Javier A. Kypuros

University of Texas, Pan American University, Edinburg, Texas

Kevin J. Melcher

Glenn Research Center, Cleveland, Ohio

A Reduced Model for Prediction of Thermal

and Rotational Effects on Turbine

Tip Clearance

NASA/TM—2003-212226

March 2003

https://ntrs.nasa.gov/search.jsp?R=20030032933 2018-05-15T15:58:23+00:00Z
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Javier A. Kypuros

University of Texas, Pan American University, Edinburg, Texas

Kevin J. Melcher

Glenn Research Center, Cleveland, Ohio

A Reduced Model for Prediction of Thermal

and Rotational Effects on Turbine

Tip Clearance

NASA/TM—2003-212226

March 2003

National Aeronautics and

Space Administration

Glenn Research Center
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Available electronically at http://gltrs.grc.nasa.gov

The Propulsion and Power Program at

NASA Glenn Research Center sponsored this work.
A Reduced Model for the Prediction of Thermal and Rotational 

Effects on Turbine Tip Clearance 

Abstract 

This paper describes a dynamic model that was developed to predict changes in turbine tip 

clearance – the radial distance between the end of a turbine blade and the abradable tip seal. The clearance 

is estimated by using a first principles approach to model the thermal and mechanical effects of engine 

operating conditions on the turbine subcomponents. These effects are summed to determine the resulting 

clearance. The model is demonstrated via a ground idle to maximum power transient and a lapse-rate 

takeoff transient. Results show the model demonstrates the expected “pinch point” behavior. The paper 

concludes by identifying knowledge gaps and suggesting additional research to improve the model. 

Nomenclature 

Symbol Units Description

A m2 area 

E Pa Young’s modulus 

F N force 

L m unstressed blade length 

N RPM rotational speed 

P Pa pressure 

T °C temperature 

V m3 volume 

c J/kg-°C specific heat 

h W/m2

-°C convection heat transfer coefficient 

k W/m-°C thermal conductivity 

l m stressed blade length 

q W heat transfer rate 

r m radius 

t sec time 

u m deformation 

w m width 

α 1/°C thermal expansion coefficient 

δ tip clearance 

ρ kg/m3 density 

ν Poisson’s ratio 

ω rad/sec angular speed 

σ Pa stress 

ε m/m strain 

η film cooling effectiveness 

π ratio of the perimeter of a circle to its diameter 

Javier A. Kypuros

University of Texas, Pan American University

Edinburg, Texas 78539

Kevin J. Melcher

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

NASA/TM—2003-212226 1
1. Introduction 

Turbine blade tip clearance continues to be a concern in the design and control of gas turbines. 

Ever increasing demands for improved efficiency and higher operating temperatures require more 

stringent tolerances on turbine tip clearance. A successfully implemented active tip clearance control is 

expected to have benefits of increased efficiency, reduced specific fuel consumption, and additional 

service life. 

The focus of this study is to develop a simplified model from first principles that grossly captures 

the dynamic change in turbine tip clearance during take-off. Such a model can aid in identifying and 

understanding the primary mechanisms involved in the turbine tip clearance problem and their inherent 

frequency response. This work serves to identify knowledge gaps that must be addressed in order to 

facilitate design and development of an actuator and control strategy for active clearance control. 

Furthermore, this work is a step towards a higher fidelity model suitable for virtual prototyping*

 of a 

controller/actuation mechanism. The purpose of beginning with this first-principles approach is to 

eliminate the necessity for a finite-element model that would require detailed information about geometry 

specific to a particular engine system. The model developed herein is intended to be more generic. 

The main objectives of this study can be summarized as follows: 

1. Identify and characterize the primary mechanisms involved in turbine tip clearance dynamics; 

2. Identify the known and unknown parameters necessary to predict turbine tip clearance; and 

3. Develop a simplified model that can be used to: 

(a) reasonably represent turbine tip clearance during common gas turbine transients (takeoff, reburst, etc.), 

(b) determine actuator performance requirements for active clearance control, and 

(c) serve as a baseline for virtual prototyping of a controller and actuation system. 

Previous NASA efforts in collaboration with industry have quantified tip clearance transients 

using both experimental and numerical results [1, 2, 3]. One example is the Energy Efficient Engine 

studies [1] conducted in the late 1970’s and early 1980’s. Under these studies, detailed analyses and 

experiments were performed to identify the dynamic characteristics of mechanisms impacting turbine tip 

clearance. Due to proprietary issues, these reports, however, do not describe analysis used to derive 

numerical results. Much of this work has been reserved for classified reports which document analyses 

performed using proprietary finite-element tools, and few publications document the combined effects of 

both thermal and mechanical stresses experienced by the rotor, blade, and shroud. Little has been 

published on the stress analysis used to predict deflections of the rotor, blade, or shroud. Most of the 

finite-element studies in the area focus on long term effects like creep and crack propagation and do not 

focus on prediction of deflections due to thermal and rotational transients during flight. Significant 

experimental and numerical work has been published on the film cooling of turbine blades [4, 5, 6, 7, 8]. 

For NASA to conduct fundamental research that will facilitate future technologies in active turbine tip 

clearance control, a valid model or a test rig is necessary. The test rig can be potentially costly to build 

and run. A model-based simulation is less costly, and is useful in designing the test rig and associated 

experiments. If sufficiently accurate, it can be used to virtually prototype the controller/actuator 

mechanism. 

This document is divided into four chapters, introduction, model development, implementation 

and results, and discussion. The section on model development is presented in three sub-sections based 

on the three system elements composing the turbine – the shroud, rotor, and blade. Each element is 

presented by describing the parameters and boundary conditions necessary for analysis and by detailing 

assumptions used to facilitate results. Analyses used to determine thermal and mechanical deflections are 

described for each element. This is followed by a section in which parameter values are defined for 

simulation purposes; and preliminary results of clearance predictions are presented and discussed. The 

 

*

Virtual prototyping refers to the process by which models and simulations are employed to virtually

design and test mechanisms and/or controls. 

NASA/TM—2003-212226 2
paper concludes with a discussion in which the knowledge gaps of most concern are identified and 

summarized, and suggestions for further model development are presented. 

2. Model Development 

As shown in Figure 1, the proposed model incorporates three basic elements – the shroud, turbine 

rotor (or disk), and turbine blade. In order to predict deflections of each element due to thermal and 

mechanical stresses the temperature, pressure, and force distributions in each element must be modeled. 

The block diagram shows the parameters necessary to calculate these temperature and stress distributions. 

An engine model, not discussed in this report, is employed to provide speed, temperature, and pressure 

transients for the shroud, rotor, and blade sub-models. Each sub-model predicts deflections due to 

thermal and mechanical stresses. As shown in Equation 1, the relative change in the time-varying 

geometry of each sub-model is then used to calculate the overall change in tip clearance. 

[ ]

[ ])()()(

)()()()(

bbrrssa 2121021

bladerotorshroud

uuLuuruur

tltrtrt

+++++−++=

δ +−=

 (1) 

Here, rshroud(t), rrotor(t), and lblade(t) are, respectively, the shroud inner radius, rotor outer radius, and blade 

length as a function of time. Note that, ra, r0, and L are the initial geometric state of the shroud, rotor, and 

blade respectively; while the subscripted u’s denote deformations due to thermal and mechanical forces. 

