Engineering, 2010, 2, 733-739
doi:10.4236/eng.2010.29095 Published Online September 2010 (http://www.SciRP.org/journal/eng)
A Non-Dimensional Consideration in Combustor Axial
Stress Computations
Ebene Ufot1,2, Barinaadaa Thaddeus Lebele-Alawa1*, Ibiba Emmanuel Douglas1, Kelvin D. H. Bob-Manuel1
1Faculty of Engineering, Rivers State University of Science and Technology, Port Harcourt, Nigeria
2Department of Mechanical Engineering, University of Uyo, Uyo, Nigeria
Received June 23, 2010; revised August 5, 2010; accepted August 5, 2010
Abstract
Thermal stresses in the combustor of gas-turbines are computed using non-dimensional parameters. Buck-
ingham pi theorem was used to arrange the listed relevant parameters into non-dimensional groups. In testing
the validity of the functional relation of the non-dimensional independent parameters, use is made of the
prevailing temperatures of the combustor in operation. A computer program was used to enhance computa-
tions. The results showed an interesting way of influencing the axial stresses. To reduce stresses in gas-tur-
bine combustors, a method of varying the independent parameter that is of radius ratio oriented and thickness
dependent was adopted. This showed a reduction of the axial stresses to minimal levels using the parameters.
Plots were made and a point of inflection that manifested itself in the presentation of the axial stress function
was further investigated upon. It turned out to be a point of abnormal stress level and out-of-trend tempera-
ture profile. The use of non-dimensional consideration proved adequate in the computation of axial stresses.
The results showed a 2 percent difference from existing values of stresses got from a transient thermal load-
ing of a combustor.
Keywords: Thermal, Stresses, Combustor, Gas-Turbine
1. Introduction
Thermally induced axial stresses or shocks occur in ma-
terials when they are heated or cooled. It affects the op-
erations of gas-turbines due to the large components sub-
jected to stresses. Many structural elements of hollow
cylinders subjected to rapid internal heating crack or
deform due to thermally induced axial stresses produced
in them. Tret’yachenko, et al. [1] carried out investiga-
tions on thermal stresses of hollow cylinders drum of
unilateral internal heating. The main aim of their inves-
tigations was to obtain graphs that could be used to esti-
mate the stress level in cylindrical structural elements.
Kumar and Rajgopalan, [2] performed non-dimensional
stress analysis on cylindrical objects. They obtained plots
of non-dimensional tangential stress against non-dimen-
sional length of cylinder. Their presentations showed the
differences in the values of functions from various radial
positions.
The analytical solution for computing the radial and
circumferential stresses in a functionally graded material
(FGM) thick cylindrical vessel under influence of inter-
nal pressure and temperature was presented by Abrinia
et al. [3]. FGMs are fabricated by continuously changing
the volume fraction of two basic materials (usually ce-
ramics and metals) in one or more directions.
Nomenclature
Ax.Str.Funct Axial stress function-σzth/E
C Non-dimensional axial stress parameter
for average temperature
C1 Non-dimensional axial stress parameter
as ratio of radius dependent
C2 Non-dimensional axial stress parameter
as ratio of temperature to
average temperature
E Young’s modulus of elasticity
T Temperature in Kelvin
Ti Temperature of the bulk airstream of
combustion products
Tm Average temperature in combustor
Tma Maximum average temperature in com-
bustor

E. UFOT ET AL. 734
Twa Combustor outer wall temperature
Twi Combustor internal wall temperature
Tsurr Temperature of the surroundings
r Radial distance from centre of cylinder
ra Radius to outer wall
ri Radius to inner wall
Greek letters
α Coefficient of thermal expansion
σ Axial stress
ν Poisson’s ratio
Suffixes
r radial stress
φ tangential stress
zth Thermal stress in axial (z) direction
In the analysis, the effect of non-homogeneity in FGM
thick cylinder could be implemented by choosing a di-
mensionless parameter, β, which could be assigned an
arbitrary value affecting the stresses in the cylinder.
Various values of  were used to demonstrate the effect
of in-homogeneity on the stress distribution. They con-
cluded that by changing the values of , the properties of
FGM could be so modified that the lowest stress levels
were reached.
Kubo et al. [4] investigated a multidisciplinary prob-
lem of heat conduction, elastic deformation, heat transfer,
liquid flow. They used inverse method for determining
the optimum thermal load history which reduced tran-
sient thermal stress. Temperature history functions were
introduced to ensure the continuity of the temperature
increasing rate. The multidisciplinary complex problem
was decomposed into heat transfer and thermal stress
problems. Ootoa et al. [5] determined the temperature
and thermal stress distribution in the cross-section of a
non-homogenous hollow circular cylinder due to a mov-
ing heat source in the axial direction, and found that; the
maximum temperature occurs at the region through
which the moving heat source passed. All of the above
used non-dimensional functions of stress and found that
it very conveniently influenced the stresses so computed.
Principal axes that could be considered in thermal stress-
es in hollow cylinders are axial, radial and the tangential.
In this work only axial stresses will be considered. A
non-dimensional approach of expression of the pertinent
parameters has been adopted. This makes the expressions
more compact and allows a wider understanding of the
properties in further considerations. Dimensional analy-
sis using the Buckingham  theorem has been applied in
analyzing the stress problems. This therefore presents the
axial stresses in few independent non-dimensional pa-
rameters. Thus it has enhanced the establishment of in-
fluencing factors to reduce stresses in materials. The in-
volvement of a computer code has enabled fast computa-
tion of the axial stresses.
2. Materials and Methods
From the equation of thermally induced axial stress [6]
one can re-work it to the required expression
Axial Stress, σzth = ν(σrth + σφth) - EαthT(r) (1)
where σrth and σφth are given as:
Radial Stress,
222
222
[()()]
1
rth
thi a
mm
ai
Errr
Tra Tr
rrr






