# Effect of Ingress on Flow and Heat Transfer Upstream and Downstream of a Rotating Turbine Disc

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Brief Review of Relevant Research

## 3. Experimental Rig and Instrumentation

_{ϕ}= 7.2 × 10

^{5}and 10

^{6}). The Mach numbers at the vane exit were 0.3 and 0.4. A dynamometer was used to absorb the generated power. The characteristic diameter of the disc was 380 mm and the height of the annulus was 25 mm. Static pressure taps on the hub platform, spanning the vane exit both upstream and downstream of the rotor, provided the pressure variation in the annulus; these are described in detail by Patinios et al. [8].

^{3}, k

_{s}= 0.2 W/m∙K and c

_{p}= 1200 J/kg∙K). The inner surface was insulated using Rohacell 51 foam (ρ = 75 kg/m

^{3}, k

_{s}= 0.03 W/m∙K and c

_{p}= 1225 J/kg∙K). On the stator surfaces, Rohacell was used to minimise the heat loss, and polycarbonate backing plates were used for assembly purposes.

_{2}was measured in both the annulus (unseeded) and at entry to the wheel-space, c

_{o}, using a dual-channel multi-gas analyser (9000MGA from Signal Group, Camberley, Surrey, UK). The radial variation of the concentration along the stator (c

_{s}) was measured through 15 hypodermic tubes of diameter 1.6 mm, located between 0.65 ≤ r/b ≤ 0.993. Reference points were located at two radii upstream (r/b = 0.924 and 0.850) and downstream (r/b = 0.941 and 0.800). Carbon dioxide was extracted using a pump to an infrared analyzer; typically, gas concentration measurements were time-averaged over 10 seconds. The uncertainty in gas concentration was 0.015%, as discussed in detail by Patinios et al. [8].

## 4. Theoretical Models and Data Analysis

#### 4.1. Adiabatic Effectiveness of the Rotor

_{r}, can be defined as:

_{i}is the total enthalpy of the ingress and H

_{o}is the total enthalpy of the purge flow.

_{r}is related to the sealing effectiveness, ε

_{s}, and the buffer parameter, Ψ, by:

_{s}decreases.

_{T}

_{,o}is the turbulent flow parameter, which is defined as:

_{o}, is also used and defined as:

_{o}and λ

_{T}

_{,o}is given by:

_{T}

_{,o}≈ Φ

_{o}. Therefore, Equation (8) can be rewritten as:

#### 4.2. Sealing Effectiveness of the Stator

_{i}is the concentration of the ingress that is measured in the annulus, c

_{s}is the concentration measurements on the stator and c

_{o}is the concentration of the purge air.

_{min}is the minimum value of Φ

_{o}needed to prevent ingress, and Γ

_{c}is the ratio of the discharge coefficients for ingress and egress.

#### 4.3. Calculation of Buffer Parameter Ψ

_{i}, H

_{o}and H

_{s}are the total enthalpies of the annulus flow, the purge flow and the stator boundary layer flow, respectively, ${\dot{m}}_{o}$ is the purge mass flow rate, ε

_{s}the stator effectiveness measured using the concentration, and W

_{r}the work exerted by the rotating disc from r = a to r = b.

_{s,t}, T

_{i,t}and T

_{o,t}are the total-temperatures of the stator boundary layer flow, the annulus flow and the purge flow, respectively.

_{o,t}, then the total-temperature in the stator boundary layer will be changed by ΔT

_{s,t}where:

_{s,t}and ΔT

_{r,ad}can be calculated from transient temperature measurements in the wheel-spaces and on the rotor surfaces, and ε

_{s}can be determined from concentration measurements. The value of the buffer parameter, Ψ, can then be acquired using:

_{s}< 1.

#### 4.4. Maximum Likelihood Estimation (MLE) Analysis of Transient Temperature Measurements

_{s,t}and ΔT

_{r,ad}from the transient temperatures measured in the wheel-spaces and on the rotor surfaces.

