5.0 Introduction
Sealing technology is critical to modern high-performance gas turbine designs whose internal flow systems involve a variety of static (between stator surfaces) and dynamic (between stator and rotor surfaces) seals. Some of these seals are designed to minimize parasitic leakage flows, while others simultaneously meter the cooling flows as needed for the downstream components under various operating conditions. They endure severe operating and boundary conditions. The key design goals for seals used in gas turbines are: (1) to achieve pressure sealing between stationary and rotating surfaces with minimum leakage flow, (2) to minimally impact turbine and compressor efficiencies, (3) to generate acceptable windage temperature rise across them, (4) to achieve acceptable rotor thrust across them, and (5) to impact rotor dynamics within acceptable limits. Key design parameters that influence each seal performance include seal geometry, drag characteristics of the stator and rotor surfaces; and operating parameters such as inlet pressure, temperature, swirl, seal pressure ratio, fluid viscosity, and leakage flow compressibility.
During a gas turbine operation, as a result of the radial thermal and centrifugal growth of the rotor and the radial thermal growth of the adjacent stator, the seal clearance between them decreases and may lead to interference and excessive material loss through rubbing. These rubs typically result in the permanent removal of effective sealing material, increasing leakage. To avoid a rub, dynamic seals with rigid geometry; for example, straight-through labyrinth seals, inevitably have a larger build clearance. To better accommodate rotor excursions, the performance of other legacy labyrinth seals is improved by having teeth on the rotor and a layer of abradable or honeycomb material on the stator. In contrast, compliant seals track rotor movement with minimal loss of effective sealing material when a rub is experienced. The gas turbine returns to steady-state performance with minimal increase in leakage. Because brush seals can follow rotor movement with minimal wear, they fall into the category of contacting compliant seals, reducing leakage gaps that occur from large thermal differences between rotating and stationary components. The sealing technology used in modern gas turbines is trending toward the development of noncontacting compliant seals.
Chupp et al. (2006, 2014) provide an extensive and authoritative review of standard static and dynamics seals and advanced seal designs used in turbomachinery, including a rich bibliography for those who are interested in further advancing the seal technology. Rotordynamics considerations of seals are extensively presented in Childs (1993). The main thrust of this chapter is on the thermofluids modeling of standard dynamic seals used in gas turbines, enabling the calculation of seal leakage flow rate and the associated windage temperature rise for 1-D flow network simulation of secondary cooling and sealing flows. Toward this objective, we limit our discussion in this chapter to the labyrinth seals. For the other types of seals, the interested readers may refer to the references and bibliography at the end of this chapter.
5.1 Straight-Through and Stepped-Tooth Designs
As the name suggests, a labyrinth seal creates a complex flow passage to increase pressure loss across it and reduces leakage flow rate for a given clearance. Being the most widely used in gas turbines, these seals are inexpensive, noncontacting, and they operate under a wide range of pressures, temperatures, and rotor speeds. Some of their disadvantages include inevitable wear that enlarges clearance and worsens leakage.
Three most commonly used design configurations of a labyrinth seal are schematically shown in Figure 5.1. The straight-through design with nominal labyrinth is the most basic one with nearly equal inlet and outlet radii. Stepped designs are used to handle leakage through an inclined rotor-stator interface, and they offer higher flow resistance than a straight-through design.
Figure 5.1 (a) Straight-through design, (b) step-up design, and (c) step-down design.
5.1.1 Flow Physics
Figure 5.2a shows typical streamlines in a straight-through labyrinth seal. As the flow jets out the clearance gap between the rotating tooth and stator, it undergoes a sudden radially inward expansion in the downstream pocket formed between two adjacent teeth. The entrainment of the fluid for jet expansion creates flow recirculation in the pocket. Exceptfor rotation, the flow field in each pocket is similar to that of a driven cavity flow. Rotation produces radial pumping along each tooth, generating in each pocket additional shear layers, which are primarily responsible to destroy most of the incoming dynamic pressure, resulting in excess entropy production and loss in total pressure. The wall jet flow along the stator surface (land) suffers minimum loss in dynamic pressure and mainly contributes to the “kinetic energy carry-over factor” for the next tooth passage. An ideal labyrinth seal aims at total loss in dynamic pressure associated with its through-flow into each pocket, enabling a 1-D tube-and-tank modeling of the seal. This makes the seal more predicable in terms of its leakage flow characteristics. For an accurate assessment of the carry-over factor for each seal design, one must resort to either experimental data or high-fidelity 3-D CFD simulation.
Figure 5.2 (a) Expected streamlines in a straight-through labyrinth seal and (b) seal geometric parameters.
An inevitable consequence of rotation is the generation of windage in the seal as a result of work transfer from the rotor surface. Windage is an undesirable feature of each seal and must be accurately quantified to calculate the rise in coolant air temperature. Air enters each seal pocket at a swirl factor close to 0.5. As a result of the preponderance of rotor surface over the stator surface in each pocket, the air is expected to exit the pocket at a swirl factor greater than 0.5.