(2)
Tangential Stress,
 
2 2 2
22
2( )()()
1
i
i
th
th a
mm
a
Er rr TraTr Tr
r rr










(3)
Turning Axial Stress Equation to a Non-Dimen-
sional Equation:
From Equation (1) above,
() ()
zth rth
E
Tr


 


 
22
22 22
1
() ()
22 22
1
***()*
1
th ai
Z
thmammam rrthm
aai
Errr rr ra
TTrTTT EaT
rr
rr rr


  

 

 
2
22
() ()
22 22
2** **
1
thi ai
mamarth r
ai ai
Errrrrra
TTTET
r
rr rr
 



 


 

 


2
2
() ()
222
1**2_
1
th a
marth r
ai
Er
TrT ET
rrr
 






2
22 2 2
() ()
22
1**()
1
th
maiirth r
ai
Era Txr rrrTET
r rr
 










E. UFOT ET AL. 735
()
2
2
11
1
th math
r
ET E
T
ri
ra
 


 







()
2
2
*
11
r
th
zthma
ma
T
ETT
ri
ra








 



(4)
Hence, from Equation (2) above, some functional rela-
tion must exist in the form of


,()
,,,,
ZthZththa imar
ErrTT   (5)
Dimensionless Consideration on Axial Stress.
Sorting out the inherent parameters:
Forming a table of parameters: Table 1.
The Buckingham  theorem proves that in a physical
problem including n quantities in which there are m di-
mensions, the quantities can be arranged into n – m in-
dependent dimensionless parameters.
Hence some functional relation must exist in the form
as expressed in Equation (5)

()
,,,,
zthzththaimar
ErrTT   
Applying the Buckingham

theorem, there are
three

groups to be found, so Axial Stress,
Z
th

can be more compactly stated as a function of these three
non-dimensional parameters [7]. Thus from Equation (5)
1. Zasa nondimensional parameter
E

 (Axial
Stress Function)

()
2
1
2
0
2.
2
3.
1
4. *1
r
ma
i
a
th
ma
TasC
T
as C
r
r
TasC







Actually, the main three independent dimensionless
parameters can be seen as:
0102
,* *
ZCCandCC
E

But Equation (3) can be written as:
01 2
() ()
Zth CCr Cr
E





 (6)
3. Results and Discussions
The results of the non-dimensional axial stress computa-
tions are shown as program results below, and the corre-
sponding wall thickness values are given in Table 2. The
non-dimensional consideration gave a wider view of the
axial stress. In Figure 1 and Figure 2, the values were
plotted against a non-dimensional temperature parameter,
EU402-AXIAL STRESS FUNCTION_PROGRAM RESULTS
VarNr, ra, Twa, Twi
1 38 2549.6 2584.1
T(0), T(1), T(2), T(3), T(4)
2549.6 2558.2 2566.9 2575.5 2584.1
Tm(0), Tm(1), Tm(2), Tm(3), Tm(4)
194.8 176.5 130.2 80.7 27.8
VarNr C1 C2 C0 AxialStress AxialStressFunct:Ax-Str/E
1 3.96 13.09 0 .00 .000000
1 3.96 13.09 0.00345 −6618.35 −.031516
1 3.96 13.14 0 .00 .000000
1 3.96 13.14 0.00345 −6650.44 −.031669
1 3.96 13.18 0 .00 .000000
1 3.96 13.18 0.00345 −6682.52 −.031822
1 3.96 13.22 0 .00 .000000
1 3.96 13.22 0.00345 −6714.61 −.031974
1 3.96 13.27 0 .00 .000000
1 3.96 13.27 0.00345 −6746.69 −.032127