_{exp}≈ $\mathcal{N}$ (T

_{true}, σ

^{2}), where $\mathcal{N}$ denotes a normal distribution. T

_{exp}and T

_{true}are the measured and true values of the temperatures. The likelihood function is expressed as:

_{exp}. Each experiment lasted around 15 minutes, and only the measurements in the last 7.5 minutes were used. The true temperature was assumed to be an exponential function where:

_{inf}, D and E are optimised to minimize $l$. The optimised standard deviation between the measured and true temperatures, σ, is about 0.1 °C. Note that, when t → ∞, T

_{true}→ T

_{inf}. By applying the analysis to the transient air temperature measurements (T

_{s,exp}) and the transient disc surface temperature measurements (T

_{r,exp}), T

_{s,inf}and T

_{r,inf}can be acquired. Therefore, the temperature differences between the initial steady-state and the steady-state when t → ∞ can be calculated using:

_{s,inf}and T

_{r,inf}are the extrapolated temperatures at infinity, and T

_{s,ini}and T

_{r,ini}are the initial temperatures.

_{o}= 0.0265. T

_{s,inf}, T

_{s,ini}, T

_{r,inf}and T

_{r,ini}are also shown using the indicated symbols.

## 5. Discussion of Experimental Results

#### 5.1. Swirl Ratios in the Core

_{t}= 0, and λ

_{t}is the local turbulent flow parameter defined as:

_{ϕ}= 10

^{6}and 7.2 × 10

^{5}. The values of β at r/b = 0.825 and 0.958, where the temperatures were measured, were obtained via interpolation of the experimental data. Figure 5 and Figure 6 show the measured values of β/β* versus λ

_{t}and β* versus r/b, respectively, in both wheel-spaces. The swirl ratio in the core increased with increasing radius and with decreasing purge. For the range of λ

_{t}relevant to the heat transfer measurements, the results presented in these figures were limited to λ

_{t}< 0.080 and 0.046 for the upstream and downstream cases, respectively. The swirl correlation, which was linear over these limited ranges, is given by:

`→`1 as r/b

`→`1. For the downstream case, where the swirl in the annulus was low, the angular momentum of the ingested fluid is believed to be the reason for the smaller values of β*.

#### 5.2. Sealing Effectiveness

_{s}, obtained using measurements of concentration on both the upstream and downstream stators. The abscissa is Φ

_{o}and the ε

_{s}results were correlated using Equation (14). The values of Γ

_{c}and Φ

_{min}were 1.32 and 0.0526, respectively, for the upstream case and 0.0895 and 0.0400 downstream. As found in previous experiments, e.g., Reference [8], there was no significant variation of effectiveness with radius. For the upstream wheel-space, the fits between the effectiveness curve and the experimental data were not as good as those obtained in previous experiments. This is attributed, in part at least, to the polycarbonate seals and the layer of Rohacell on the underside of the seal attached to the stator. It is shown below that the sealing effectiveness for the downstream wheel-space was significantly higher than that in the upstream wheel-space. This was attributed to the reduced circumferential pressure variation in the downstream annulus.

#### 5.3. Adiabatic Effectiveness of the Rotor

_{r}, is related to the stator effectiveness, ε

_{s}, and the buffer parameter, Ψ, by Equation (2). Ψ can be determined using the transient temperature measurements, as shown in Section 4.3 and Section 4.4. Distributions of Ψ with Φ

_{o}for the wheel-spaces upstream and downstream of the rotor are plotted in Figure 8. It was deduced from Equation (5) that, for a given superposed flow, Ψ increases as ${\dot{m}}_{s}$ decreases; meanwhile, ${\dot{m}}_{s}$, which is affected by the swirl ratio in the wheel-space core, is equal to the entrainment flow rate into the boundary layer on the rotor. The lower swirl ratio in the downstream wheel-space indicates lower ${\dot{m}}_{s}$ and hence a higher buffer ratio. In both wheel-spaces, there is a critical Φ

_{o}at which Ψ = 1; above this critical Φ

_{o}, ingress has no effect on the rotor temperature. As shown in Reference [7], Ψ should be a function of r/b, but the measurements shown in Figure 8 do not have sufficient accuracy to show a sensitivity to radius.