Figure 1. Block diagram for tip clearance model

Shroud

Parameters:

• geometry of unstressed shroud (w, ra, rb, and rc) 

• material properties of thermal barrier (ka, ca, and ρa) 

• material properties of shroud (αb, kb, cb, and ρb) 

• convection coefficient of thermal barrier (ha) 

• convection coefficient of shroud (hb) 

Inputs:

• turbine inlet airflow temperature (Tturbine) 

• turbine inlet airflow pressure (Pturbine) 

• compressor discharge temperature (Tcompressor) 

• compressor discharge pressure (Pcompressor) 

Outputs:

• shroud/thermal barrier inner radius (rshroud) 

• average shroud temperature (Tshroud) 

Turbine Rotor

Parameters:

• unstressed rotor geometry (r) 

• material properties (α, c, E, k, ρ and ν) 

• rotor convection coefficient (h) 

Inputs:

• compressor discharge temperature (Tcompressor) 

• angular speed (ω) 

Outputs:

• rotor tip radius (rrotor) 

• average rotor temperature (Trotor) 

Turbine Blades

Parameters:

• geometry of unstressed blade (L and A) 

• material properties (α, c, E, k, ρ, and ν) 

• convection coefficient for film cooling (h) 

• film cooling effectiveness (η) 

Inputs:

• temperature of primary airflow (Tturbine) 

• temperature of film cooling airflow (Tcompressor) 

• angular speed (ω) 

Outputs:

• stressed blade length (lblade) 

• blade temperature (Tblade)

Turbine Engine Transient

Outputs:

• compressor/turbine angular speed (ω) 

• compressor discharge pressure (Pcompressor) 

• compressor exit temperature (Tcompressor) 

• turbine inlet pressure (Pturbine) 

• turbine inlet temperature (Tturbine) 

+ - - Clearance

rshroud

lblade

rrotor

NASA/TM—2003-212226 3
Equations for calculating these deflections will be given in the subsections that follow. 

2.1. Shroud 

As shown in Figure 2, it is assumed that the general geometry of the shroud is a ring-like 

structure where the inner surface is coated with an abradable material that also functions as a thermal 

barrier. However, the shroud is not a single continuous ring-like structure. Instead it is made up of a 

series of arcs joined together to form a ring, which may negate the usefulness of this assumption. This 

issue shall be addressed in § 2.1.4 with an alternative formulation of the shroud growth due to thermal 

stresses on the casing to which the shroud is attached. The inner surface of the abradable material at 

radius ra is exposed to heated gas at a temperature approximately equal to the turbine inlet temperature, 

Tturbine. The outer surface of the shroud at radius rb is exposed to compressor discharge air which is 

assumed to be at the compressor discharge temperature, Tcompressor. As shown in Figure 2, pressure 

distributions on the inner and outer surface of the shroud are adapted from Lattime and Steinetz [9]. For 

simplicity, pressure in the tip clearance region is assumed to vary linearly between the turbine inlet and 

exit pressures. Compressor bleed air is used to purge the space between the shroud and the casing. To 

facilitate results, it is assumed that the temperature difference between the compressor discharge and the 

shroud outer surface is negligible. 

For the purposes of this study, the abradable material is presumed to function primarily as a 

thermal barrier and not as a structural member. It is assumed, therefore, that the abradable material acts 

as a coating and does not impose a significant stress on the inner surface of the superalloy layer as the 

shroud grows due to thermal loads. Additionally, because the shroud is attached to the case in the form of 

a series of arcs, it is assumed that the abradable material layer maintains its approximate thickness even 

when thermally stressed (i.e. the relative distance between the bond radius and inner wall, rc–ra, is 

Figure 2. Shroud schematic

ra

rb

rc

Tcompressor & hb

Tturbine & ha

abradable material (ρa, ka, & ca)

super alloy (ρb, kb, cb, & αb)

Tb

Ta

Tc

qr

ln(rc/ra)

2πkaw

ln(rb/rc)

2πkbw

w is width of

shroud into the

page

0.8Pcompressor

0.7Pcompressor

0.3Pcompressor

w

A 

A 

Gas Flow →

Section A-A 

Pin 

Pout 

PPurge

NASA/TM—2003-212226 4
assumed to remain relatively constant). Due to its conductive properties, the abradable layer dissipates 

much of the turbine heat, thus reducing the temperature otherwise experienced by the superalloy surface 

at radius rc. This temperature, Tc, is the thermal boundary condition on the inner surface of the alloy used 

to calculate the deflection due to thermal stresses. The heat transfer at the inner surface of the abradable 

material and outer surface of the alloy are assumed to be uniform (quasi-steady) to facilitate a simple 

model. Heat is convected to the inner surface of the abradable material at wall temperature Ta with a heat 

transfer coefficient of ha. The convection coefficient at the outer surface of the shroud is hb

 and the 

associated surface temperature is Tb. 

2.1.1. Shroud Heat Transfer 

To facilitate a closed form solution for the heat transfer across the shroud, the responses near the 

shroud surfaces are assumed to be governed by a semi-infinite formulation of the transient heat 

conduction equation where the temperature at the surface, Tw, is given by the classical solution [6, 7]: 

2

1 exp erfc w i

r i

TT h h

t t

T T ck ρ ρck

−     = −     −    

 (2) 

where Tr and Ti are the reference temperature and initial wall temperature, respectively. This formulation 

is appropriate as long as the transient temperature penetration does not exceed the thickness of the wall 

material.*

 For a single-flow system, the reference temperature is the temperature of the mainstream flow 

which for the inner surface of the shroud is assumed to be Tturbine and for the outer surface Tcompressor. This 

formulation is used to calculate the inner and outer shroud wall temperatures, Ta and Tb, respectively. 

Using a thermal resistance formulation [10] as shown in Figure 2, one can estimate the bond 

temperature, Tc, at the interface between the abradable material and the superalloy: 

.

)/ln(

)/ln( 1

2

)/ln(

2

)/ln(

2

)/ln(

b

a

ac

cb

ba

ac

a

ac

ca

b

cb

a

ac

ba

k

k

rr

rr

TT TT

wk

rr

TT

wk

rr

wk

rr

TT

+

− ⇒ −= − =

+

−

π π π

 (3) 

Additionally, by representing the superalloy layer of the shroud as a hollow cylinder and 

assuming one-dimensional heat conduction in the radial direction under quasi-steady state conditions, the 

heat diffusion equation simplifies to 

0 1  =



 





dr

dT kr

dr

d

r

which, when integrated and combined with the boundary conditions T(rc)=Tc and T(rb)=Tb, results in the 

radial temperature distribution 

.

)/ln(

)/ln( )()(

cb

c

cbc

rr

rr

−+= TTTrT (4) 

2.1.2. Shroud Deflection due to Thermal Stresses, us1

To enable prediction of shroud deflection due to thermal stresses, it is assumed that the stresses 

and displacement do not vary over the width of the shroud. The formulation presented herein is 

equivalent to that given in [11] for a thin circular disk with a hollow center (like a ring). According to 

Timoshenko [11], the radial and tangential stresses (σr and σθ) must satisfy the equation of equilibrium 

= 0 − +

dr r

dσ r σ r σ θ (5) 

 

*

The application of the semi-infinite solution should be justified in future work and is used here primarily 

to facilitate initial results. 

NASA/TM—2003-212226 5
The strain components in the radial and tangential directions are 

dr

du

E

T rr )( =−+=

1

α σ νσ θ ε

and 

r

u

E

T r )( =−+=

1

ε θ α σ θ νσ

The above strain equations can be solved for the radial and tangential stresses: 

[ ] T E

ε rr νε ν α

ν

θ σ )1( 1 2 +−+ − =

and 

[ ] T E

ε νε r ν α

ν

θ θ σ )1( 1 2 +−+ − =

Given the above stress equations and noting that εr = du/dr and εθ= u/r the equilibrium equation 

(Equation 5) may be written as 

.)1( )(1

dr

dT

dr

rud

rdr

d

+= ν α 



 





Integrating twice yields the radial deflection 

r

C rCTrdr

r

ru r

rc

2

1

1 )1()( +++= ∫ ν α (6) 

where rc is the lower limit of the radius or in this case the unstressed inner wall radius of the shroud 

superalloy layer. Assuming no thermal stress at the inner and outer boundaries of the superalloy portion 

of the shroud (surfaces stresses due to pressure will be considered separately in § 2.1.3) and recalling the 

radial temperature distribution (Equation 4), C1 and C2 are: 















