Construction Of Axial-flow Turbine Blades

Abstract

Axial-flow turbine nozzles and moving blades which employ hollow blade means, the blades having a wall thickness distribution on both sides, suction side and pressure side, of the hollow portion which is selected in accordance with the distribution of effective local heat transfer coefficients along the blade surface in the chordwise direction, whereby the temperature distribution in said hollow blades responds almost uniformly to the temperature change of the motive fluid.

Description

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to improvements in turbine blades and nozzles and more particularly to a construction of axial-flow turbine blades for a gas turbine or a steam turbine which is subjected to frequent starts and stops.

2. Description of the Prior Art

Much research on fluid cooled turbine blades has been carried out and many inventions have been made. However, almost all of such research and inventions have been intended to make the blade temperature uniform and to keep it lower under steady state conditions. Effective cooling methods of a turbine blade which is subject to severe thermal conditions which have been adopted successfully are impingement cooling or film cooling at the leading edge and film cooling at the trailing edge. Therefore, heat resistance of the blade has been considerably improved under steady state conditions. Generally speaking, heat capacities at the leading edge and the trailing edge are relatively small compared with heat capacities in the chordwise direction of the middle part of the blade. Therefore, if the blade is subject to a sudden change of the temperature of the motive fluid, each part of the blade shows different response and excessive thermal stresses come about at the leading edge and/or the trailing edge, and for such reason, many examples of blades with cracks are found.

SUMMARY OF THE INVENTION

It is an object of the present invention to eliminate the above-mentioned disadvantage existing in the conventional form of turbine blade and to provide a hollow turbine blade, the wall thickness distribution of which corresponds to the effective local heat transfer coefficient distribution of said blade. Similar principles are applicable to turbine nozzles.

It is another object of the present invention to provide a method for determining the turbine blade thickness distribution.

It is a further object of the present invention to provide a method for reducing an effective local heat transfer coefficient, in the case that the blade thickness is restricted by the aerodynamic performances of the blade at the trailing edge region.

These and other objects of the present invention will be apparent when the reference is made to the following description and accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a cross-sectional, end view of an axial-flow turbine blade constructed in accordance with the present invention, and the cooling thereof comprises impingement cooling at the leading edge, convection cooling in the mid-chord region and film cooling at the trailing edge;

FIG. 2 is a graph illustrating the distribution of the effective local heat transfer coefficients;

FIG. 3 is a schematic diagram used for the calculation of the non-steady state, one-dimensional temperature;

FIG. 4 is a graph illustrating the non-steady state temperature distribution at the leading edge of a blade with elapsed time;

FIG. 5 is a graph of the blade thickness distribution in the chordwise direction, i.e., leading edge to trailing edge direction, of the outer shell of the blade;

FIG. 6 is a cross-sectional, end view of another example of an axial-flow turbine blade constructed in accordance with the present invention, in which film cooling is used at the leading edge region and the trailing edge region;

FIG. 7 is a graph illustrating the distribution, of non-steady state thermal stresses without the use of the invention; and

FIG. 8 is a graph illustrating the distribution of non-steady state thermal stresses with the use of the present invention.

DETAILED DESCRIPTION OF THE INVENTION

The present invention provides a method for making the shell thickness distribution on the pressure side and the suction side of the hollow blade correspond to the distribution of the effective local heat transfer coefficients along the blade surface in the chordwise direction, i.e., the direction from the leading edge to the trailing edge. According to the method, the temperature at each part of the blade changes almost uniformly even in the case of the transient operation such as starting, stopping, acceleration and deceleration. Therefore, no excessive thermal stresses occur in the blade, and consequently, the durability of the blade constructed by such method is remarkably increased compared with that of the conventional hollow blade which is constructed without taking into consideration the transient operation.