E. UFOT ET AL. 736
-0.032 14
-0.032 12
-0.032 1
-0.032 08
-0.032 06
00.00050.0010.0015 0.002 0.0025 0.003 0.0035 0.004
C
0
-Values[-]
Non-dimensional
Axial Stress[-]
σz/E
Figure 1. Non-dimensional axial stress versus C0-values.
-6
-5
-4
-3
-2
-1
0
00.511.522.533.5
WALL THICKNESS[cm]
AXIAL STRESS/C0-Values
(TANGENT)
σz/C0
Figure 2. Axial stress/C0 (Tangent) versus wall thickness showing a point of inflection at a thickness of 1.44 cm.
Table 1. Dimensions of independent parameters for axial stress.
S/N Input parameters Dimensions
1 Axial stress, 2
2
2
1
*
1
th Nmm
mkg
s







M L-1 T
-2
2 E,
2
2
1
*
Nkg
m
s
m






M L-1 T
-2
3 th


1
K
K
-1
4 Tma [ K ] K
5 T(r) [ K ] K
6 ra [m] L
7 ri [m] L
n - m = 3
Table 2. Axial stress function versus wall thickness.
Var.Nr ra r
i Wall
Thickness Tsurr T
i T wa T wi Ax.Str.Fct.
σzth(max)
1 38 35 3 620 2620 2549.6 2584.1 −0.032127
5 37 35 2 620 2620 2561.9 2584.7 −0.032106
7 36 35 1 620 2620 2574.0 2585.4 −0.032086
10 35.5 35 0.5 620 2620 2580.0 2585.7 −0.032075
18 35.25 35 0.25 620 2620 2586.0 2586.5 −0.032073
16
(AFAM) 75 74.5 0.25 605 2578 2535.5 2536.0 −0.031447

E. UFOT ET AL. 737
CO and against the wall thickness, respectively. In Fig-
ure 3 the axial stress function is plotted against the radial
nodal positions. All the presentations show an increasing
tendency with increased wall thickness.
The Figure 4 above shows a method of influencing
the stress levels in materials:
By increasing the independent parameter, C1, the
stress levels can be reduced
where C12
2
.
1i
a
r
r




In other words, the radial ratio, ri/ra should be increas-
ed.
In Equation (4),




01 2
/
zECCrCr
 
where:

()
2
12
0
.
2
1
And,
.* 1
r
ma
i
a
th
ma
T
CT
C
r
r
CT







If C1 is to be high,
Then the ratio ri/ra must be high
i.e., ai rr
i.e., ra should be reduced to the lowest value applicable.
A table of values can be formed for ri = 35 cm:
See Table 3, above. The results for internal wall tem-
peratures are presented in Table 4.
Table 3. A non-dimensional parameter, C1 by various ex-
ternal wall radius of model.
ra
[cm] ri/ra
2
1i
a
r
r



2
2/1 i
a
r
r






 

38 0.92 0.1536 3.96
37 0.946 0.1052 5.70
36 0.97 0.055 10.95
35.5 0.986 0.028 21.42
35.25 0.993 0.014 42.86
Table 4. Axial stress function versus internal wall tempera-
tures.
Time
[secs]
Twi
[K]
Axial Stress
σzth
[MPa] (Ufot,
2010)
Axial Stress
Function
(σzth/E) [-]
Axial Stress
Function
(Present
model)
70 2027.8−5820.0 −0.027714 −0.027113
140 2298.2−6351.0 −0.030243 −0.029990
210 2392.4−6531.0 −0.031100 −0.031005
350 2429.9−6601.0 −0.031433 −0.031413
490 2433.0−6607.0 −0.031462 −0.031447
630 2433.3−6608.0 −0.031467 −0.031450
922.12584.1−6746.7 −0.032127 −0.032127
Ax.Fct1
Ax. Fct2
Ax. Fct3
Ax. Fct4
Ax. Fct5
-0.0322
-0.0321
-0.032
-0.0319
-0.0318
-0.0317
-0.0316
-0.0315
-0.0314
0.91 0.920.93 0.94 0.95 0.960.97 0.98 0.9911.01
NODAL(RADIAL) POSITIONS[-]
AXIAL STRESS FUNCTION(MAX)[
Ax.Fct1
Ax.Fct1
Ax. Fct2
Ax. Fct3
Ax. Fct4
Ax. Fct5
Figure 3. Axial stress function (Max) versus radial nodal positions radial nodal positions r = ri/ra.
3.96
5.7
10.95
10.95
-0. 0 321 3
-0. 0 321 2
-0. 0 321 1
-0.0321
-0. 0 320 9
-0. 0 320 8
-0. 0 320 7
024681012
C1-Values [-]
Non-dimensional
Axial Stress [-]
σz/E
Figure 4. Non-dimensional axial stress versus C1.

