Groove Shape Optimization on Dry Gas Seals

*Masayuki Ochiai and Yuki Sato*

### **Abstract**

In this paper, a topological optimum design for the shape of a groove in a dry gas seal is described. Dry gas seals are widely used in high speed and high pressure rotating machinery such as gas turbines, compressors, and so on because of their high reliability compared to other types of seals. However, recent requirements for reducing emission with further control of leakage are in order. With this background, we propose applying topological optimization to the groove shape in a dry gas seal to reduce its leakage while keeping its stiffness for safe operation. First, the method of topological optimum design as applied to the groove of a dry gas seal is explained via numerical analysis. Next, results of the topological optimization are shown via categorizing an optimum shape map. Finally, the mechanism of reducing the gas leakage with an optimized seal is discussed based on the prediction of the flow field using a CFD analysis.

**Keywords:** mechanical seals, dry gas seals, groove shape, optimization, gas lubrication

### **1. Introduction**

Non-contact dry gas seals with a grooved pattern on a seal face can maintain a film thickness of just a few micrometers. Therefore, these seals have better sealing performance when compared to typical labyrinth seals [1]. Dry gas seals are used in many turbomachinery, such as in gas and steam turbines, turbochargers, and compressors. Moreover, they are applied to high-speed operation and under high-pressure differences.

Recently, to reduce energy consumption, more enhancements toward efficient turbomachinery are required. To solve this problem, one effective way is by enhancing the sealing characteristics of seals. Many types of grooved dry gas seals have been developed [1]. Spiral grooved seals are widely used because of their good sealing ability. Lately, a significant amount of research on spiral grooved dry gas seals focused on analytical methods [2–8], dynamic force characteristics [9–11], thermal effects considerations [12], and CFD analysis considering the turbulent flow [13] have been performed.

On the other hand, the optimum design of the grooves is one of the effective ways to enhance the seal characteristics. The optimum design methods have been also applied to gas film bearings. Lin and Satomi [14] and Hashimoto and Ochiai [15, 16] applied an optimum design method to spiral groove thrust bearing towards enhancing performance characteristics from variations in groove depth, groove angle, and so on. Moreover, an experimental verification was conducted comparing the novel configuration against a conventional designed spiral groove bearing. However, it was found that the effectiveness of the optimization is limited because these studies have not been changed the groove shapes which were based on a spiral path.

Under this circumstance, Hashimoto and Ochiai [17] proposed a topological optimum design method for a grooved thrust gas bearing. In this method, the groove shape could be changed freely using a cubic spline function. Novel groove shapes were found in this study. The effectiveness and the applicability of the method were verified theoretically and experimentally. Moreover, Hashimoto and Namba [18] found the best groove shapes against various objective functions such as film thickness, friction torque, and dynamic axial stiffness. Also, the effect of the new groove shape on sealing characteristics of FDB(Fluid dynamic bearing) was studied previously and discussed by authors [19].

To date, many researchers have treated spirally grooved shape dry gas seals. On the other hand, recently, the optimum design of groove shape on the dry gas seal was proposed by authors, and comparison of the flow visualization was presented [21]. However, the process of the optimum design has not been mentioned and also it has not been studied for a wide range of operation conditions. Therefore, in this study, the application of the topological optimum design to the dry gas seal instead of the thrust bearings to find an optimum groove shape that enhances the seal leakage restriction and its dynamic stiffness is presented. Moreover, it is important to know the optimum groove shapes under various conditions, therefore, in this study, we tried to make a categorization map of the seal's optimum shape based on the results of the optimum design calculations under a wide range of operating conditions. Furthermore, CFD analysis is conducted and compared with the experimental flow visualizations for verification, while the rationale for reducing the gas leakage with an optimized seal is presented.

## **2. Topological optimization methods**

**Figure 1** shows the typical structure of a dry gas seal cartridge. It consists of a rotating shaft, a ring with grooves on its face, a stationary ring, support springs, and housing. The gas film is generated by the hydrodynamic effect induced on the

**Figure 1.** *Components of a non-contacting dry gas seal.*


#### **Table 1.**

*Dry gas seal physical parameters.*

grooves of the face. The film thickness is determined by the force balance between the support springs and the hydrodynamic gas film force. The film thickness can be changed by changing the support springs. The seal leakage is a function of the film thickness, the gas pressure differential between the inner side and outer side of the seal chamber, the viscosity of the gas, and the groove shape mounted on the face.

In the design of dry gas seals, it is important to minimize the gas leakage towards enhancing the efficiency of turbomachinery. Simultaneously, enhancing the dynamic stiffness of a gas film is an important factor for its safe operation, at high speed in particular. Because turbomachinery is likely to be exposed to some outer disturbance such as earthquakes, a hard contact of the rotor on the seal surface leads to serious damage to the mechanical system.