The said effective local heat transfer coefficient .alpha. is a constant of proportionality, defined by the following equation,

q = .alpha.(Tg - Tb) (1)

where q represents heat flux, Tg represents the recovery temperature of a motive fluid and Tb represents the blade surface temperature. In the case of film cooling and transpiration cooling, the local heat transfer coefficient .alpha.' is expressed as follows,

q = .alpha.'(Taw - Tb) (2)

where Taw represents adiabatic wall temperature of the blade. It will be noted that with the cooling of the turbine blade by secondary fluid, Tg is always higher than Taw. Therefore, from equations (1) and (2) one obtains the relation that .alpha.<.alpha.'. This relation means that if (Tg - Tb) is introduced as a standard temperature difference even in the case of a film cooling or a transpiration cooling, the value of effective local heat transfer coefficient is lower than the local heat transfer coefficient, whereas the effective local heat transfer coefficient .alpha. coincides with the conventional local heat transfer coefficient without secondary air cooling. Therefore, the factors can be taken into account by equation (1) independently of the cooling methods.

In what follows, theoretical background, blade construction and effects of the present invention are given together with the description of the figures.

The blade illustrated in cross-section in FIG. 1 has an outer shell 1 having a lower pressure or suction surface wall and a high pressure or pressure surface wall, the suction side being designated by the numeral 3 and the pressure side being designated by the numeral 4, and a cooling fluid insert or duct 2 is within the shell 1 and has its outer wall spaced from the inner wall of the shell 1. The insert 2 has an opening 2a for directing cooling fluid against the leading edge portion of the shell 1, and the fluid flows rearwardly of the blade between the outer wall of the insert 2 and the inner wall of the shell 1 and is exhausted through the channel 1a.

Reference numeral 5 designates one of the small elements or portions of the outer shell 1 which is used for the application of numerical calculations. Reference numerals 6 and 7 designate main air flow side and cooling air flow side of the hollow blade respectively.

In FIG. 1, the intersections of the extensions of the wall of the impingement hole 2a of said insert 2 and the inner surface of the said outer shell 1 is designated by the letter P. Extensions which are .theta. = 50.degree. on both sides of the impingement hole center line and which go through the center of the circle which contains the blade leading edge will intersect the inner surface of the said outer shell 1 at Q. The main flow is divided into two parts, suction side surface flow and pressure side surface flow, at the outer surface stagnation point R. On the other hand, the cooling air impinges on the inner surface stagnation point S which is located at the inner side of the shell 1, and opposite to the point R.

If there occurs a sudden temperature change of the motive fluid impinging on convection cooled turbine blades, such as in FIG. 1, almost all of the heat flow is transferred in the shell thickness direction. Therefore, the heat flow by conduction, both in the chordwise direction and in the spanwise direction, can be neglected. Consequently, non-steady state temperature in said small element 5 is obtained analytically from the fundamental equation for one-dimensional, non-steady state heat conduction, for which the distribution of the local heat transfer coefficients and the temperature distribution in the ambient fluid are needed as boundary conditions. Incidentally, the heat flow by conduction both in the chordwise direction and in the spanwise direction are neglected in the present calculations, but if these heat flows are also considered in determining the temperature distribution, an even more effective blade will be realized.

The local heat transfer coefficients .alpha..sub.gx in the main air flow side 6 along the outer surface of the shell 1 and .alpha..sub.cx in the cooling air flow side 7 along the inner surface of the shell 1, in the chordwise direction can be calculated from the empirical equations explained below. The empirical equation on the convective heat transfer is universally described with some dimensionless numbers as follows,

Nu.sub.x = c.sup.. Re.sub.x.sup.m Pr.sup.n (3)

where Nu.sub.x represents a local Nusselt number, Re.sub.x represents a Reynolds number, Pr represents a Prandtl number. These three numbers are described in detail in a proper textbook of Heat Transfer, e.g., "Heat & Mass Transfer" by Eckert, Drake, McGraw-Hill, or "Heat Transmission" by McAdams, McGraw-Hill, and c, m and n are numerical constants. Then, the local heat transfer coefficient .alpha..sub.x can be obtained by substituting Nu.sub.x = .alpha..sub.x.sup.. X/.lambda., and Re.sub. x = UX /.nu. into equation (3) and adopting the values of c, m and n which are suitable to the portion of the blade surface considered, where .alpha..sub.x represents the heat transfer coefficient, X stands for a representative length, .lambda. represents the thermal conductivity of the fluid, U represents the velocity of fluid and .nu. represents the kinetic viscosity of fluid.

