Strength o f Materials, Vol. 31, No. 4, 1999

P R O D U C T I O N S E C T I O N

T E M P E R A T U R E D I S P L A C E M E N T S IN N A T U R A L L Y T W IST E D T U R B I N E BLADES

U N D E R C O N D I T I O N S OF N O N U N I F O R M H E A T I N G

A. I. Petrenko UDC 539.388

We deduce and analyze design-basis formulas for the evaluation of the degrees of temperature torsion, temperature deflections in the planes of minimum and maximum stiffness, and temperature elongation of the pen of a naturally twisted turbine blade under the conditions of nonuniform heating. We present a practical example of the numerical calculations of temperature displacements in the peripheral section of the pen of a blade subjected to thermal cycling. In this case, the process of nonuniform heating of the naturally twisted blade is performed by a gas flow with cyclically varying temperature. To find the temperature fields in sections of the blade, we determine the local heat transfer coefficients along the perimeter of the profile separately for the periods of heating and cooling.

It is known that, in nonstationary temperature modes typical of, e.g., vehicular gas-turbine engines, temperature drops along the profile of turbine blades can be significant. In this case, the degree of temperature bending of the pen of the blade in the plane of its minimum stiffness is quite high and the degree of temperature bending in the plane of maximum stiffness becomes noticeable. In the presence of natural torsion (typical of the greater part of turbine blades of gas-turbine engines) and nonuniform distribution of temperature across the profile, the pen of the blade also undergoes temperature torsion. It is clear that the temperature torsion of the pen of a naturally twisted turbine blade correlates with its temperature bending and elongation and that the components of these displacements are not independent.

The aim of the present work is to determine approximate values of the temperature displacements in naturally twisted turbine blades under the conditions of nonuniform heating and describe the scheme of numerical evaluation of these displacements in the stage of design of turbine blades.

The fact that our calculations are approximate is explained by the following factors: the pen of the turbine blade is regarded as a twisted bar with elongated asymmetric profile (for the engineering theory of twisted bars, see [1]), the centers of mass of the cross sections are reduced according to the principle of proportionality of the mass of an elementary area dF of a section to the modulus of elasticity E, we neglect the facts that these centers of mass are located not on the straight axis, that this axis does not coincide with the line of centers of stiffness of the sections and the axis of natural torsion of the pen, and that torsion is constrained near the place of fixing of the pen, and the variability of sections along the length of the pen of the blade is taken into account by reducing all their parameters to a current section, which is admissible for practical calculations only in the case of smooth variation of the area of cross sections.

A blade and a coordinate system used in numerical calculations are depicted in Fig. 1. As an immobile coordinate system, we use a rectangular coordinate system xyr whose x- and y-axes are located in the plane of the root section of the pen and the r-axis is directed along the axis o f the pen. The origin of coordinates O and the directions of the x- and y-axes are chosen to satisfy the conditions

f ExdF = O, f EydF = 0, and f ExydF = O. F F F

The ~- and r/-axes of a moving coordinate system ~r/r rotate together with the section as the system moves along the r-axis and continue to play the role of reduced central principal axes. The origin of coordinates O l and the directions of these axes are chosen to satisfy the conditions

Institute of Problems of Strength, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 4, pp. 116 - 124, July - August, 1999. Original article submitted December 7, 1995.

0039-2316/99/3104-0417522.00 �9 Kluwer Academic/Plenum Publishers 417

F ;

Fig. 1. Schematic diagram of the blade together with the coordinate system used in numerical calculations.

E~dF = O, [ EqdF = O, and j" E~qdF = O. F F F

The angle a of natural torsion of the pen of the blade is actually the angle between its current and root sections measured as the angle between the projection of the x-axis onto the current section and the q-axis.

We write the static equations in the form

fodF=O, ~edF=O, fqodF=O, and da~apZdF=KdO, dr J dr (1)

F F F F

where o is the level of normal stresses, F is the cross-sectional area, p is the distance from a given point of the section to the r-axis, and 0 is the angle of rotation of the section about the r-axis in the process of torsional deformation. The first three equations correspond to the absence of external axial loads along the r-axis and external bending moments about the axes ~ and q. The fourth equation means that, in the twisted pen of the blade, the moment of the projections of normal stresses in inclined fibers onto the plane of the cross section, in the absence of external torsional moments, is expressed via the relative angle of torsional deformation of the pen in this section with the coefficient of proportionality K [2]. If the shear modulus G is variable, then the coefficient of proportionality can be written in the form

Here, K = G m J t . r .

