**4. Calculation method of seal characteristics**

The analysis method to calculate the seal characteristics is shown below. During the optimum design calculations, the groove shape should be changed continuously from its original spiral groove shape into other shapes. Therefore, a boundary-fitted coordinate system is adopted as the numerical calculation method [15]. Moreover, a divergence formulation method is implemented. A Reynolds equivalent equation obtained from flow balance as shown in **Figure 3** is used to obtain the pressure distributions on the seal face. This is because the geometry has a step over which there is a discontinuous pressure gradient between the groove and the land areas.

The Reynolds equivalent equation [16] is

$$\mathbf{Q}\_{2I}^{\xi} + \mathbf{Q}\_{1III}^{\xi} - \mathbf{Q}\_{2I}^{\xi} - \mathbf{Q}\_{1IV}^{\xi} + \mathbf{Q}\_{2I}^{\eta} + \mathbf{Q}\_{1II}^{\eta} - \mathbf{Q}\_{2III}^{\eta} - \mathbf{Q}\_{1IV}^{\eta} = \mathbf{Q}^{\Gamma} \tag{7}$$

Subscripts 1, 2, and *I* to *IV* indicate the areas in the control volume shown in **Figure 3**.

where the mass flow rates through the various boundaries are

$$Q^{\xi} = \int\_{\eta\_1}^{\eta\_2} \rho \left( -A\_0 \frac{\partial p}{\partial \xi} + B\_0 \frac{\partial p}{\partial \eta} + D\_0 + E\_0 \right) d\eta \tag{8}$$

$$Q^{\eta} = \int\_{\xi\_1}^{\xi\_2} \rho \left( B\_0 \frac{\partial p}{\partial \xi} - C\_0 \frac{\partial p}{\partial \eta} + F\_0 + G\_0 \right) d\xi \tag{9}$$

**Figure 3.** *Control volume and flow rates.*

*Tribology of Machine Elements - Fundamentals and Applications*

$$Q^{\Gamma} = \int\_{\xi\_1}^{\xi\_2} \int\_{\eta\_1}^{\eta\_2} \frac{\partial(\rho h)}{\partial t} |f\_0| d\eta d\xi \tag{10}$$

The coefficients of *Ao* to *Jo* are

$$A\_o = a\_o \frac{h^3}{12\mu l}, B\_o = b\_o \frac{h^3}{12\mu l}, C\_o = c\_o \frac{h^3}{12\mu l}, D\_o = -\frac{roo\_\eta h}{2} r\_\eta,$$

$$E\_o = \frac{\rho r o\_\eta^2 h^3}{40\mu} r \theta\_\eta, F\_o = \frac{roo\_\eta h}{2} r\_\xi, G\_o = \frac{\rho r o\_\eta^2 h^3}{40\mu} r \theta\_\xi,$$

$$a\_o = (r\theta\_\xi)^2 + r\_\eta^2, b\_o = (r\theta\_\xi)(r\theta\_\eta) + r\_\xi r\_2, c\_o = (r\theta\_\xi)^2 + r\_\xi^2,$$

$$f\_o = r\_\xi (r\theta\_\eta) - r\_\eta (r\theta\_\xi), r\_\xi = \frac{\partial r}{\partial \xi}, r\_\eta = \frac{\partial r}{\partial \eta},$$

$$r\theta\_\xi = r \frac{\partial \theta}{\partial \xi}, r\theta\_\eta = r \frac{\partial \theta}{\partial \eta}$$

Assuming a small amplitude vibration of the seal with frequency *ωf*, the gas film thickness and pressure are expressed as follows.

$$\begin{aligned} h &= h\_0 + \varepsilon e^{j o\_f t} \\ p &= p\_0 + \varepsilon p\_t e^{j o\_f t} \end{aligned} \tag{12}$$

where, *ε* is an amplitude of vibration and *p0* and *pt* express a static component and dynamic pressure components, respectively.

Substituting Eq. (12) into Eq. (7) and neglecting seconds terms of *ε*, the following equations regarding the 0th field and 1st field *ε* are obtained.

$$F\_0(p\_0) = Q\_{200}^\xi + Q\_{1\text{III}0}^\xi - Q\_{2\text{II}0}^\xi - Q\_{1\text{IV}0}^\xi + Q\_{2\text{I}0}^\eta + Q\_{1\text{II}0}^\eta - Q\_{2\text{III}0}^\eta - Q\_{1\text{IV}0}^\eta \tag{13}$$

$$F\_t(p\_t, p\_0) = \mathbf{Q}\_{\text{2lt}}^{\xi} + \mathbf{Q}\_{\text{1IIIt}}^{\xi} - \mathbf{Q}\_{\text{2llt}}^{\xi} - \mathbf{Q}\_{\text{1IVt}}^{\xi} + \mathbf{Q}\_{\text{2lt}}^{\eta} + \mathbf{Q}\_{\text{1IIt}}^{\eta} - \mathbf{Q}\_{\text{2IIIt}}^{\eta} - \mathbf{Q}\_{\text{1IVt}}^{\eta} \tag{14}$$

Discretizing Eqs. (13) and (14), and then solving the equations numerically, the static and dynamic components of the gas pressure fields are obtained. Finally, the gas leakage rate *q* is calculated from the static pressure distributions as

$$q = \int\_0^{2\pi} -\frac{\rho h^3}{12\mu} \frac{\partial p\_0}{\partial r} \bigg|\_{r=r\_i} r d\theta \tag{15}$$

Moreover, assuming the simple vibration model of a dry gas seal shown in **Figure 4**, the dynamic stiffness *K* is obtained from the calculated dynamic pressure distributions as,

$$K = \sqrt{\mathbf{k}^2 + \left(\alpha\_f c\right)^2} \tag{16}$$

$$k = \int\_{0}^{2\pi} \int\_{r\_i}^{r\_s} \text{Re}\left\{-p\_t\right\} r dr d\theta \tag{17}$$

$$\mathcal{L} = \int\_0^{2\pi} \int\_{r\_i}^{r\_o} \operatorname{Im} \{-p\_t\} r dr d\theta \tag{18}$$

*Groove Shape Optimization on Dry Gas Seals DOI: http://dx.doi.org/10.5772/intechopen.103088*

**Figure 4.**

*Simple vibration model of the dry gas seal.*