According to this procedure, the values of the heat transfer coefficients are calculated from each empirical equation applied to the blade portion identified hereinafter.

a. Main air flow side (.alpha..sub.gx)

i. the leading edge stagnation point R and its neiborhood region:

In the case of the turbine blade under consideration, the leading edge region can be considered as a circular cylinder in the range from the leading edge stagnation point R to .theta. = 60.degree.. Therefore, the following empirical equation by Schmidt and Wenner (see Forschung, 12, 65 (1941)) is used for the heat transfer coefficients along the circumference of a circular cylinder,

.alpha..sub.gx = 1.14(.lambda.g/d.sub.l)Pr.sup.0.4 .sqroot.(U.sub.1 d.sub.l /.nu.g) {1 - [(.theta./90)] .sup.3 } (4)

with 0.degree. .ltoreq. .theta. .ltoreq. 60.degree., where .lambda..sub.g represents a thermal conductivity of main flow, d.sub.l represents a leading edge outer diameter, U.sub.1 represents an inlet velocity of the main air flow, .nu..sub.g represents the kinematic viscosity of the main flow and .theta. represents the angle from the leading edge stagnation point R as shown in FIG. 1.

ii. the mid-chord region and trailing edge region:

In the region from the point .theta. = 60.degree. to the trailing edge along the blade surface, the required empirical equation is adopted from that for the laminar boundary layer along a flat plate,

.alpha..sub.gx = K.sub.g U.sub.sx.sup.0.5 X.sup.-.sup.0.5

with

K.sub.g = .alpha..sub.g60 U.sub.s60.sup.-.sup.0.5 [(.pi./6) d.sub.l ].sup.0.5 (5)

where U.sub.sx represents the local velocity of the main air flow, X represents the distance from the leading edge stagnation point along the blade surface in the chordwise direction, .alpha..sub.g60 represents a heat transfer coefficient at the point .theta. = 60.degree., and U.sub.s60 represents the main air flow velocity at the point .theta. = 60.degree.. When the transition point is reached on the blade surface, the following empirical equation is adopted in the rearward direction from the transition point,

.alpha..sub.gx = 0.0296 .lambda..sub.g .nu..sub.g.sup.-.sup.0.8 Pr.sup.1/3 U.sub.sx.sup.0.8 X.sup.-.sup.0.2 (6)

this equation is derived from the equation for the turbulent boundary layer along a flat plate.

b. Cooling air flow side (.alpha..sub.cx)

i. the leading edge stagnation point S and the adjacent region:

The heat transfer coefficient .alpha..sub.cstg at the leading edge stagnation point S in the cooling air flow side is obtained from the blade surface temperature T.sub.bstg at the stagnation point and heat transfer coefficient .alpha..sub.gstg at the stagnation point in the main air flow side. Namely, .alpha..sub.cstg is calculated from the following equation,

.alpha..sub.cstg = .alpha..sub.gstg S.sub.g /S.sub.c [T.sub.go -T.sub.bstg /T.sub.bstg -T.sub.co ] (7)

where S.sub.g and S.sub.c represent surface heat transfer area in the main air flow side and in the cooling air flow side, respectively. T.sub.go represents the main air flow inlet temperature and T.sub.co represents the cooling air inlet temperature. .alpha..sub.gstg is obtained from equation (4). The value of .alpha..sub.cstg is applied to the region from the stagnation point S to the point P shown in FIG. 1.

ii. the region adjacent to the leading edge area:

The stream flow of the cooling air in the region adjacent to the leading edge area can be considered to be equivalent to that of the jet flow which impinges upon a flat plate. Therefore, the following empirical equation is applied to the region from the point P to the point Q shown in FIG. 1:

.alpha..sub.cx = K.sub.cp.sup.. U.sub.cx.sup.2/3.sup.. X.sub.i.sup.-.sup.1/ 3

with

K.sub.cp = .alpha..sub.cstg U.sub.cp.sup.-.sup.2/3 X.sub.ip.sup.1/3 (8)

where U.sub.cx represents the local velocity of the cooling air flow, x.sub.i represents the distance in the chordwise direction from the leading edge stagnation point S in the cooling air flow side along the inner surface of the outer shell 1, and U.sub.cp and X.sub.ip are values of U.sub.cx and X.sub.ip at the point P, respectively.