F

and J t . r is the reduced torsional stiffness of the elongated cross section equal to [3]

1 !G(q)O 3 (q)dq, J t . r = 3G m

where 6(q) is the current thickness of the section and s is the length of the mean line of the profile. On the basis of the kinematic hypothesis, according to which a section warped as a result of torsion does not

change its shape in the processes of bending and tension, we can write the expression for relative axial strains in the form [1, 4]

dw o + d~p ~ _ dg q + __ ----:-dct dr dr dr dr clr p2,

6 = (2)

where w 0 is the displacement of the section toward the r-axis of the blade and ~p and ~o are angles of rotation of the section about the q- and ~-axes, respectively.

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By combining Hooke's law with the law of thermal expansion, we get

a e = ~- + fiT, (3)

where fl is the coefficient of linear thermal expansion and T is the increment of temperature at a given point of the section.

By equating the right-hand sides of relations (2) and (3), we arrive at the expression for o. Substituting this expression in the static equations (1), we obtain the following system of four equations with four unknowns dO/dr, dVd/dr, dqg/dr, and dw o/dr:

dO da I Ep2 dF _ I EflTdF =O, dw~ I EdF + - - dr F dr dr F F

dO da ; E~p 2 dF - I E~flTdF = O, d-~ f E~Z dF + - - - - a r - dr dr F F F

dO da I Erlp 2 dF I ErlflTdF = O, _dqg f Erl2 dF + - - - dr a dr dr F F F

dw 0 da f Ep 2 d f + ~ # I E~O 2dE dq9 do: I EY/p 2dE dr dr a F ar ar F dr dr f

+d-~ dal2 [Ep4dF-Tr Ep2flTdF-Gm4rdO=o. drl, dr ) F " dr

(4)

To make the expression for the angle of torsion of a current section of the pen in the process of nonuniform heating (the angle of temperature torsion) less cumbersome, we denote

F F F F F F

F F F F F

We now solve the system of equations (4) and successively find, for the current section, the relative angle of torsion dO~dr, the relative angles of rotation dT)/dr and d~o/dr in the planes of minimum and maximum stiffness, respectively, and the relative elongation dw o/dr of the r-axis of the pen of the blade. Further, we separate the variables and integrate the expressions obtained as a result over the length of the pen. It follows from the condition of rigid fixing of the root section of the pen of the blade that, in this section, the angle of temperature torsion, the angles of rotation in the planes of minimum and maximum stiffness, and the temperature elongation of the axis o f the pen are equal to zero. This means that the constants of integration are also equal to zero. As a result, we obtain the following formulas for the determination of the angle of temperature torsion O r of the current section of the pen, current temperature deflections of the axis of the pen in the planes of minimum ~r and maximum r/r stiffness, and temperature elongation w0r of the axis of the pen in the current section r:

r

OT= I ro

"~ / .2 /

dr " + J-Py + -- p-~ - Jp +Gm Jt r Jrl J~

ar, (5)

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r r I E~flTdF

rot o I E~ 2 dF F

dr 2 - I I ] -~r da dr I E~ 2 dF dr2' ro ro ~, F

(6)

r r I ErlflTdF

F

,,r dr S / dr 2 + f~,._2dF dr2, ro r~ ~ F

(7)

r I Ef lTdF

wOT = I F ro I E d F

F

2d ) r dOda! Ep dr.

dr - -~r dr I EdF F

(8)

For the upper limit r = R, we arrive at the corresponding expressions for the peripheral section of the pen. The analysis of relations (5)--(8) demonstrates that, in the absence of natural torsion of the blades, their

temperature torsion is also absent. Temperature deflections in the planes of minimum and maximum stiffness and the temperature elongation of the axis of the pen of a naturally twisted blade are represented as algebraic sums of two components. In each case, one component determines the displacements of the untwisted blade if all other conditions are equal and the second component is a correction factor used to take into account the influence of natural torsion of the blade. Temperature displacements of the peripheral section of the pen depend on the combination of geometric characteristics of the profile o f the blade and the distribution of temperature over its pen. If all other conditions are equal, then the temperature displacements of the peripheral section of the pen increase with its length.

To perform computations according to relations (5)-(8), it is necessary to know the temperature fields in the entire volume of the pen for all stages of the temperature mode. The displacements were computed for different stages of the temperature mode, i.e., the problem was studied in the quasistatic statement. The integrands in relations (5)-(8) can be found for several cross sections along the length of the blade. Since the integrands are continuous along the r-axis of the blade, their missing values can be found by interpolation. The validity of this approach was established in [5].