Both a low gas leakage and a high gas film stiffness are trade-off relations, being difficult to optimize both parameters at the same time. Therefore, in this study, sufficient stiffness is selected for safety. The whole structure of the dry gas seal with the gas film is modeled as spring and damper as shown later. Therefore, from the calculation of a linear vibration waveform, the minimum film thickness is obtained. Under the conditions presented in **Table 1**, Ref. [19], the required gas lubricated film stiffness is defined. Because the leakage rate is strongly affected by film thickness, the value is fixed as 5 μm in this optimization as shown in **Table 1**.

### **3. Optimum design formula**

The optimization method in this study is based on Hashimoto and Ochiai's topological optimum design theory [17]. The outline of the method is as follows. The initial groove geometry is the usual spiral groove shape, and then, cubic spline interpolation functions are applied to the initial geometry with 4 grids. Moving the grids on the same circumferences changes the groove shape. Applying the optimum design method, an optimized seal groove shape is obtained. Simultaneously, the number of grooves *N*, the seal radius *Rs*, the groove depth *hg*, and the groove width ratio *α* are set as the optimized design valuables in this study. Therefore, the design vector of parameters is

$$\mathbf{X} = \left(\rho\_1, \rho\_2, \rho\_3, \rho\_4, N, R\_\circ, h\_\mathfrak{g}, a\right) \tag{1}$$

where, the *ϕ*<sup>1</sup> to *ϕ*<sup>4</sup> mean the angles from an initial spiral groove shape, shown in **Figure 2a**.

#### *Tribology of Machine Elements - Fundamentals and Applications*

**Figure 2.** *Geometry of a seal and optimum design variables.*

In the optimum design, the objective functions should be defined. Obviously, the most important one is to minimize the leakage *q*. Therefore, the objective function is set as

$$f\_1(\mathbf{X}) = q \tag{2}$$

Moreover, even if a lesser leakage design is available, it is impractical to have a lesser dynamic stiffness simultaneously. Since dry gas seals are usually used under high speed and high-pressure differential conditions, sudden contact on the seal faces may lead to serious accidents. Therefore, the dynamic stiffness *K* must be used as the second objective function,

$$f\_2(\mathbf{X}) = \mathbf{K} \tag{3}$$

The constraint relationships in this optimization are

$$\mathbf{g}\_i(\mathbf{X}) \le \mathbf{0} (i = \mathbf{1} \sim \mathbf{19}) \tag{4}$$

where

$$\begin{aligned} \mathcal{g}\_1 &= \rho\_{1\text{min}} - \rho\_1, \mathcal{g}\_2 = \rho\_1 - \rho\_{1\text{max}}, \mathcal{g}\_3 = \rho\_{2\text{min}} - \rho\_2, \\ \mathcal{g}\_4 &= \rho\_2 - \rho\_{2\text{max}}, \mathcal{g}\_5 = \rho\_{3\text{min}} - \rho\_3, \mathcal{g}\_6 = \rho\_3 - \rho\_{3\text{max}}, \\ \mathcal{g}\_7 &= \rho\_{4\text{min}} - \rho\_4, \mathcal{g}\_8 = \rho\_4 - \rho\_{4\text{max}}, \mathcal{g}\_9 = N\_{\text{min}} - N, \\ \mathcal{g}\_{10} &= N - N\_{\text{max}}, \mathcal{g}\_{11} = R\_{\text{min}} - R\_s, \mathcal{g}\_{12} = R\_S - R\_{\text{sum}}, \\ \mathcal{g}\_{13} &= h\_{\text{gmin}} - h\_{\text{g}}, \mathcal{g}\_{14} = h\_{\text{g}} - h\_{\text{gmax}}, \mathcal{g}\_{15} = a\_{\text{min}} - a, \\ \mathcal{g}\_{16} &= a - a\_{\text{max}}, \mathcal{g}\_{17} = -c, \mathcal{g}\_{18} = -N, \mathcal{g}\_{19} = -W \end{aligned} \tag{5}$$

The *g*<sup>1</sup> to *g*<sup>16</sup> indicate the upper and lower limit of the design variables and the *g*<sup>17</sup> is to avoid negative damping.

The optimum design problem is formulated as

$$\text{Find } \mathbf{X} \text{ to minimize } f\_1(\mathbf{X}) \text{ and } \text{maximize } f\_2(\mathbf{X})$$

$$\text{subject to } \mathbf{g}\_i(\mathbf{X}) \le \mathbf{0} (i = 1 \sim 19) \tag{6}$$