iii. the mid-chord region and trailing edge region:

In the region of the blade which is rearward from the point Q, the empirical equation for the turbulent boundary layer along a flat plate is applied,

.alpha..sub.cx = K.sub.cq U.sub.cx.sup.0.8 X.sub.i.sup.-.sup.0.2

with

K.sub.cq = .alpha..sub.cq U.sub.cq.sup.-.sup.0.8 X.sub.iq.sup.0.2 (9)

where .alpha..sub.cq, U.sub.cq and X.sub.iq are values of .alpha..sub.cx, U.sub.cx and X.sub.i at the point Q, respectively. Making use of equations (4) - (9), the local heat transfer coefficients .alpha..sub.gx and .alpha..sub.cx were calculated for the turbine blade shown in FIG. 1 under the following conditions: turbine inlet temperature T.sub.go = 1,150.degree. C, cooling air inlet temperature T.sub.co = 500.degree. C, main flow inlet velocity U.sub.1 = 114 m/sec and cooling air weight flow ratio Wc/Wg = 2 percent, where Wc is cooling air weight flow rate and Wg is main air weight flow rate.

FIG. 2 is a graph which shows the distribution of the blade surface local heat transfer coefficients .alpha..sub.gx, .alpha..sub.cx using the methods of calculation just described. In this figure the ordinate is the heat transfer coefficient, the abscissa is the distance along the outer blade surface and the origin corresponds to the leading edge stagnation points R and S.

FIG. 3 is a schematic diagram referred to for the calculation of the non-steady state temperature in a small element of the blade, such as the small element 5. Let the blade shell thickness be l, and assume that the y axis is oriented in the blade shell thickness direction with its origin located at the blade surface in the main air flow side. T.sub.g represents the main air flow temperature, and T.sub.c represents cooling air flow temperature. T.sub.A and T.sub.B represents the blade surface temperatures at y = 0 and y = l respectively, under steady state conditions. To represents the temperature of the entire region kept in an equilibrium state that is realized before heating or after cooling. Temperature T(y,t) at an arbitrary position and arbitrary time in the small element can be obtained from the following fundamental equation:

.delta.T/.delta.t = a(.delta..sup.2 T/.delta..sub.y 2) (10)

where t is the elapsed time after a sudden temperature change of the motive fluid and a represents the thermal diffusivity of the blade material. The analytical solution of equation (10) with the following boundary conditions and initial conditions is already set forth in an article by I. Fujii and N. Isshiki appearing in Vol. 35 No. 271 for March 1969 of the publication "TRANSACTIONS OF THE J.S.M.E."

Boundary conditions and initial conditions: In the case of heating (H)

at y = O,

.alpha..sub.gx (T.sub.g - T(o,t)) = (-.lambda..delta.T/.delta. y) y = o

at y = l,

.alpha..sub.cx (T(l,t) - T.sub.c) = (-.lambda..delta.T/.delta. y) y = l (11)

at t = O,

T(y,O) = To

at t = .infin.,

T(y,.infin.) = T.sub.A - (T.sub.A - T.sub.B)y/l In the case of cooling (C)

at y = O,

.alpha..sub.gx (T(o,t) - T.sub.o) = (.lambda..delta.T/.delta. y) y = o

at y = l,

.alpha..sub.cx (T(l,t) - T.sub.o) = (-.lambda..delta.T/.delta. y) y = l (12)

at t = O,

T(y,O) = T.sub.A - (T.sub.A - T.sub.B )y/l

at t = .infin.,

T(y,.infin.) = T.sub.o

where .lambda. represents the thermal conductivity of the blade material, and T.sub.A and T.sub.B are described as follows,

T.sub.A = T.sub.g [1 + (.alpha..sub.cx l/.lambda.) + (.alpha..sub.cx /.alpha..sub.gx)(T.sub.c /T.sub.g)]/[1 + (.alpha..sub.cx l/.lambda.) + (.alpha..sub.cx /.alpha..sub.gx)] (13)