As an example, we consider the procedure of evaluation of the angle of torsion and displacements of the peripheral section of the pen of the blade under the conditions of nonuniform heating. The nonstationary thermal mode of operation of vehicular gas-turbine engines is characterized by the alternation of the types of changes in temperature, i.e., in fact, by the consecutive periods of heating and cooling. Moreover, since the volume between the turbine and the compressor is small, on passing from one mode of operation to another, parallel with the consumption of fuel, the temperature and pressure of gases change almost instantaneously [6]. Therefore, in practical calculations, it is possible to split the nonstationary thermal mode of operation of gas-turbine engines into stages of finite duration for which the parameters of the gas are constant and analyze the process of heat exchange between the gas and the pen of the blade for each of these stages separately. In this case, the problem of nonstationary heat conduction for the pen of the blade should be solved successively under the assumption that the initial condition for each stage of the nonstationary mode coincides with the final distribution of temperature in the pen attained in the previous stage.

As an object of numerical calculations, we use a turbine blade made of KhN51VMTYuKFR heat-resistant alloy with a length of the pen of 45 mm. The length of the chord in the mean section is equal to 27 mm, the angle of torsion of the peripheral section about the root section is equal to 0.45 rad, and the maximum thickness of the profile in the mean section is 4.3 mm. The temperature of the gas in the process of heating is equal to 1150~ and, in the process of cooling,

to 300~ The total pressure is equal to 1.3.105 and 1.16.105 Pa, respectively. The velocity of incoming flow in the

grating is 435 m/s in the process of heating and 215 m/s in the process of cooling. It is worth noting that these and similar data are, as a rule, known in the stage of design of turbine blades. To make the situation more illustrative, we consider the following pulsating mode of operation with a period of thermal cycling of 15 sec: 5 sec of heating followed by 10 sec of cooling. In this case, the process of heat exchange is analyzed under boundary conditions of the third kind, i.e., the temperature of the gas and the heat transfer coefficients on the surface of the pen are known. To do this, we

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a, W / m e .deg

2500 /~

,ooo., II

soo , ' . 'l

0 o.~ o.q ' o.6 o.s Back Saddle

Fig. 2. Distributions of the local heat transfer coefficients along the perimeter of the profile in the processes of heating (solid line) and cooling (dashed line).

7~0 ?f5

~ 2 ~ 7 1 5

9~o'cL ~ a ' . , , , , j \~To

q?O*..A ",.,,"- h "~ i \ ~90

~&O'C

Fig. 3. Temperature fields in the section of the blade at the end of the processes of heating (a) and cooling (b).

preliminarily constructed the diagrams of velocities of the gas along the contour of the profile by the method o f Samoilovich and Sherstyuk [7, 8]. Then these diagrams were used to determine the local heat transfer coefficients along the profile by the method of Zysina-Molozhen [9].

The distributions of the local heat transfer coefficients along the perimeter of the profile in the mean section of the pen are displayed in Fig. 2 for the heating and cooling modes. Similar calculations were carried out for sections close to the root and peripheral sections. The two-dimensional boundary-value problem of nonstationary heat conduction was solved for different sections by the finite-element method for known time dependences of the temperature of the gas and local heat transfer coefficients on the profile under the assumption that the heat flows along the length of the pen can be neglected [ 10]. For simplicity, we took a uniform distribution of temperature across the section of the pen as initial. The temperature fields were computed up to the attainment of a stable cyclic temperature mode.

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0 T ' 104 , r a d / c m

0

- 8

-1 6

- 2 4

16

8

/

0 T " 10 4 , r a d / c m

20/.,.__., \ 1~ /

0~ 4

-30 i

b

8 12

~ J

l ' , S

Fig. 4. Behavior of the relative angle of torsion along the length of the pen (a) and the angle of torsion of the peripheral section of the pen (b) for different stages of a temperature cycle.

6 r

0.2

-0.2

-0.4

-0.6

m m

\ \

I I

/ \ /

/

8 !10 12 14 ~, S

Fig. 5. Displacements of the peripheral section of the pen of the blade in the planes of minimum (1) and maximum (2) stiffness.

As an example, in Fig. 3, we present the temperature fields in the midsection of the blade at the end of the processes of heating and cooling. It should be emphasized that, under the indicated boundary conditions, the experimental distribution of temperature in this section of the blade established in [11] on a gas-dynamical stand for a stable mode of thermal cycling practically coincides with the predicted distribution. The existing insignificant difference between the values of temperature can be explained by the delayed action of the wire thermocouples used in these tests.