 − 













 − − + − = )( )/ln(2

1

2

)1(

2 2

2

1 cb

cb cb

b

c TT

rr rr

r TC α ν (7) 

and 

.)( )/ln(2

1

2

)1(

2 2

2 2

2















 − 













 − − +

+ = cb

cb cb

b

c

c TT

rr rr

r T r C α ν (8) 

Because the abradable material is thought to serve mainly as a thermal barrier, the shroud 

deflection is primarily due to the superalloy layer and thus should be measured from the unstressed bond 

radius, rc. The thickness of the abradable material layer can later be subtracted since, as previously 

mentioned, it is assumed to remain relatively constant. Thus the shroud radial deflection due to thermal 

stresses is 

.)( )/ln(2

1 )( 2 2

2

1















 − 













 − − +== cb

cb cb

b

cccs TT

rr rr

r

α Trruu (9) 

2.1.3. Shroud Deflection due to Pressure Differential, us2

To predict deflection of the shroud at the bond radius, rc, due to pressure differential, 

Timoshenko’s hollow cylinder equation is adapted [11]: 

[ ].)()1()()1( )(

1 2 22

2 2 2 iobcbocic

cb

s PPrrrPrPr

rrE

u −+−−− − = ν ν (10) 

NASA/TM—2003-212226 6
In Equation 10, Pi is the mean internal pressure and Po is the outer surface pressure. Per Figure 2, they 

are equivalent to (Pin+ Pout)/2 and Ppurge, respectively. 

2.1.4. Alternative Shroud Growth Formulation 

Alternatively, the deflection of the shroud can be thought of as limited by the casing growth. The 

shroud can be then modeled as a series of arcs rigidly attached to the casing as shown in Figure 3. The 

casing grows due to thermal and pressure differentials. Similar to the previous formulation, the surface 

temperatures on the outer and inner surfaces of the casing, Te and Td, can be calculated from mainstream 

flow temperatures and experimentally or numerically determined convection coefficients using 

Equation 2. The outer casing wall, re, is assumed to be exposed to ambient temperature and pressure (T∞

and P∞). If the casing is grossly modeled as a hoop-like structure then deflection due to thermal 

differential can be derived as was done in § 2.1.2. The deflection due to thermal stresses would, 

alternatively, be as shown in Equation 11. 

.)( )/ln(2

1

22

2

1















 − 













 − − += de

de de

e

dds TT

rr rr

r

α Tru (11) 

Similarly, the alternative deflection due to pressure differential can be derived as 

[ ].)()1()()1( )(

1 2 22

2 2 2 ioedeodid

de

s PPrrrPrPr

rrE

u −+−−− − = ν ν (12) 

where Pi is now the pressure inside the cavity, Ppurge, and Po is the ambient pressure, P∞. 

In this model, a higher bandwidth mechanical clearance control system is presumed to replace the 

slower thermal clearance control systems found on many current commercial engines. Therefore, active 

cooling of the case is not considered. The deformation of the structure that connects the shroud to the 

case is another effect that is not developed here. The structure deforms as the differential pressure across 

the shroud changes, and so, should be included in the alternative formulation. 

Tcompressor

re

rb

ra

rd

Tturbine

T8

Td & hd

Tb & hb Tb

& hb

Te & he

casing (ρc, kc, cc, & αc)

shroud supports

shroud

shroud support

T8 Tturbine

Tcompressor &

0.8Pcompressor

cavity between

shroud and casing

& P8

& P8

Figure 3. Shroud/casing schematic

NASA/TM—2003-212226 7
2.2. Rotor 

Detailed studies of rotor-stator systems have been conducted by Owen [12]. However, in the 

attempt to arrive at a simplified model representing the gross clearance dynamics, this initial effort has 

focused on modeling the first-order effects. For this initial study, the rotor structure is modeled as a 

rotating disk of uniform thickness, and only deflections due to changes in centrifugal forces and purge air 

temperature are considered. Compressor discharge air is generally used to purge and cool the cavity 

around the rotor. Only the small tip surface of the rotor is exposed to the high temperature core flow. 

Thermal expansion is assumed to be dominated by heat transfer between the rotor disk and the 

compressor discharge air, at temperature Tcompressor. Thus, as suggested by Figure 4, the rotor is assumed 

to be insulated to heat transfer from the turbine blades simplifying the analysis. Compressor discharge air 

flows over the majority of the rotor’s surface area. 

2.2.1. Rotor Deflection due to Thermal Stresses, ur1

As with the outer shroud surface temperature, the rotor surface temperature is assumed to be 

governed by the semi-infinite formulation (i.e., Equation 2) where the reference temperature is the 

compressor discharge temperature. Because most of the rotor’s surface is exposed to the discharge 

temperature, the temperature through the width of the rotor is assumed to be relatively constant across the 

surface and approximately equal to the surface temperature (i.e., Trotor=Tcompressor). Note that in this 

simplified model, windage heating, which may produce a significant temperature gradient between the 

hub and the tip, is ignored. 

As with the outer shroud surface, the heat transfer convection coefficient is not known, but it is 

assumed to be similar in magnitude to that given for the inner shroud surface. Since the rotor temperature 

is assumed to be relatively constant throughout, the thermal strain analysis is significantly simplified: 

dr

du

ε r αT == and .

r

u

ε θ αT ==

Therefore, the approximate rotor deflection due to thermal stresses is 

r = αTru 01 (13) 

where αr and r0 are the rotor thermal expansion coefficient and unstressed radius, respectively. 

Figure 4. Rotor schematic

insulated 

(assumption)

r0

Trotor & hr

ρr, Er

, νr, cr, kr & αr

(a) frontal view (b) side view

ωr

Core Flow →

Core Flow →

Tcompressor

Cooling Flow

Tcompressor

Cooling Flow

Rotor 

 

NASA/TM—2003-212226 8
2.2.2. Rotor Deflection due to Centrifugal Forces, ur2

To estimate the rotor tip radial deflection due to centrifugal forces, a simplified form of the 

rotating flat disk equation from reference 11 is used: 

3

0

2

2 )1( 4

1

r

E

u −= ν ρ rrr ω (14) 

where νr, ρ r, and E r are material properties and ω=(2π/60)N is the rotor angular speed in rad/sec. 

2.3. Blades 

Of the three basic components, the most documented is the turbine blade. The blades are stressed 

by both centrifugal forces and thermal expansion. Because of the high temperatures within the turbine 

stage the turbine blades are film cooled by compressor discharge air. As shown in Figure 5, compressor 

discharge air that flows through the blade exits film cooling holes on the leading edge of the blade and is 

introduced into the boundary layer at the surface of the blade. The blades are of length L and crosssectional area Ac. 

2.3.1. Film Cooling of Turbine Blades 

Film cooling is employed to reduce the heat transfer from hot air in the turbine at the turbine inlet 

temperature, Tturbine, to prevent surface temperatures in excess of the blade material’s melting point. In 

film cooling, a secondary flow is introduced into the boundary layer at the surface of the blade [13]. The 

temperature of the film cooling fluid, Tf, is much lower than the mainstream temperature, Tm. The blade 

surface temperature, Tblade, is determined using a semi-infinite formulation similar to Equation 2 with the 

exception that the reference temperature, Tr, is not the mainstream temperature as in two temperature 

flows. The reference temperature for film cooling is generally unknown and depends on the supply 

temperatures of the two interacting flows and the degree of mixing that occurs [6, 7]. In such cases, the 

reference temperature and convection coefficient, h, must be determined experimentally or numerically. 

Using experimental data, a film cooling effectiveness, similar to that shown in Equation 15, is determined 

in terms of the two flow temperatures, Tm and Tf [7]. 

L

ρ, E, ν, c, k & α

Αc

Tcompressor

Tturbine

Tblade & h

Tturbine

film cooling hole

blade surface

film cooling hole

Figure 5. Blade schematic

NASA/TM—2003-212226 9
.

mf

mr

TT

TT

−

− η = (15) 

In the case of the turbine blades, the mainstream fluid is at Tturbine and the film cooling flow is 

assumed to be at the compressor discharge temperature, Tcompressor. 