5.1.2 Leakage Mass Flow Rate
The well-known formula of Martin (1908) to compute mass flow rate through a multitooth straight-through seal is given by
(5.1)where CdCd is the discharge coefficient, AA the geometric area available for the seal flow, and ββ the gland factor, which is given by
(5.2)In Equations 5.1 and 5.2, Pt(0)Pt0 and Tt(0)Tt0 are the total pressure and total temperature, respectively, at the inlet, and Ps(n)Psn is the static pressure at the exit of the seal with nn teeth. Martin’s formula assumes that the flow throughout the seal is subsonic, the dynamic pressure through each tooth is completely lost in the downstream cavity, and the total temperature remains nearly constant.
For the seal geometry shown in Figure 5.2b, Egli (1935) proposed the following equation to estimate leakage mass flow rate:
(5.3)where
Ct≡Ct≡ Seal throttling coefficient
Cc≡Cc≡ Seal carry-over coefficient
Cr≡Cr≡ Seal contraction coefficient
For CtCt, CcCc, and CrCr, Aungier (2000) converted the graphical representations in Egli (1935) into the following easy-to-use equations:
(5.4)
(5.5)
(5.6)where
Equations 5.4, 5.5, and 5.6 are shown plotted in Figure 5.3.
Figure 5.3 (a) Seal contraction ratio, (b) seal throttling coefficient, and (c) seal carry-over coefficient.
Vermes (1961) further extended Martin’s work and modified Equation 5.1 to compute leakage flow rate for straight, stepped, and combination seals. In particular, to account for the change in total temperature through the seal as a result of windage, he augmented the numerical coefficient from 5.68 to 5.76 in Equation 5.1. Furthermore, using Zabriskie and Sternlicht (1959), he proposed the following modified leakage equation:
(5.7)where αα is called the kinetic energy carry-over factor, which is related to seal geometry. With the aid of the boundary layer analysis, he proposed the following equation to compute αα:
(5.8)where pp is the seal pitch, bb the tooth tip width, and ss the seal clearance. Note that the seal flow total temperature can only be changed either by heat transfer or windage (work transfer), and it should not depend upon the kinetic energy carry-over factor. Only for an incompressible flow, the dynamic pressure, which equals 0.5ρV20.5ρV2, can be interpreted as the flow kinetic energy per unit volume.
For the seal geometry shown in Figure 5.4a, McGreehan and Ko (1989) proposed the following leakage mass flow rate equation, which is a modified form of Equation 5.7:
(5.9)where KLKL is the seal configuration factor, which is like an overall discharge coefficient for the seal.
For the seal geometry shown in Figure 5.4b, Zimmermann and Wolff (1998) proposed the following seal mass flow rate equation:
(5.10)where CdCd is the discharge coefficient, and k2k2, the corrected carry-over coefficient given by k2 = kk1k2=kk1, where the carry-over coefficient kk, first proposed by Hodkinson (1939), is expressed in terms of seal geometry as
(5.11)and the correction factor k1k1 is given by
(5.12)Equations 5.11 and 5.12 are shown plotted in Figure 5.5a and 5.5b, respectively. While the variations of the carry-over coefficient kk shown in Figure 5.11a exhibit acceptable tends for a seal with the number of teeth ranging from 2 to 15, the corrected carry-over coefficient k2k2 shown in Figure 5.5c seems suspect for 0 < s/p < 0.080<s/p<0.08. Note that both kk and k2k2 are akin to CcCc used in Equation 5.3, which is shown in Figure 5.3c.
Figure 5.5 (a) Seal carry-over coefficient k1k1, (b) correction factor kk, and (c) corrected carry-over coefficient k2k2.
5.2 Tooth-by-Tooth Modeling
Various empirical correlations proposed in the foregoing section generally compute significantly different values (see Problem 5.1) for the seal leakage mass flow rate. This is to be expected as the leakage mass flow rate depends on both the seal operating conditions and various geometric features, which are difficult to accurately capture in a universal empirical correlation. Furthermore, these correlations do not explicitly account for the swirl velocity variation and windage temperature rise in these seals, somewhat impacting their leakage mass flow rate. Nevertheless, these empirical correlations are still valuable in providing an initial estimate of the leakage mass flow rate to enable an iterative solution method based on the tooth-by-tooth distributed modeling of the labyrinth seal discussed next in this section.
The tooth-by-tooth distributed modeling of a six-tooth labyrinth seal is schematically shown in Figure 5.6. This modeling methodology from seal inlet to outlet allows us to incorporate seal geometry variations, changes in total temperature as a result of both heat transfer and windage, and forced vortex pressure changes as a result of the change in radius between two adjacent teeth, as found in a stepped labyrinth seal design. The prediction results in this case include not only the seal leakage mass flow rate but also the total temperature and swirl factor at the seal exit.
Figure 5.6 Tooth-by-tooth modeling of a six-tooth labyrinth seal.