The curves reflecting the behavior of the relative angles of torsion of sections of the blade along its length computed by using the integrand from relation (5) for different stages of the temperature cycle are presented in Fig. 4a. It is easy to see that some curves lie above the abscissa and the other curves lie below this line. This means that the direction of torsion changes in each temperature cycle when we pass from heating to cooling. The character of the curves shows that a relatively thin-walled upper one-third part of the pen makes the main contribution to the temperature torsion of the peripheral section. Thus, as in the case of thin-walled bars [ 12], the thickness of the walls of the pens of naturally twisted turbine blades is a decisive factor for their temperature torsion. According to these curves, by using relation (5), we found the angle of torsion of the peripheral section of the pen for different times. The behavior of this angle in the course of a temperature cycle is illustrated by Fig. 4b. It is easy to see that the angle of temperature torsion passes through zero at the beginning of the processes of heating and cooling. In the process of heating, the angle of torsion of the pen increases. In the process of cooling, it decreases.

The curves of displacements of the peripheral section of the pen of the blade in the course of a temperature cycle are plotted in Fig. 5 according to formulas (6) and (7). Curves 1 and 2 correspond, respectively, to the temperature displacements of the peripheral section in the planes of minimum and maximum stiffness. The dashed line 1 is plotted

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without taking into account the influence of the natural torsion of the pen on the temperature displacements of the peripheral section of the pen in the plane of minimum stiffness. The fact that the difference between the values of displacements for the same times is insignificant means that the indicated influence is negligibly weak for the analyzed relatively short blade. The results of calculations show that the influence of the natural torsion of the pen of the blade can be neglected for the temperature deflections in the plane of maximum stiffness and temperature elongations.

In conclusion, we note that, in practice, in the presence of temperature gradients across the section of the pen, parallel with thermal strains induced in the pen by various temperature elongations of its elements, one must also take into account additional thermal strains formed in turbine blades in the presence restrictions (constraints) imposed on their temperature twists and deflections, in particular, in blades with shroud connections, especially if the technology of mounting is not perfect. This is important not only for relatively long blades whose length is 4-5 times longerr than their chord but also for relatively short blades whose length is longer than the length of the chord only by a factor of 1.5-2. Thus, due to the nonuniform heating of short blades, by varying the degree of constraints imposed on the displacements of the peripheral section of the pen, one may obtain durabilities that differ by an order of magnitude for the same temperature cycle [13].

REFERENCES

.

2.

3.

4.

5.

6.

7.

8.

9.

10. 11.

12. 13.

I. A. Birger and Yu. G. Panovko (editors), Strength, Stability, and Vibrations. A Handbook [in Russian], Vol. 1, Mashinostroenie, Moscow (1968). " I. A. Birger, B. F. Shorr, and R. M. Shneiderovich, Strength Analysis of Machine Parts [in Russian], Mashinostroenie, Moscow (1966). I. A. Birger and B. F. Shorr (editors), Thermal Strength of Machine Parts [in Russian], Mashinostroenie, Moscow (1975). B. F. Shorr, "On the theory of twisted nonuniformly heated rods," Izv. Akad. Nauk SSSR, Mekh. Mashinostr., No. 1, 141-151 (1960). G. S. Pisarenko and A. I. Petrenko, "Investigation of the strained state of turbine blades caused by nonuniform heating," ProbL Proehn., No. 11, 3-6 (1975). I. I. Kirillov, Automatic Control of Steam Turbines and Gas-Turbine Installations [in Russian], Mashino- stroenie, Leningrad (1988). M. E. Deich and G. S. Samoilovich, Fundamentals of the Aerodynamics of Axial Turbomachines [in Russian], Mashgiz, Moscow (I 959). A. N. Sherstyuk, Numerical Analysis o f Flows in the Elements of Turbomaehines [in Russian], Mashino- stroenie, Moscow (1967). L. M. Zysina-Molozhen, L. V. Zysin, and M. P. Polyak, Heat Exchange in Turbomachines [in Russian], Mashinostroenie, Leningrad (1974). L. J. Segerlind, Applied Finite Element Analysis, Wiley, New York (1976). G. S. Pisarenko (editor), Strength of Materials and Structural Elements under Extreme Conditions [in Russian], Vol. 2, Naukova Dumka, Kiev (1980). V. Z. Vlasov, Thin-Walled Elastic Rods [in Russian], Fizmatgiz, Moscow (1959). G. S. Pisarenko and A. I. Pctrenko, "On a procedure for testing turbine blades for thermal fatigue," ProbL Prochn., No. 6, 100--105 (1976).

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