2.3.2. Blade Deflection due to Thermal Stresses, ub1

The blade is relatively thin, and thus the thermal gradient through the thickness of the blade is 

assumed negligible. Therefore, the blade is assumed to have a uniform temperature equal to the surface 

temperature, Tblade. This simplifies the thermal stress analysis and results in a simple linear relation: 

b1 = αTLu . (16) 

2.3.3. Blade Deflection due to Centrifugal Forces, ub2

Like the rotor, the blade is stressed by centrifugal forces. To simplify results, the maximum blade 

and rotor growth are thought to be sufficiently small to assume that the centrifugal force, Fc, is primarily a 

function of the rotor’s angular velocity, ω(t), and does not vary significantly due to the rotor and blade 

growth. The deflection can hence be determined from a simple stress strain relation: 

( )

( ) .

)(/

22

2

2

0

2

0

E

rLL Lu

L

L

E

rLL

EA

rLm

E

AF

E

b

c

cc

ρ ω

σ ω ρ ω

ε

+

⇒ =∆=

∆ = + = + ===

 (17) 

2.4. Clearance Calculation 

Having obtained expressions for the time varying deflections of the various turbine sub-models 

due to thermal and mechanical forces, these can now be summed to obtain the resulting tip clearance. 

Equation 1, reproduced here for convenience, shows how the relative change in the time-varying 

geometry of each sub-model is used to calculate the overall change in tip clearance. 

[ ]

[ ])()()(

)()()()(

bbrrssa 2121021

bladerotorshroud

uuLuuruur

tltrtrt

+++++−++=

δ +−=

Here, δ(t) is the overall tip clearance as a function of time. Variables rshroud, rrotor, and lblade are, 

respectively, the shroud inner radius, rotor outer radius, and blade length as a function of time. Note that, 

ra, r0, and L are the initial geometric state of the shroud, rotor, and blade, respectively; while the 

subscripted u’s denote time-dependent results of previously described deformation calculations. 

3. Model Implementation and Results 

 In this section specific parameters used in implementing a dynamic simulation containing the 

previously presented tip clearance model are discussed, along with results from that simulation. Details 

of the actual implementation are not given here, but are given in the Appendix included at the end of this 

report. 

3.1. Implementation 

 In this section, values are given for the various parameters required to implement the tip clearance 

model described in this paper. The parameters include geometry, material properties, and heat transfer 

coefficients. A consistent set of data containing all of the necessary model parameters was unavailable. 

Therefore, it was necessary to use estimates for many of the parameters. What follows is a discussion of 

NASA/TM—2003-212226 10
the sources, confidence level, and associated assumptions used in determining the various parameter 

values. 

3.1.1. Shroud 

Pressure distributions on the inner and outer surface of the shroud as shown in Figure 2 are 

adapted from Lattime and Steinetz [9]. For simplicity, pressure in the tip clearance region is assumed to 

vary linearly between the turbine inlet and exit pressures. Per Lattime and Steinetz [9], these pressures 

are approximately 70% and 30% of compressor discharge, respectively. The resulting average pressure, 

0.5Pcompressor, is used on the inner surface of the shroud to determine the shroud deflection due to pressure 

differential. Compressor bleed air is also used to purge the space between the shroud and the casing. The 

resultant pressure on the outer shroud surface is assumed to have an even distribution approximately 80% 

of Pcompressor. In this preliminary model, delays between the pressure signal at the compressor exit and the 

subsequent signal in the turbine are assumed negligible. 

The shroud parameters and material properties are given in Table 1. Subscript a denotes 

materials and surface conditions of the abradable material, while subscript b denotes the same for the 

superalloy. For the purposes of this model the superalloy layer is assumed to be made of Inconel 718 and 

the abradable material of a partially stabilized Zirconium Oxide. Parameter values are rated on a level of 

confidence from 1 to 5; 5 indicating near certainty. Experimental approximations of the convection 

coefficient on the inner surface of the shroud (i.e., in the tip clearance region) are given in [8] and match 

reported experimental results for the blade tip [4, 5, 6, 7]. The convection heat transfer coefficient for the 

outer surface of the shroud is not known at this time but to facilitate results is assumed to be similar in 

magnitude to that on the inner shroud surface. 

For the alternative shroud growth formulation, the inner casing wall, rd, is exposed to the 

compressor discharge temperature, Tcompressor, and approximately 80% of the compressor discharge 

pressure, 0.8Pcompressor. Convection coefficients for the case are not readily available. 

3.1.2. Rotor 

Rotor disks in most modern gas turbines are fabricated using newer members of the René family 

of superalloys. Some rotors [1] are fabricated from René 95 which has properties similar to Inconel 718. 

For this work, properties of Inconel 718 are used. Table 2 gives parameters used in the rotor sub-model. 

Table 1. Shroud parameters 

Parameter Description Value Confidence 

ra thermal barrier inner radius 38 cm (~15 in) 4 

rb shroud outer radius 38.6 cm (~15.2 in) 3 

rc shroud inner (bond) radius 48.3 cm (~15.1 in) 3 

w shroud width 5.1 cm (~2 in) 4 

ρa abradable material density 5030 kg/m3 3 

ka abradable material thermal conductivity 2.93 W/m-°C 3 

ca abradable material specific heat 456 J/Kg-°C 3 

ρb shroud density 8220 kg/m3 5 

kb shroud thermal conductivity 24.9 W/m-°C 5 

cb shroud specific heat 645 J/Kg-°C 5 

αb shroud thermal expansion coefficient at 538°C 15.1x10-6 1/°C 5 

ha convection coefficient at inner surface 500 W/m2

-°C 4 

hb convection coefficient at outer surface 1000 W/m2

-°C 2 

NASA/TM—2003-212226 11
3.1.3. Blades 

The blade parameters, material properties, and parameter confidence levels are given in Table 3. 

For this initial model, blades are assumed to be made of a material similar to Inconel 718. Turbine blades 

found in modern gas turbine engines are generally fabricated as single crystals using the René family of 

superalloys. The incorporation into the clearance model of properties for these superalloys would be 

appropriate for future studies. 

Experimental values for both the convection coefficient and effectiveness are presented in [6, 7] 

and numerical estimations of the convection coefficient can be found in [14, 15]. For the purpose of this 

study, a bulk or blade-surface-averaged effectiveness and convection coefficient are used. Given the 

experimentally determined convection coefficient and effectiveness, Equations 2 and 15 can be used to 

determine the blade surface temperature, Tblade. 

3.2. Results 

In this section the transient clearance is predicted for a ground idle-to-maximum power transient 

and for a lapse-rate take-off transient. As shown in the upper plot of Figure 6, the ground-idle to

maximum power transient consists of a step in the throttle setting between the two operating conditions. 

A lapse-rate take-off transient involves ramping the throttle setting from ground-idle to about 90% of 

maximum power, performance of final flight checks, and then ramping up to maximum power. The 

throttle transient for the lapse-rate takeoff is shown in the upper plot of Figure 8. 