For an incompressible flow, the mass flow rate at a section depends on the difference between the total pressure and static pressure at the section. In case of a compressible flow, however, the mass flow rate depends upon both the total pressure and total-to-static pressure ratio, and the seal may feature a choked flow condition if the Mach number over a tooth becomes unity. For a labyrinth seal with uniform clearance across each tooth, we can make the following argument that the seal can only choke at the last tooth. In steady state, the mass flow rate remains constant throughout the seal. Over each seal tooth with constant leakage flow area, discharge coefficient, swirl factor, and total temperature, the mass flow rate will be proportional to the total pressure and total-pressure mass flow function, which is a function of Mach number, which in turn is a function of total-to-static pressure ratio (see Chapter 2). When the flow chokes at a section, its Mach number becomes unity, yielding a constant value of the total-pressure mass flow function. Because the total pressure decreases monotonically across the labyrinth seal, the maximum leakage mass flow rate through it must correspond to the minimum total pressure, which occurs at the last tooth.
Let us assume that the six-tooth labyrinth seal shown in Figure 5.6 is operating under ideal conditions such that the entire dynamic pressure generated at each tooth is dissipated in the downstream cavity, implying zero kinetic energy carry-over factor (α = 0α=0). In addition, under adiabatic conditions with no rotation, the total air temperature remains constant throughout the seal. Accordingly, the static pressure in each cavity becomes the total pressure for the flow over the downstream tooth. This implies that the static-pressure mass flow function over one tooth must equal the total-pressure mass flow function over the next tooth, as shown in Figure 5.7a for a choked air flow through the seal. This figure also shows how Mach number increases from seal inlet to outlet. While the air flow remains subsonic from tooth 1 to tooth 5, it increases rapidly to unity over tooth 6, the last tooth. The corresponding variation in total-to-static pressure ratio from tooth 1 to tooth 6 is shown in Figure 5.7b.
Figure 5.7 (a) Mach number variation across a six-tooth choked ideal labyrinth seal and (b) tooth-wise pressure ratio variation across the seal.
The overall total-to-static pressure ratio for a choked air flow through an ideal labyrinth seal varies with the number of teeth. This variation is depicted in Figure 5.8. We can represent the curve in this figure by the following cubic polynomial:
(5.13)
Figure 5.8 Overall total-to-static pressure ratio variation with nn for the choked flow through an ideal labyrinth seal.
For a labyrinth seal with nn number of teeth, if the overall pressure ratio is equal or higher than that computed by Equation 5.13, the seal flow will choke at the last tooth, and if it is lower, the flow will remain subsonic throughout the seal. For the six-tooth labyrinth seal, the overall critical pressure ratio is exactly calculated to be 3.556, while the curve-fit Equation 5.13 yields 3.564, which is within an acceptable error of 0.23 percent.
5.2.1 Orifice-Cavity Model
For a given labyrinth seal with known geometry, the typical design objective is to calculate the seal leakage mass flow rate when the total pressure, total temperature, and swirl factor are specified at the seal inlet, and the static pressure is specified at the seal outlet. The orifice-cavity model shown in Figure 5.9 forms the basis for the tooth-by-tooth modeling of a multitooth labyrinth seal shown in Figure 5.6. For the rotor-stator cavity control volume, which is formed between two consecutive teeth together with the stator surface, shown in Figure 5.9, we present here a methodology to compute the compressible flow properties at outlet j + 1j+1, knowing their values at inlet jj. Note that, if the seal operates under the choked flow condition, which typically occurs at its last tooth, the specified outlet static pressure boundary condition becomes irrelevant. In this case, we should use M = 1M=1 as the new boundary condition at the last tooth. In steady state, the air leakage mass flow rate over each seal tooth remains constant.
Figure 5.9 Orifice-cavity model control volume formed between tooth j and tooth j + 1 in a multitooth labyrinth seal.
With an initial estimate of the seal leakage mass flow rate ṁ
, which may be obtained using any of the empirical equations presented in the previous section, we summarize here an iterative solution method for the orifice-cavity model shown in Figure 5.9.
1. Assume Mach number M(j)Mj and calculate the total-pressure mass flow function at section jj
F̂ftj=Mjκ1+κ−12Mj2κ+1κ−1
2. Knowing the empirical discharge coefficient Cd(j)Cdj, calculate the velocity coefficient CV(j) = Vx(j)/V(j)CVj=Vxj/Vj from the equation
CVj=ṁRTtjAjCdjF̂ftjPtj
where A(j)Aj is the mechanical flow area at jj.
3. Obtain Vj=Vθj2/1−CVj2
where Vθ(j) = r(j)Sf(j)ΩVθj=rjSfjΩ.
4. Calculate the static temperature Ts(j)Tsj
Tsj=Ttj−Vj22cp
5. Calculate the Mach number M˜j
M˜j=VjκRTsj
6. Repeat steps from 1 to 5 until M(j)Mj equals M˜j
within an acceptable tolerance.
7. Calculate the static pressure Ps(j)Psj
Psj=PtjTtjTsjκκ−1