_{r}with Φ

_{o}for both wheel-spaces are plotted in Figure 9; the curves of ε

_{s}, correlated using Equation (14), are also shown in this figure. It can be seen that Δε, the difference between ε

_{r}and ε

_{s}, was larger for the upstream wheel-space than for the downstream one. It was shown from Equation (7) that Δε increases as Ψ increases and as ε

_{s}decreases; in Figure 9, the effect of the change in ε

_{s}between the wheel-spaces was larger than the effect of the change in Ψ. In both wheel-spaces, ε

_{r}reached unity at a non-dimensional purge flow rate Φ

_{o}< Φ

_{min}, i.e., ε

_{r}= 1 when ε

_{s}< 1.

## 6. Conclusions

- The sealing effectiveness for the downstream wheel-space was larger than for the upstream one; this was attributed to the fact that the circumferential variation of pressure (which is the driver of externally-induced ingress) in the annulus downstream of the blades was much smaller than that upstream.
- In the core of both wheel-spaces, the swirl ratio, which increased with increasing radius and with decreasing purge flow rate, could be correlated with the local turbulent flow parameter, λ
_{t}. However, the swirl ratio in the downstream wheel-space was smaller than that in the upstream one; this was attributed to the fact that the swirl in the annulus downstream of the blades was much smaller than that upstream. - A maximum likelihood estimation analysis was applied successfully to extrapolate transient temperature measurements in the core and rotor surfaces to determine the buffer parameter, Ψ, in both upstream and downstream wheel-spaces.
- Using the calculated values of Ψ for both wheel-spaces, there was mainly good qualitative agreement between the experimental and theoretical variations of rotor effectiveness with non-dimensional purge.
- It was shown that the buffer effect, Δε, of the purge flow was larger for the upstream seal than for the downstream one; this was attributed to the fact that the sealing effectiveness for the upstream wheel-space was lower than the downstream one.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

a | inner radius of rotor |

A,B | constants |

b | outer radius of rotor |

c | concentration |

c_{p} | specific heat at constant pressure |

C,D,E | constants |

C_{w,o} | nondimensional flow rate (= $\dot{m}$_{o}/μb) |

f | function relationship between β and β* |

f’ | empirical constant |

G | gap ratio (= S/b) |

G_{c} | seal-clearance ratio (= s_{c,ax/}b) |

H | total enthalpy |

H’ | total enthalpy of rotor when frictional effects are neglected |

k_{s} | thermal conductivity of solid |

l | negative logarithm of likelihood function |

$\dot{m}$ | mass flow rate |

n | number of data points |

P | likelihood function |

r | radius |

R | recovery factor |

Re_{ϕ} | rotational Reynolds number (= ρΩb^{2}/μ) |

s_{c,ax} | axial clearance of seals |

s_{c,rad} | radial clearance of seals |

s_{overlap} | axial overlap of seals |

S | axial space between rotor and stator |

t | time |

T | temperature |

U | bulk-mean velocity through rim-seal clearance (= $\dot{m}$_{o}/2πρbs_{c,ax}) |

V_{ϕ} | tangential velocity in wheel-spaces |

W_{r} | work done by the rotor from r = a to r = b |

β | swirl ratio in wheel-space (= V_{ϕ}/Ωr) |

β* | swirl ratio when λ_{t} = 0 |

Γ_{c} | ratio of discharge coefficients |

ΔT | difference between initial and extrapolated steady-state temperatures |

Δε | buffer effect (= ε_{r} − ε_{s}) |

ε_{r} | adiabatic effectiveness of rotor (= (H’ − H_{i})/(H_{o} − H_{i})) |

ε_{s} | concentration effectiveness of stator (= (c_{s} − c_{i})/(c_{o} − c_{i})) |