For the ground idle-to-maximum power transient, the time history of the engine speed is shown 

on the upper axes of Figure 6. Associated temperatures transients for the high pressure turbine and the 

Table 2. Rotor parameters 

Parameter Description Value Confidence 

r0 unstressed radius 32.9 cm (~13 in) 3 

wr width 5.1 cm (~2 in) 3 

ρr density 8220 kg/m3 4 

Εr modulus of elasticity 139 GPa 4 

νr Poisson’s ratio 0.4 4 

kr thermal conductivity 24.9 W/m-°C 4 

cr specific heat 645 J/Kg-°C 4 

αr thermal expansion coefficient 15.1x10-6 1/°C 4 

hr convection coefficient at surface 500 W/m2

-°C 2 

Table 3. Blade parameters 

Parameter Description Value Confidence 

L unstressed length 5 cm (~1.97 in) 3 

Ac cross-sectional area 6 cm2

 (~0.93 in2) 3 

ρ density 8220 kg/m3 5 

Ε modulus of elasticity 139 GPa 5 

ν Poisson’s ratio 0.4 4 

k thermal conductivity 24.9 W/m-°C 5 

c specific heat 645 J/Kg-°C 5 

α thermal expansion coefficient at 

871 °C 

16.4x10-6 1/°C 5 

h convection coefficient at surface 1000 W/m2

-°C 3 

η film cooling effectiveness 0.5 3 

NASA/TM—2003-212226 12
0 10 20 30 40 50 60 70 80 90 100

0.7

0.95

1.2

1.45

1.7

x 104

N (rp

m)

N (rpm)

Throttle

0 10 20 30 40 50 60 70 80 90 100

G/I

Max Throttle Setting

0 10 20 30 40 50 60 70 80 90 100

0

1

2

3

4

Deflection (

m

m)

Shroud

Rotor

Blade

0 10 20 30 40 50 60 70 80 90 100

0

50

100

150 Deflection (

mils)

0 10 20 30 40 50 60 70 80 90 100

0.5

0.75

1

1.25

1.5

Clearance (

m

m)

Time (seconds)

0 10 20 30 40 50 60 70 80 90 100

20

30

40

50

Clearance (

mils)

Figure 6. Deflections and clearance for transient from ground idle (G/I) to maximum power 

0 10 20 30 40 50 60 70 80 90 100

200

600

1000

1400

1800

Temperature ( oC)

Tturbine

T

compressor

0 10 20 30 40 50 60 70 80 90 100

1000

2000

3000

Temperature ( oF)

0 10 20 30 40 50 60 70 80 90 100

200

600

1000

1400

Temperature ( oC)

Ta

Tc

Tb

0 10 20 30 40 50 60 70 80 90 100

500

1000

1500

2000

2500 Temperature ( oF)

0 10 20 30 40 50 60 70 80 90 100

200

400

600

800

1000

Temperature ( oC)

Time (seconds)

Tshroud

Trotor

Tblade

0 10 20 30 40 50 60 70 80 90 100

500

1000

1500

Temperature ( oF)

Figure 7. Temperatures for transient from ground idle (G/I) to maximum power 

NASA/TM—2003-212226 13
high pressure compressor are shown on the upper axes of Figure 7. Similar plots for the lapse-rate-takeoff transient are shown on the upper axes of Figure 8 and Figure 9. Recall from Figure 1, that these 

parameters serve as inputs to the shroud, rotor, and blade sub-models. It is important to note that a 

transient simulation of a large commercial engine was not available. Instead, a simulation of a small 

military engine was used to generate the input transients. The transients were then scaled (as shown) to 

approximate the dynamics of a large commercial engine, for which the clearance model was developed. 

In addition to the transient temperature inputs, Figure 7, shows the shroud temperatures and the 

basic element temperatures. Note that the shroud temperature, Tshroud, plotted on the lower axis, is just the 

average of the bond and outer wall temperatures, Tc and Tb, respectively. As evidenced by the figure, the 

dynamic temperatures generally follow the expected trend – a relatively slow response compared to the 

input temperature transients. The outer wall temperature approaches an expected steady-state value 

between 1200-1300 °F. However, the bond temperature does not appear to approach an expected value of 

near 1800 °F. This is likely due to three primary reasons: 

1. the convection coefficient for the outer shroud surface is not known or well documented so the 

assumed value may be incorrect, 

2. the actual properties of the abradable material are not known. A change in the thermal 

conductivity of the abradable material significantly affects the bond temperature, and 

3. the input engine transients may not match those for a larger commercial engine. 

Data on the lower axes also show that the blade wall temperature, Tblade, stays well below Inconel 718’s 

melting point of 2370 °F. 

Figure 6 shows the individual deflections and overall clearance with reference to the engine speed 

transient. As expected the rotor initially responds more quickly due to the centrifugal forces induced by 

the engine speed transient. The shroud catches up and grows more rapidly due to thermal stresses than 

0 10 20 30 40 50 60 70 80 90 100

0.7

0.95

1.2

1.45

1.7

x 104

N (rp

m)

N (rpm)

Throttle

0 10 20 30 40 50 60 70 80 90 100

G/I

Max Throttle Setting

0 10 20 30 40 50 60 70 80 90 100

0

1

2

3

4

Deflection (

m

m)

Shroud

Rotor

Blade

0 10 20 30 40 50 60 70 80 90 100

0

50

100

150 Deflection (

mils)

0 10 20 30 40 50 60 70 80 90 100

0.5

0.75

1

1.25

1.5

Clearance (

m

m)

Time (seconds)

0 10 20 30 40 50 60 70 80 90 100

20

30

40

50

Clearance (

mils)

Figure 8. Deflections and clearance for lapse-rate takeoff transient 

NASA/TM—2003-212226 14
0 10 20 30 40 50 60 70 80 90 100

200

600

1000

1400

1800

Temperature ( oC)

Tturbine

T

compressor

0 10 20 30 40 50 60 70 80 90 100

1000

2000

3000

Temperature ( oF)

0 10 20 30 40 50 60 70 80 90 100

200

600

1000

1400

Temperature ( oC)

Ta

Tc

Tb

0 10 20 30 40 50 60 70 80 90 100

500

1000

1500

2000

2500 Temperature ( oF)

0 10 20 30 40 50 60 70 80 90 100

200

400

600

800

1000

Temperature ( oC)

Time (seconds)

Tshroud

Trotor

Tblade

0 10 20 30 40 50 60 70 80 90 100

500

1000

1500

Temperature ( oF)

Figure 9. Temperatures for lapse-rate takeoff transient 

either the rotor or blades. According to this formulation, the blades contribute the least to the clearance 

change. Like the rotor, they grow initially quickly due to centrifugal forces, but their growth is 

significantly less than the rotor which should be expected considering the relatively short length of the 

blades compared to the rotor radius. The predicted clearance generally follows the expected trend for a 

take-off-transient. The clearance is expected to reduce about 30 mils and the recovery due to the thermal 

growth of the shroud is expected to be about 10 mils. The model predicts a reduction of approximately 20 

mils and a recovery of nearly 5 mils. Exact ranges for a given engine would depend on the parameters 

and transient conditions specific to that engine. Nonetheless, the results are promising. Similar 

comments can be made of results garnered for the lapse-rate-take-off transient (refer to Figure 8 and 

Figure 9). 

4. Discussion 

The goal of this effort is to develop a relatively simple physics-based model with sufficient 

accuracy for control development research. The formulation presented here, though grossly simplified, 

appears to reasonably predict the expected change in turbine tip clearance during a take-off-transient. 

Moreover, the predicted temperatures and deflection respond, for the most part, as physically expected. 

Discrepancies do exist, but most deviations from expected response can likely be attributed to 

generalized or inaccurate information for many of the parameters used in this model. Studies by Owen 

[12] provide detailed information on various mechanisms effecting tip clearance, particularly with respect 

to the rotor and shroud. It is not the purpose of this initial model to incorporate all of the details 

contained therein. However, these studies provide a basis for incorporating additional details that could 

improve the accuracy of clearance calculation. 

NASA/TM—2003-212226 15
The true shroud thermo-mechanical response is particularly unclear, especially where interactions 

with the case are concerned. The turbine blades are relatively straight-forward, and, in terms of heat 

transfer, have been studied extensively. The blade analysis presented is, therefore, expected to be more 

accurate than that of the shroud or rotor. The rotor can be further complicated due to its rather non-trivial 

geometry and the fact that, in some gas turbines, the rotor is cooled by air at different temperatures on 

either of its sides. 

In light of this study and the questions raised, the following tasks must be addressed to further the 

application and usefulness of this physics-based modeling approach: 

• Confirm materials and associated properties. Because materials are application dependent, the 

material properties of Inconel 718 have been used to approximate those of various superalloys 

used in the manufacture turbine components. A target engine/turbine should be identified and 

accurate material properties used to further develop and validate this model. In particular, the 

abradable material must be more closely researched. The temperature dependence of many of 

the material properties should also be considered in future models. 