E. UFOT ET AL. 738
σmax 13
σmax 1
σmax 5
σmax 7
σmax 10
-0.03213
-0.03212
-0.03211
-0. 0321
-0.03209
-0.03208
-0.03207
00.5 11.5 22.5 33.5
Wall Thickness [cm]
Axial Stress Function
(Max) [-]
Series1
σmax1
σmax5
σmax7
σmax10
σmax13
Figure 5. Axial stress function versus wall thickness - showing value position of point of inflection: σmax13.
-0. 033
-0. 032
-0. 031
-0.0 3
-0. 029
-0. 028
-0. 027
-0. 026
05001000 1500 20002500 3000
Internal Wall Temperatures[K]
Axial Stress Function[-]
EU-2010
present model
Figure 6. Axial stress function versus internal wall temperatures.
Figure 1 is a presentation of a non-dimensional axial
stress against C0-values. The non-dimensional tempera-
ture parameter, C0, is the maximum average temperature
dependent parameter. The stress shows a tendency of
increasing with the C0. That means to reduce the stress in
the material, the maximum average temperature in the
material must be reduced. And this maximum tempera-
ture is noted to be increasing with wall thickness. There-
fore, in the final analysis, the wall thickness for minimal
thermal stresses should be as small as applicable.
Figure 2 shows a point of Inflection in the presenta-
tion of axial stress/C0 with wall thickness. Further con-
sideration of this point of Inflection occurring at a wall
thickness of 1.44 cm, shows an abnormal trends of tem-
peratures and stresses in the material. In a cross-sectional
view of the non-dimensional stress profile in the material
shows linear relationship with the non-dimensional radial
positions. In Figure 4, a non-dimensional axial stress is
shown varying against C1-values. The indicated C1-value
on the trend is showing that with very high values of C1
the stress tends minimal. It is a definite method of influ-
encing the stress levels in materials. It can be shown with
Figure 5 that to obtain very minimal axial stresses in
material, the wall thickness should be reduced. As is
shown in the figure, the stress values are reducing as the
wall thickness is reduced. Figure 6 shows that the Axial
Stress Function increases with increased internal wall
temperatures.
4. Conclusions
The work is very adequate in computing the thermal ax-
ial stress in combustors and cylindrical pipes at the in-
stance of known wall surface temperatures. With the
Non-dimensional consideration in the combustor axial
Stress computation it is possible to indicate ways of mi-
nimizing the thermal axial stress in the material.
5. References
[1] G. N. Tret’yachenko, S. Karpinos and L. E. Kiyashko,
“Thermal Stress State of Hollow Cylinders during Uni-
lateral Internal Heating,” UDC 539.319.624.074.4, 1976.
[2] I. J. Kumar and D. Rajgopalan, “Thermal Stresses in a
Hollow Cylinder Due to a Sinusoidal Surface Heating
Source,” V. R. Thiruvenkatachar, F.N.I. Defence Science
Laboratory, Delhi 6 I, 1969.
[3] K. Abrinia, H. Naee, F. Sadeghi and F. Djavanroodi,
“New Analysis for the FGM Thick Cylinders under
Combined Pressure and Temperature Loading,” © 2008
Science Publications Tehran, Amirabad Shomali St., Te-
hran, Iran 852, 2008.
[4] S. Kubo, K. Uchida, T. Ishizaka and S. Ioka, “Determina-
tion of the Optimum Temperature History of Inlet Water
for Minimizing Thermal Stresses in a Pipe by Multiphys-
ics Inverse Analysis,” Journal of Physics: Conference Se-
ries, Vol. 135, 2008, p. 012058.
[5] Y. Ootao, T. Akai and Y. Tanigawa, “Three-Dimensional

E. UFOT ET AL. 739
Transient Thermal Stress Analysis of a Non-Homoge-
neous Hollow Circular Cylinder Due to a Moving Heat
Source in the Axial Direction,” Journal of Thermal
Stresses, Vol. 18, No. 5, 1995, pp. 497-512.
[6] S. Fronius and G. Trankner, “Taschenbuch Maschinenbau,
Band 1/īī Grundlagen,” 3rd Edition, VEB Verlag Technik,
Berlin, 1975.
[7] E. Ufot, “Modeling Thermal Stresses in the Combustor of
Gas Turbine,” Ph.D. Dissertation, Rivers State University
of Science and Technology, Port Harcourt, 2010.