T.sub.B = T.sub.g [1 + (.alpha..sub.cx l/.lambda.)(T.sub.c /T.sub.g + (.alpha..sub.cx /.alpha..sub.gx)(T.sub.c /T.sub.g)]/[1 + (.alpha..sub.cx l/.lambda.) + (.alpha..sub.cx /.alpha..sub.gx)] (14)

The results for the non-steady state, one-dimensional temperature distributions are then:

In the case of heating (H)

T(y,t) = T.sub.N + T.sub.A - (T.sub.A - T.sub.B)y/l (15)

In the case of cooling (C)

T(y,t) = T.sub.O - T.sub.N (16)

where T.sub.N represents the non-steady term of the temperature, and is expressed as follows, ##SPC1##

x{[C.sub.1 + C.sub.2 l + (C.sub.2 K.sub.g /.alpha..sub.n.sup.2)] sin(.alpha..sub.n l) - [(C.sub.1 K.sub.g /.alpha..sub.n) - (C.sub.2 /.alpha..sub.n) + (C.sub.2 K.sub.g l/.alpha..sub.n)]cos (.alpha..sub.n l) + (C.sub.1 K.sub.g /.alpha..sub.n) - (C.sub.2 /.alpha..sub.n)} (17)

where C.sub.1, C.sub.2, K.sub.g and K.sub.c are constants defined by the following equations,

C.sub.1 = T.sub.O - T.sub.A, C.sub.2 = (T.sub.A - T.sub.B)/l, K.sub.g = .alpha..sub.gx /.lambda. and K.sub.c = .alpha..sub.cx /.lambda. and .alpha..sub.n is a positive root in the equation:

tan(.alpha..sub.n l) = [(K.sub.c + K.sub.g).alpha..sub.n /.alpha..sub.n.sup.2 - K.sub.c K.sub.g ]

First of all, the non-steady state temperature at the leading edge (x = O,y = O) was calculated under the following conditions: T.sub.go = 1,150.degree. C, T.sub.co = 500.degree. C, a = 4.44 .times. 10.sup.-.sup.6 m.sup.2 /sec., .lambda. = 15 Kcal/mh.degree. C, .alpha..sub.gx = 1,360 Kcal/m.sup.2 h.degree. C, .alpha..sub.cx = 1,520 Kcal/m.sup.2 h.degree. C, l = 2mm and chord length = 32 mm. Then, the calculations were carried out also at the point (x = O,y = l/2) and (x = O,y = l) according to the same procedure. These results are plotted in FIG. 4 where the ordinate is dimensionless temperature T - T.sub.c /T.sub.g - T.sub.c, the abscissa is elapsed time t, and the symbols (H) and (C) correspond to the case of heating and cooling respectively. As is evident from equations (15) - (17) and FIG. 4, the response of temperature T(y,t) is not exactly the same as the first-order response to the step input used in the linear dynamic system, but its trend is very similar. The time constant .tau. of the blade temperature T(y,t) is defined by the same method as is used in said first-order response, namely is the elapsed time when 63.2 percent of the value at the steady state, T(y,.infin.), is reached.

If the transient temperature response were the same in every part of the blade, thermal stresses which come about under the transient operating conditions can be considerably reduced. In order to reduce the thermal stresses, it is very effective to make the shell thickness l along the blade surface so that the time constant .tau. may be considered much the same in every part of the blade. Let the arithmetic mean value of time constants calculated at y = O, y = l/2 and y = l of the leading edge be a representative time constant .tau.m. Then, the blade thickness at each position in the chordwise direction is determined so that its time constant will be equal to .tau.m, where the time constant at each position is also the arithmetic mean value of the three points y = O, y = l/2 and y = l.

The blade thickness distribution in the chordwise direction calculated by the said procedure is shown by the graph of FIG. 5. In this figure, the ordinate is the blade thickness l, the abscissa is the distance along the blade surface and the origin corresponds to the leading edge. In this case, the representative time constant .tau.m is equal to 2.391 sec.