• Determine unknown convection coefficients. The outer shroud surface and rotor surfaces are 

cooled by compressor discharge air and the convection process is not well documented. 

• Determine role of interaction between casing and shroud. The shroud growth is likely limited 

by the casing. Documentation on casing heat transfer is not readily available. 

• Quantify roles of windage heating and heat transfer at rotor tip play in determining rotor 

growth. It was assumed that the rotor temperature is primarily determined by heat transfer from 

cooling air discharged from the compressor. However, the effects of windage heating should be 

investigated as it can cause significant variation in the radial temperature distribution of the 

rotor. Significant heat may also be transferred from the heated blades to the circumferential tip 

surface of the rotor. 

• Identify thermal and mechanical response of abradable material. The abradable material was 

assumed to function mostly as a thermal barrier. Its exact material properties are not known. It 

was assumed that the material did not respond like an alloy resulting in a thermal expansion. 

This assumption must be critically examined. 

Future work should involve more detailed research into those parameters and mechanisms least 

documented. The current model should also be refined and its fidelity quantified in order to assess its 

usefulness for the design of a tip clearance control system. This assessment would be most meaningful if 

real engine data were used as a basis for comparison. 

References

1. Halila, E., Lenahan, D., and Thomas, T., “Energy Efficient, High Pressure Turbine Test 

Hardware Detailed Design Report,” NASA CR-167955, 1982.

2. Howard, W. and Fasching, W., “CF6 Jet Engine Diagnostics Program: High Pressure Turbine 

Roundness/Clearance Investigation,” NASA CR-165581, 1982.

3. Olsson, W. and Martin, R., “B747/JT9D Flight Loads and Their Effect on Engine Running 

Clearance and Performance Deterioration,” Nacelle Aerodynamic and Inertial Loads 

(NAIL)/JT9D Jet Engine Diagnostics Programs. NASA CR-165573, 1982.

4. Azad, G., Han, J., Shuye, T., and Boyle, R., “Heat transfer and pressure distributions on a gas 

turbine blade tip,” ASME Journal of Turbomachinery, Vol. 122, 2000, pp. 717–724.

5. Azad, G., Han, J., and Boyle, R., “Heat transfer and flow on the squealer tip of a gas turbine 

blade,” ASME Journal of Turbomachinery, Vol. 122, 2000, pp. 725–732.

6. Kim, Y., Downs, J., Soechting, F., Abdel-Messeh, W., Steuber, G., and Tanrikut, S., “A 

summary of the cooled turbine blade tip heat transfer and film effectiveness investigations 

performed by Dr. D.E. Metzger,” ASME Journal of Turbomachinery, Vol. 117, 1995, pp. 1–11.

NASA/TM—2003-212226 16
7. Kim, Y. and Metzger, D., “Heat transfer and effectiveness on film cooled turbine blade tip 

models,” ASME Journal of Turbomachinery, Vol. 117, 1995, pp. 12–21.

8. Kumada, M., Iwata, S., Obata, M., and Watanabe, O., “Tip clearance effect on heat transfer and 

leakage flows on the shroud-wall surface in an axial flow turbine,” ASME Journal of 

Turbomachinery, Vol. 116, 1994, pp. 39–45.

9. Lattime, S. and Steinetz, B., “Turbine engine clearance control systems: current practices and 

future directions,” AIAA2002-3790, NASA TM-2002-211794, 38th AIAA/ASME/ASEE Joint 

Propulsion Conference and Exhibit, July 7-10, 2002.

10. Incropera, F. and DeWitt, D., Fundamentals of Heat and Mass Transfer, John Wiley & Sons, 

New York, NY, 1996.

11. Timoshenko, S. and Goodier, J., Theory of Elasticity, McGraw-Hill Book Company, New York, 

NY, 1970.

12. Owen, J. and Rogers, R., Flow and Heat Transfer in Rotating Disk Systems: Volume 1 – RotorStator Systems, John Wiley & Sons Inc., New York, NY, 1989.

13. Goldstein, R., “Film cooling,” Advances in Heat Transfer, Vol. 7, 1971, pp. 321–379.

14. Medic, G. and Durbin, P., “Toward improved prediction of heat transfer on turbine blades,” 

ASME Journal of Turbomachinery, Vol. 124, 2002, pp. 187–192.

15. Medic, G. and Durbin, P., “Toward improved film cooling prediction,” ASME Journal of 

Turbomachinery, Vol. 124, 2002, pp. 193–199. 

NASA/TM—2003-212226 17

Appendix 

This appendix includes the Matlab m-file and a series of block diagrams from the Simulink model 

used to create the plots presented in § 3. As shown below, the m-file, GI2MaxPower.m, contains all 

parameters required for the clearance model used to produce the results. Most parameters are 

documented within the comments of the m-file. Input transients used in the calculations were loaded 

from file, GI2MaxPower.mat, into the Matlab workspace. The MAT-file includes engine speed, 

temperature, and pressure transients used to produce the results shown in Figure 6 and Figure 7. To 

generate the results shown in Figure 8 and, a different MAT-file containing input transients for a lapserate takeoff was used. The input transients, not given here, were scaled in an attempt to make them more 

representative of a larger gas turbine. The Simulink models were organized to resemble the block 

diagram shown in Figure 1. 

GI2MaxPower.m: Matlab m-file 

% GI2MaxPower.m (created 071102, updated 011603/plotting)

% Turbine tip clearance parameters

% Parameter file for calculation of expansion due to

% thermal and mechanical stresses.

clear

help GI2MaxPower

% Load MAT-files

%load eng3

load GI2MaxPower %Ground Idle to Max Power Transient

%Define Geometry, Thermal & Mechanical Properties,

% Turbine dynamic inputs

RtoK = 0.555556; % R to K, temperature conversion

time = eng.time*2; % slow down transient input

RPM = eng.xnh*2.45 - 21500; % increase engine speed range

t41_K = eng.t41*RtoK*0.85; % K, turbine inlet temperature

% needed to scale compressor temperature to make sure it matched

% published temperatures for cooling air around shroud (1250 F)

t3_K = eng.t3*RtoK*1.25; % K, compressor discharge temperature

t41_C = t41_K - 273.15; % C, turbine inlet temperature

t3_C = t3_K - 273.15; % C, turbine inlet temperature

Po = eng.p3*6894.76*0.8; % Pa, compressor discharge pressure

Pi = (0.5/0.7)*eng.p4*6894.76; % Pa, average turbine stage pressure

P3 = eng.p3*6894.76; % Pa, compressor exit pressure

% Unstressed parameters

r0 = 32.9/100; % m, rotor tip radius

L = 0.050; % m, blade unstressed length

Ac = 6e-4; % m^2, blade cross-sectional area

P = 0.110; % m, blade perimeter

R0 = r0+L+0.0014; % m, thermal barrier inner radius

% Properties of Inconnel 718

rho = 8.22/1000*100^3; % kg/m^3, density

nu = 0.4; % poissons ratio

c = 645; % J/Kg-C, specific heat

alpha1 = 15.1e-6; % 1/K, thermal expansion coef. at 538 C

alpha2 = 16.4e-6; % 1/K, thermal expansion coef. at 871 C

NASA/TM—2003-212226 19
k = 24.9; % W/m-K, thermal conductivity at 871 C

E = 139e9; % Pa, modulus of elasticity at 870 C

%%%%%%%%%%%%%%%%%%%

% Shroud parameters

w = 2*0.0254; % shroud with

a = R0; % m, thermal barrier inner radius

b=a+ 0.25*0.0254; % m, shroud outer radius

rc = (a+b)/2; % m, shroud inner radius

Aa = 2*pi*a*w; % m^2, thermal barrier inner surface area

Ab = 2*pi*b*w; % m^2, shroud outer surface area

V1 = pi*rc^2*w - pi*a^2*w; % m^3, thermal barrier volume

V2 = pi*b^2*w - pi*rc^2*w; % m^3, shroud volume

ha = 500; % W/m^2-K, conv. coef. thermal barrier

hb = 1000; % W/m^2-K, conv. coef. shroud

% Zirconium Oxide

rhoa = 5.03/1000*100^3; % kg/m^3

ca = 0.109*4184; % J/kg-K

ka = 2.93; % W/m-K

alphaa = 8.2e-6; % 1/K, thermal expansion coef. at 1000 C

% Inconel 718

rhob = rho; % kg/m^3, density

cb = c; % J/Kg-C, specific heat

kb = k; % W/m-K, conductivity at 871 C

alphab = alpha1; % 1/K, thermal expansion coef. at 538 C

% Initial conditions

Ta0 = t41_C(1);

Tb0 = t3_C(1);

%%%%%%%%%%%%%%%%%%%

% Blade parameters

V = Ac*L; % m^3, volume

m = rho*V; % kg, mass

As = L*P; % m^2, surface temperature

h = 1000; % W/m^2-K, film convection coefficient

eta = 0.5; % bulk film cooling effectivness

alpha = alpha2; % 1/K, thermal expansion coef.

