**5. Topological optimization results**

Using the method mentioned above, topological optimum calculations were conducted. The calculation conditions are shown in **Tables 1** and **2**, and **Figure 2**. As shown in **Table 1**, the mass of stator *<sup>m</sup>* = 1.0 kg, the support spring *<sup>k</sup>* = 5.0 <sup>10</sup><sup>5</sup> N/m, the steady-state clearance *cr* = 5 μm are defined in the reference literature [19]. The assumed disturbance of *f* = 5G is defined from the magnitude of earthquakes. The Compressibility number *Λ* and outer side pressure *P0* are set under a wide range of operation conditions. Because we would like to find the general optimum shape of the seal groove under various operating conditions, not a limited condition. The parameters for optimum design calculations are set as shown in **Table 2**. These set values are defined by representative of the dry gas seals. The groove numbers are discrete values. The other optimum design variables of the groove depth, the angle amount, the groove width, the inner and outer radius are continuous values, and their maximum and maximum values are set as concentration conditions.

By solving the above optimum design problem, a multi objective genetic algorithm is used as this in a multi objective optimization [20].

**Figure 5** shows the optimization results for operation with a compressibility number *Λ* = 500 and inlet pressure *Po* = 2.5 MPa. The vertical axis shows the gas leakage flow rate as the 1st objective function and the horizontal axis shows the dynamic stiffness as the 2nd objective function. In this figure, the red line at the


#### **Table 2.** *Parameters for optimum design study.*

#### **Figure 5.**

*The case of Pareto optimum solutions (Pi = 2.5 MPa, Λ = 500).*


#### **Table 3.**

*Characteristic values.*

bottom part denotes the Pareto solution curve. Here, the Pareto result means the cloud of optimum solutions from the multi-optimization results. From this result, it is confirmed that there is trade-off relations between the objective functions of gas leakage and the dynamic stiffness. The leakage minimized seal has less dynamic stiffness, whereas the dynamic stiffness maximized seal has inferior sealing characteristic. Therefore, in this study, the allowable dynamic stiffness which means avoiding the contact of the seal surface against the outer disturbance as criteria is set. Because the gas leakage should be reduced as much as possible under a safe operation. Comparing the gas film thickness and the linear impulse response by using the vibration model as shown in **Figure 4**, we can recognize whether the contact occurs or not by outer disturbance. The critical dynamic stiffness can be defined from the criteria. Moreover, in this study, we defined the allowable

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

dynamic stiffness adopting a safety factor of 3 which means three times of the critical dynamic stiffness. In this manner, the optimized groove shapes as shown in **Figure 5** were obtained and the characteristic values of optimized seals are shown in **Table 3**.

The initial shape of the spiral groove seal labeled (A) does not have the desired characteristics of both low gas leakage and high dynamic stiffness. Comparing the shapes of (A) through (D), from the point of view of minimizing the gas leakage, the shape of the groove is quite different from the initial spiral groove as shown in **Figure 5B**. The optimized shape has a bending curve in the vicinity of the outer diameter of the seal face. On the other hand, from the viewpoint of maximizing the dynamic stiffness, the shape of the groove, as shown in **Figure 5C** is similar to the spiral groove shape in **Figure 5A**. This is because a high positive dynamic pressure is required. It is well known that the spiral groove shape can effectively generate high positive pressure.

Thus, considering an allowable dynamic stiffness, the optimized shape as shown in **Figure 5D** is similar to the shape that minimizes gas leakage with a bending curve. However, the length of the bending curve is no longer that of the leakage minimized seal. This is due to gas flow around the outer vicinity of the gas seal face. The gas flow from the outer high pressure is retarded by the effect of the curved shape of the grooves. From these results, the most interesting thing is that quite a different shape is obtained for the case reducing gas leakage only. However, the results are valid only for the case of *Λ* = 500 and inlet pressure *Po* = 1.0 MPa.

**Figure 6** depicts the tendency of change in the shape of the dry gas seal face on the Pareto optimum solution. Orienting the low leakage design, the strong bending shape in the outer vicinity and the wide plane region in the inner side are obtained. This bending shape reduces the leakage to the inner side of the seal by pump-out effect from the inner to the outer circumference side. On the other hand, emphasizing the stiffness design, it is found that the bending tendency goes weak and finally the shape goes to the spiral shape gradually.