Θ_{o} | ratio of flow parameters (= Φ_{o}/Φ_{min}) |

λ_{T,o} | turbulent flow parameter (= C_{w,o} Re_{ϕ}^{−0.8}) |

λ_{t} | local turbulent flow parameter (= λ_{T,o}(r/b)^{−13/5}) |

μ | dynamic viscosity |

ρ | density |

σ | standard deviation from maximum likelihood estimation |

Φ_{o} | nondimensional flow parameter (= U/$\Omega b)$ |

Φ_{min} | value of Φ_{o} when there is no ingress |

Ψ | buffer parameter (= f’($\dot{m}$_{o}/$\dot{m}$_{r})) |

Ω | angular velocity of rotor |

Subscripts | |

ad | adiabatic |

exp | experimental measurements |

i | ingress |

inf | extrapolated steady-state temperature |

ini | initial steady-state temperature |

true | “true” values from maximum likelihood estimation |

o | purge flow |

r | rotor |

s | stator |

t | total values |

## References

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**Figure 4.**Transient temperature measurements and “true” temperature from MLE: upstream wheel-space and Φ

_{o}= 0.0265.

**Figure 5.**Variation of β/β* with λ

_{t}for upstream and downstream wheel-spaces (symbols: data; lines: swirl correlations).

**Figure 7.**Variation of ε

_{s}with Φ

_{o}for upstream and downstream wheel-spaces (Re

_{ϕ}= 7.2 × 10

^{5}). (Symbols: data; lines: Externally-induced effectiveness correlation).

**Figure 8.**Variation of ψ with Φ

_{o}in wheel-spaces upstream and downstream of the rotor (Re

_{ϕ}= 7.2 × 10

^{5}). (Symbols: data; lines: fitted theoretical curves).

**Figure 9.**Variation of ε

_{r}with Φ

_{o}in wheel-spaces upstream and downstream of the rotor (Re

_{ϕ}= 7.2 × 10

^{5}). (Symbols: data; broken lines: EI effectiveness correlation; solid line: adiabatic rotor effectiveness using the fitted Ψ).

Parameters | Values |
---|---|

b (outer radius of the disc) | 190 mm |

S (axial width of both wheel-spaces) | 10 mm |

s_{c,ax} (axial clearance) | 2 mm |

s_{c,rad} (radial clearance) | 1.28 mm |

s_{overlap} (axial overlap) | 1.86 mm |

G_{c} (seal-clearance ratio, = s_{c,ax}/b) | 0.0105 |

G (gap ratio, = S/b) | 0.0526 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Tang, H.; Cho, G.; Patinios, M.; Scobie, J.A.; Sangan, C.M.; Owen, J.M.; Lock, G.D.
Effect of Ingress on Flow and Heat Transfer Upstream and Downstream of a Rotating Turbine Disc. *Aerospace* **2019**, *6*, 49.
https://doi.org/10.3390/aerospace6050049

**AMA Style**

Tang H, Cho G, Patinios M, Scobie JA, Sangan CM, Owen JM, Lock GD.
Effect of Ingress on Flow and Heat Transfer Upstream and Downstream of a Rotating Turbine Disc. *Aerospace*. 2019; 6(5):49.
https://doi.org/10.3390/aerospace6050049

**Chicago/Turabian Style**

Tang, Hui, GeonHwan Cho, Mario Patinios, James A. Scobie, Carl M. Sangan, J. Michael Owen, and Gary D. Lock.
2019. "Effect of Ingress on Flow and Heat Transfer Upstream and Downstream of a Rotating Turbine Disc" *Aerospace* 6, no. 5: 49.
https://doi.org/10.3390/aerospace6050049