% Initial conditions

w0 = RPM(1)*2*pi/60;

Ti_blade = (1 - eta)*t41_C(1) + eta*t3_C(1);

%%%%%%%%%%%%%%%%%%%

% rotor parameters

hr = 500; % W/m^2-K, convection coefficient

wr = w; % m, width

Ar = 2*pi*r0^2; % m^2, convection surfe area

Vr = Ar*wr; % m^3, volume

rhor = rho; % kg/m^3, density

Er = E; % pa, modulus of elasticity

nur = nu; % poisson's ratio

alphar = alpha1; % 1/K, thermal expansion coef.

cr = c; % J/Kg-C, specific heat

kr = k; % W/m-K, thermal conductivity

% Initial Condition

Trotor0 = t3_C(1); % C, initial rotor temperature

NASA/TM—2003-212226 20
%%%%%%%%%%%%%%%%%%%

% run the simulation

[T,X,Y] = sim('ClearanceModel9',100);

RPM = Y(:,1);

TempC = (Y(:,2:6)-32)*5/9; %temperature in degrees Celcius

DeflMm = Y(:,7:9)*0.0254; %deflections in millimeters

ClearMm = Y(:,10)*0.0254; %clearance in millimeters

ShTempC = (Y(:,11:15)-32)*5/9; %shroud temperatures in degrees Celcius

%%%%%%%%%%%%%%%%%%%

% plot the results

set(0,'DefaultLineLineWidth',1.5)

time = [eng.time; 100]; PLA = [eng.pc; 50];

figure(1); clf

PLApcnt = 100*(PLA-21)/29;

subplot(3,1,1); [ha,h1,h2]=plotyy(T,RPM,time,PLApcnt);

set(ha,'Xlim',[0 100]); %Common x-axis

axes(ha(1));

set(gca,'xgrid','off','ygrid','off','box','off');

set(gca,'Ylim',[7000 17000],'Ytick',[7000:2500:17000]);

ylabel('N (rpm)');

hl=legend('N (rpm)','Throttle',0);

hlc=get(hl,'children');

axes(hl); ylim=get(hl,'Ylim');

line(get(hlc(2),'Xdata'),get(hlc(2),'Ydata')-0.45*diff(ylim), ...

'Color',get(h2,'Color'),'LineStyle',get(h2,'LineStyle'));

axes(ha(2));

Xlim = get(gca,'Xlim');

Ylim = [0 100]; YTick = [0 100]; YTlbl = ['G/I';'Max'];

set(gca,'xgrid','off','ygrid','off','box','off');

set(gca,'Ylim',Ylim,'YTick',YTick,'YTickLabel',YTlbl);

ylabel('Throttle Setting','Rotation',-

90,'VerticalAlignment','bottom');

set(h2,'LineStyle','--');

subplot(3,1,2);

%Left Y-Axis is metric units

plot(T,DeflMm(:,1),T,DeflMm(:,2),'--',T,DeflMm(:,3),'-.');

set(gca,'xgrid','off','ygrid','off','box','off');

Ylim = get(gca,'Ylim'); YTick = Ylim(1)+diff(Ylim)*[0:0.25:1];

set(gca,'YTick',YTick);

ylabel('Deflection (mm)');

hl2=legend('Shroud','Rotor','Blade');

%Right Y-axis is English Units

Xlim2 = get(gca,'Xlim');

Ylim2 = get(gca,'Ylim')*(1000/25.4);

ax2 = axes('position',get(gca,'position'));

h2=plot(Xlim2,Ylim2,'.'); set(h2,'visible','off');

set(ax2,'Xlim',Xlim2,'Ylim',Ylim2, ...

'YAxisLocation','right','color','none', ...

'xgrid','off','ygrid','off','box','off');

ylabel('Deflection (mils)','Rotation',-90,'VerticalAlignment','bottom');

NASA/TM—2003-212226 21
subplot(3,1,3);

%Left Y-Axis is metric units

plot(T,ClearMm);

set(gca,'xgrid','off','ygrid','off','box','off');

Ylim = get(gca,'Ylim');

YTick = Ylim(1)+diff(Ylim)*[0:0.25:1];

set(gca,'YTick',YTick);

ylabel('Clearance (mm)');

xlabel('Time (seconds)');

%Right Y-axis is English Units

Xlim2 = get(gca,'Xlim');

Ylim2 = get(gca,'Ylim')*(1000/25.4);

ax2 = axes('position',get(gca,'position'));

h2=plot(Xlim2,Ylim2,'.'); set(h2,'visible','off');

set(ax2,'Xlim',Xlim2,'Ylim',Ylim2, ...

'YAxisLocation','right','color','none', ...

'xgrid','off','ygrid','off','box','off');

ylabel('Clearance (mils)','Rotation',-90,'VerticalAlignment','bottom');

print -depsc2 -tiff GI2MPdeflections;

print -dmeta GI2MPdeflections;

figure(gcf+1); clf

set(gcf,'position',get(gcf-1,'position').*[1.05 0.96 1 1]);

subplot(3,1,1);

%Left Y-Axis is metric units

plot(T,TempC(:,1),T,TempC(:,2),'--');

set(gca,'xgrid','off','ygrid','off','box','off');

Xlim = [ 0 100];

Ylim = [200 1800];

YTick = Ylim(1):400:Ylim(2);

set(gca,'Xlim',Xlim,'Ylim',Ylim,'YTick',YTick);

ylabel('Temperature ( ^oC)');

legend('T_{turbine}','T_{compressor}');

%Right Y-axis is English Units

Xlim2 = get(gca,'Xlim');

Ylim2 = get(gca,'Ylim')*(9/5)+32;

ax2 = axes('position',get(gca,'position'));

h2=plot(Xlim2,Ylim2,'.'); set(h2,'visible','off');

set(ax2,'Xlim',Xlim2,'Ylim',Ylim2, ...

'YAxisLocation','right','color','none', ...

'xgrid','off','ygrid','off','box','off');

ylabel('Temperature ( ^oF)','Rotation',-90,'VerticalAlignment','bottom');

subplot(3,1,2);

%Left Y-Axis is metric units

plot(T,ShTempC(:,2),T,ShTempC(:,3),'--',T,ShTempC(:,4),'-.');

set(gca,'xgrid','off','ygrid','off','box','off');

Xlim = [ 0 100];

Ylim = [200 1400];

YTick = Ylim(1):400:Ylim(2);

set(gca,'Xlim',Xlim,'Ylim',Ylim,'YTick',YTick);

ylabel('Temperature ( ^oC)');

legend('T_a','T_c','T_b');

%Right Y-axis is English Units

Xlim2 = get(gca,'Xlim');

Ylim2 = get(gca,'Ylim')*(9/5)+32;

ax2 = axes('position',get(gca,'position'));

h2=plot(Xlim2,Ylim2,'.'); set(h2,'visible','off');

NASA/TM—2003-212226 22
set(ax2,'Xlim',Xlim2,'Ylim',Ylim2, ...

