# Effects of Vertical Glottal Duct Length on Intraglottal Pressures in the Convergent Glottis
## 3 Results
### Glottal Entrance Loss Coefficient  
The glottal entrance loss coefficient \(k_{e}\) can be used to evaluate the magnitude of the pressure drop from the trachea to the entrance of the glottis for a given transglottal pressure and glottal airflow [29,44]. This coefficient can provide a non-dimensional quantitative measure of how the pressure difference from the trachea to the glottal entrance differs from lossless Bernoulli pressures. The glottal entrance loss coefficient is given by  
\[k_{e}=2A_{en}^{2}\frac{P_{e}}{\rho U_{S}^{2}} \tag{1}\]  
where \(P_{e}\) is the pressure difference between the trachea and the entrance to the glottis, \(\rho\) is the density of the air, \(U_{S}\) is the volume flow, and \(A_{en}\) is the cross-sectional area \(L_{g}D_{e}\) at the glottal entrance location, where \(L_{g}\) is the length of the vocal folds (1.2 cm here) and \(D_{e}\) is the diameter of the glottis at the glottal entry.  
The glottal entrance loss coefficient is usually graphed as a function of Reynolds number \(R_{e}\), which is given by  
\[R_{e}=\frac{VD_{h}}{v} \tag{2}\]  
where \(V\) is the average velocity obtained by dividing the volume flow by the minimal cross-sectional area of the glottis, \(v\) is the kinematic viscosity of air, \(D_{h}\) is the hydraulic diameter, \(D_{h}=4A_{g}/S\), \(A_{g}\) is the minimal glottal area, and \(S\) is the perimeter of the glottis cross-section at the minimal glottal diameter location.  
Figure 3: (Color online) Non-dimensional intraglottal pressure distributions for \(-5^{\circ}\) convergent glottal angle and 0.308 glottal duct length for three minimal diameters (0.01, 0.04, and 0.16 cm) and three transglottal pressures (500 Pa, 1000 Pa, and 1500 Pa).  
Figure 4 shows the results of \(k_{e}\) for different vertical lengths of the glottis, glottal diameter, convergent glottal angle, and transglottal pressure. The entrance coefficient \(k_{e}\) varied from values near 1.16 to 1.84, with an overall mean value of 1.34. This value is smaller than in our previous study for the uniform glottis (\(k_{e}\) varied from 1.21 to 20.66, with a mean of 2.48) [7] and is also slightly smaller than the average number from [PERSON] et al. (1.375) for their rectangular glottis [45].  
As Figure 4 shows, the entrance coefficient differs between the cases for the minimal diameter of 0.01 cm compared to the cases for the other two diameters, 0.04 cm and 0.16 cm. For the diameter of 0.01 cm, \(k_{e}\) has a steep (decreasing) relation with \(R_{e}\) (and therefore also with transglottal pressure) up to a value of 500, with the highest values (c. 1.6 to 1.7) for the smallest convergent angle (\(-5^{\circ}\)) and the lowest values (c. 1.3 to 1.4) for the largest convergent angle (\(-20^{\circ}\)). For this small diameter, except for the smallest convergent angle, the entrance coefficient values have a tendency to increase with a longer glottal duct length.  
For the other two minimal diameters, 0.04 cm and 0.16 cm, the entrance coefficient increased as the vertical glottal duct length increased, but it was not dependent on diameter nor angle, being in the range of c. 1.28 to 1.36 for the length 0.908 cm, in the range of c. 1.15 to 1.25 for the length 0.308 cm, and between these two ranges for the length 0.608 cm. It is noted that a change from 1.15 to 1.36 is a change of 18%.  
Figure 4: Glottal entrance pressure coefficient \(k_{e}\) as a function of Reynolds number, with parameters of 0.01 cm, 0.04 cm, and 0.16 cm glottal minimal diameter, 0.108 cm, 0.308 cm, 0.608 cm, and 0.908 cm vertical glottal duct length, and 500 Pa, 1000 Pa, and 1500 Pa transglottal pressure.