While not strictly linear, these expansion trends reflect the physical mechanism of internal steam expansion driving vertical puffing and lateral contraction, characteristic of pocket formation in Lebanese flathead. To characterize the thermal expansion behavior of the bread during baking, Figure 11 illustrates the expansions plot as a function of the temperature inside the bread core. This behavior is empirically described for the X (width) and Y (height) expansion in Equations (6) and (7), respectively:  
\[DEF_{X(T_{c})}=-0.008\ T_{c}^{3}+0.1145\ T_{c}^{2}-6.601\ T_{c}+339.7454 \tag{6}\]  
Figure 6:0  
where \(T_{c}\) is the core temperature in \({}^{\circ}\)C. The regression analysis shows the following values: for \(DEF_{X(T_{c})}\), \(R^{2}\) value of 0.9933, and a Residual Standard Error of 1.3 for \(DEF_{Y(T_{c})}\). \(R^{2}\) value of 0.99682 and Residual Standard Error = 2.4. Assuming that the bread expansion is symmetrical and has an ellipsoid shape, the swelled volume can be calculated using Equation (8):  
Figure 6:1  
where \(DEF_{X}\) Is the expansion in the X (width) direction in mm, \(DEF_{Y}\) the expansion in the Y (height) direction in mm, and V the swelled volume in ml. Figure 6:2 Expansion and volume as function of the bread core temperature.