'YAxisLocation','right','color','none', ...

'xgrid','off','ygrid','off','box','off');

ylabel('Temperature ( ^oF)','Rotation',-90,'VerticalAlignment','bottom');

subplot(3,1,3);

%Left Y-Axis is metric units

plot(T,TempC(:,3),T,TempC(:,4),'--',T,TempC(:,5),'-.');

set(gca,'xgrid','off','ygrid','off','box','off');

Xlim = [ 0 100];

Ylim = [200 1000];

YTick = Ylim(1):200:Ylim(2);

set(gca,'Xlim',Xlim,'Ylim',Ylim,'YTick',YTick);

ylabel('Temperature ( ^oC)');

xlabel('Time (seconds)');

legend('T_{shroud}','T_{rotor}','T_{blade}');

%Right Y-axis is English Units

Xlim2 = get(gca,'Xlim');

Ylim2 = get(gca,'Ylim')*(9/5)+32;

ax2 = axes('position',get(gca,'position'));

h2=plot(Xlim2,Ylim2,'.'); set(h2,'visible','off');

set(ax2,'Xlim',Xlim2,'Ylim',Ylim2, ...

'YAxisLocation','right','color','none', ...

'xgrid','off','ygrid','off','box','off');

ylabel('Temperature ( ^oF)','Rotation',-90,'VerticalAlignment','bottom');

print -depsc2 -tiff GI2MPtemperatures;

print -dmeta GI2MPtemperatures;

figure(gcf-1);

NASA/TM—2003-212226 23
Figure A-1. Simulink© block diagram of TipClearance Model clearance temperatures engine speed

5

Shroud Temperatures

4

Clearance

3

Deflections

2

Temperatures

1

RPM

1e5/2.54

m to mils1

1e5/2.54

m to mils

9/5

gain

RPMTf (film/compressor temp.)

Rrotor (rotor radius)

Trotor (rotor temp.)

Turbine Rotor

Tinlet (turbine inlet)

Tcompressor (compressor)

RPM

Pi

Po

Turbine Engine Model

Tm (main stream)Tf (film coolant)RPM

Lblade (blade length)

Tblade (blade temp.)

Turbine Blades

TturbineTcompressorPiPo

Tshroud (shroud temp.)

Rshroud (shroud radius)

Shroud Temperatures (F)

Shroud

RPM, Temperatures,

Deflections, and Clearance

RshroudRrotorLblade

deflections

Deflection Calculations

32*ones(5,1)

Constant

Engine Speed (RPM)

Engine Speed (RPM)

Clearance (mil)

Clearance (mil)

Deflection (mil)

Deflection (mil)

Temperature (F)

Temperature (F)

NASA/TM—2003-212226 24
5

Po

4

Pi

3

RPM

2

Tcompressor (compressor)

1

Tinlet (turbine inlet)

[time t41_C]

Turbine inlet temp.

[time Po]

Outer Shroud pressure

[time Pi]

Inner shroud pressure

[time RPM]

Engine RPM

[time t3_C]

Compressor temp.

Figure A-2. Engine sub-model 

3

Shroud Temperatures (F) 2

Rshroud (shroud radius)

1

Tshroud (shroud temp.)

a unstressed inner shroud radius

(outer thermal barrier radius)

0.5

gain1

9/5

u[1] − (u[1]−u[2])/(1 + log(b/rc)/log(rc/a)*ka/kb)

Tc (Temperature at interface

between thermal barrier and shroud)

alphab*rc*(u[1] + (b^2/(b^2−rc^2)−1/2/log(b/rc))*(u[2]−u[1]))

Shroud deflection due to thermal stresses

1/E/(b^2−rc^2)*(−(1+nu)*rc*b^2*(u[2]−u[1])+(1−nu)*rc*(u[1]*rc^2−u[2]*b^2))

Shroud deflection due to pressure differential

Shroud

Temperatures

32*ones(5,1)

Tcompressor Tb

Calculate Tb

Tturbine Ta

Calculate Ta

4

Po

3

Pi

2

Tcompressor

1

Tturbine

Figure A-3. Shroud sub-model 

1

Ta

ha^2*u/(ka*rhoa*ca)

function1

MATLAB

Function

erfc

(u[3] − Ta0)*(1 − u[1]*u[2]) + Ta0

calculation of Ta

eu

Math

Function1

sqrt

Math

Function

Clock

1

Tturbine

Figure A-4. Inner shroud surface temperature, Ta, sub-model 

1

Tb

hb^2*u/(kb*rhob*cb)

function1

MATLAB

Function

erfc

(u[3] − Tb0)*(1 − u[1]*u[2]) + Tb0

calculation of Tb

eu

Math

Function1

sqrt

Math

Function

Clock

1

Tcompressor

Figure A-5. Outer shroud surface temperature, Tb, sub-model 

NASA/TM—2003-212226 25
2

Trotor (rotor temp.)

1

Rrotor (rotor radius)

(u − 25)*alphar*r0

rotor expansion due

to thermal stresses

(1−nur)*rhor*u^2*r0^3/(4*Er)

deflection due to mechanical stress

2*pi/60

RPM to rad/sec

r0

Constant

Tf Trotor

Calculate Trotor

2

Tf

(film/compressor

temp.)

1

RPM

Figure A-6. Rotor sub-model 

1

Trotor

hr^2*u/(kr*rhor*cr)

function1

MATLAB

Function

erfc

(u[3] − Trotor0)*(1 − u[1]*u[2]) + Trotor0

calculation of Trotor

eu

Math

Function1

sqrt

Math

Function

Clock

1

Tf

Figure A-7. Rotor surface temperature, Trotor, sub-model

2

Tblade (blade temp.)

1

Lblade (blade length)

(u − 25)*alpha*L

blade expansion due to thermal stresses 

rho*L^2*(L + r0)*u^2/E

blade expansion due to centrifugal force

2*pi/60

RPM to rad/sec

L

Constant Tm

Tf

Tblade

Calculate Tblade

3

RPM

2

Tf (film coolant)

1

Tm (main stream)

thermal deformation

centrifugal deformation

Figure A-8. Blade sub-model 

1

Tblade

h^2*u/(k*rho*c)

function1

MATLAB

Function

erfc

(1 − eta)*u[1] + eta*u[2]

calculation of Tr

(reference Temp.)

(u[3] − Ti_blade)*(1 − u[1]*u[2]) + Ti_blade

calculation of Tblade

e

u

Math

Function1

sqrt

Math

Function

Clock

2

Tf

1

Tm

Figure A-9. Blade surface temperature, Tblade, sub-model 

NASA/TM—2003-212226 26
1

deflections

u − a

shroud deflection

u − L

rotor deflection1

u − r0

rotor deflection

3

Lblade

2

Rrotor

1

Rshroud

Figure A-10. Deflection calculations sub-model 

NASA/TM—2003-212226 27
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NASA TM—2003-212226

E–13846

WBS–22–708–87–06

32

A Reduced Model for Prediction of Thermal and Rotational Effects on Turbine

Tip Clearance

Javier A. Kypuros and Kevin J. Melcher

Clearances; Active control; Turbine

Unclassified -Unlimited

Subject Category: 07 Distribution: Nonstandard

Javier A. Kypuros, University of Texas, Pan American University, Edinburg, Texas 78539; Kevin J. Melcher, NASA

Glenn Research Center. Responsible person, Kevin J. Melcher, organization code 5530, 216–433–3743.

This paper describes a dynamic model that was developed to predict changes in turbine tip clearance—the radial

distance between the end of a turbine blade and the abradable tip seal. The clearance is estimated by using a first

principles approach to model the thermal and mechanical effects of engine operating conditions on the turbine subcomponents. These effects are summed to determine the resulting clearance. The model is demonstrated via a ground idle to

maximum power transient and a lapse-rate takeoff transient. Results show the model demonstrates the expected “pinch

point” behavior. The paper concludes by identifying knowledge gaps and suggesting additional research to improve the

model.