From the point of view of the actual seal design, a wider range of operations is required. Therefore, the optimum design calculations were conducted over a wide range of conditions *Λ* = 100–750, inlet pressure *Po* = 1.0–10.0 MPa. The inlet pressure is the most important operating condition in a dry gas seal design. On the contrary, the range of compressibility numbers encompasses many operating conditions of rotational speed, film thickness, gas viscosity, and size of seal.

**Figure 6.** *Change in optimum shape tendency of the dry gas seal face.*

**Figure 7.** *Optimal design map under a wide range of conditions.*

**Figure 7** depicts the optimized shape map for a wide range of inner static pressure at the outside diameter and compressibility numbers. There are three types of shapes, one is quite similar to the spiral groove shape and applicable to a low inlet pressure range of *Po* = 1.0–4.0 MPa and low compressibility number range of*Λ* =10 to 350. Another shape has a slight bend curve like in **Figure 5B** in the range of *Po* =1.0 to 4.0 MPa and *Λ* =350 to 750. The other is a shape having a strong bending curve like in **Figure 5D** for an inlet pressure range of *Po* = 4.0–10.0 MPa and over the whole range of *Λ*.

From the results, in the case of low inner static pressure conditions and a low compressibility number (*Λ* < 350), shown in the green color area, it is a required feature of a dry gas seal to enhance its dynamic stiffness. This is due to the ability to generate a dynamic positive pressure on the seal face. Under the conditions of low inlet pressure and high compressibility number, shown in the blue color area, large film thickness, and low viscosity, etc., it is difficult to generate high dynamic pressure on the seal face. Therefore, the grooves are formed to maximize the seal dynamic stiffness.

On the other hand, for a high inlet pressure or a high compressibility number condition, shown in the red area, the allowable film stiffness could be obtained easily as its basic ability. Because the high inlet pressure condition is expected to deliver a hydrostatic effect and the high compressibility number leads to an enhancement of the hydrodynamic effect. Hence, the main object of topological optimization is to reduce gas leakage. However, for a low inner static pressure condition, the hydrostatic effect is not expected. Therefore, the bending curve shape is weak. In other words, it is found that the topological optimization for reducing gas leakage is effective in the case of a high inner static pressure condition.

#### **6. CFD analysis of visualization of the flow and discussions**

In order to consider the mechanism for reducing the gas leakage of the optimized shape, which is the interesting bending shape, a CFD analysis of the gas flow was conducted using commercial software (ANSYS FLUENT) which can solve the Navier-Stokes equation including the flow of outer side area of dry gas seal and considered to be obtained more accurate solution compared to usually used

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

Reynolds equation, which is neglecting the outer side flow of seals. In the past work of Hashimoto[18], a similar bending shape is obtained in the case of maximizing the bearing stiffness on a high-speed air bearing. However, as mentioned in the previous sections, another tendency is obtained in this case. That is, the bending shape is obtained in the case of minimizing the air leakage instead of maximizing the stiffness. Therefore, the reason why this shape is obtained is unclear.

**Figure 8** and **Table 4** report the CFD calculation model and the specifications respectively. The inner and outer radii are same as our experimental equipment [21], Moreover, the groove depths and seal clearance of the seals are 60 μm and 30 μm respectively because of their mesh size limitations. The seal radius ratio (Rs/Ro) and Groove width ratios are chosen by representative values for each seal. In addition, **Table 4** indicates the calculation conditions of CFD analysis. The inlet pressure, it means the outer side of the dry gas seal, is set as 0.11 MPa, and the rotational speed is set as 5000 rpm. These values are the same as the previous experiment. The calculations are conducted under the area of one groove pattern by using a periodic boundary condition. In addition, the calculation does not use the turbulent model and concludes choked flow. Because the Reynolds number of the gas seal flow is approximately *Re* = 26, where the representative length is the clearance 30 μm, the representative speed is peripheral speed at an outer radius of 20 m/s. This setting reduces calculation costs. Consequently, the calculation area sizes are not identical (**Table 5**).

**Figure 9** shows the predicted (I) pressure distribution and (II) velocity distribution from the CFD analysis on the middle plane of gas film thickness comparing the conventional spiral grooved seal(a) versus the optimized seal(b). In this study,

#### **Figure 8.**

*CFD analysis model.*


#### **Table 4.**

*Seal specifications of CFD analysis.*