FIG. 7 and FIG. 8 are graphs illustrating the distribution of non-steady state thermal stresses obtained from non-steady state blade temperature distribution calculated by equations (13) - (17). In these figures, the ordinate is the thermal stress .sigma.Kg/mm.sup.2, the abscissa is the distance along the blade surface, and the elapsed time t is taken as the parameter. FIG. 7 is the result obtained in the case of constant blade thickness that does not take into consideration the desirability of equal time constants. On the other hand, FIG. 8 is a graph of the results obtained by the methods of the present invention which considers the time constants and makes them substantially equal. From these two figures, it is apparent that if the transient response at every part of the blade is taken into consideration, thermal stresses can be remarkably reduced. Then, according to the present invention, crack initiation on the blade surface can be avoided for far longer times than have heretofore been accomplished, and consequently, the blade can sufficiently withstand the frequent starts and stops of the engines including the blades.

Further, in the event that the blade thickness calculated by the methods of the present invention conflict with the blade profile designed on the basis of the aerodynamic performance, especially in the trailing edge region, it is sufficient to make the effective local heat transfer coefficient correspond to the profile desired from the aerodynamic performance and then introducing a film cooling or a transpiration cooling to the relevant region.

FIG. 6 is another embodiment of the turbine blades to which the present invention is applied. The cooling thereof comprises impingement cooling and film cooling at the leading edge, convection cooling in the mid-chord region and film cooling in the trailing edge region. The outer shell 1 encloses a pair of inserts 2b and 2c. The holes 8 and 9 are made at the leading edge region for film cooling. In this embodiment, the equality of transient response of the various portions of the blade is easily realized within the required blade profile because of the application of the film cooling through the channels or holes 10 and 11 at the trailing edge.

In the embodiment shown in FIG. 1, heat flow by conduction both in the chordwise direction (x) and (-.lambda..delta.spanwise direction (z) is considered negligibly small when the calculation of the non-steady state temperature is carried out. However, if these heat flows are taken into account, then the fundamental equation (10) should be modified in the following manner.

.delta.T/.delta.t = a[.delta..sup.2 T/.delta..sub.x 2 + (.delta..sup.2 T/.delta..sub.y 2) + (.delta..sup.2 T/.delta..sub.z '

By applying this modified equation (10)', the heat resistance of the blade will be further improved.

As stated hereinbefore, the temperature of every part of the air cooled hollow blade, the thickness distribution which corresponds to the distribution of the effective local heat transfer coefficients, shows an almost uniform response to a sudden change of the motive fluid temperature. Accordingly, excessive thermal stress does not occur in the blade. For this reason, axial-flow turbine blades constructed in accordance with the present invention are very strong and resistant to frequent heat variations, such as by reason of starts and stops. In other words, the durability of the blade is remarkably increased.

Moreover, in accordance with the present invention, the turbine inlet temperature of the motive fluid can be higher, resulting in improvement of the thermal efficiency of a gas turbine or a steam turbine.

The construction of axial-flow turbine blades in accordance with the present invention is useful not only in aircraft engines, but also in marine turbines, steam turbines, automobile engines, etc. Accordingly, the present invention is extremely useful for industrial purposes.

Claims

We claim:

1. A hollow turbine part for use in a hot fluid medium, said part having a pressure surface wall and a suction surface wall and having a leading edge and a trailing edge, said walls having a thickness distribution in the direction from said leading edge to said trailing edge such that, with changes of the temperature of said fluid, the temperature response at each portion of said walls in substantially the same as the temperature response at the other portions of said walls, whereby the temperature distribution in said walls changes substantially uniformly in response to changes in temperature of said fluid.

2. A hollow turbine part as claimed in claim 1, wherein said part is a hollow blade and wherein said thickness distribution is such that each portion of said walls has a mean temperature time constant which is substantially equal to a predetermined time constant, said mean time constant at each portion of said walls being the mean of the temperature time constants at the outer surface thereof, at the inner surface thereof and at an intermediate point between the surfaces thereof, each of said outer surface, inner surface and intermediate point time constants being determined by replacing the temperature response at said outer surface, said inner surface and said point to a step change of said fluid temperature with approximately a first order response thereto, and said predetermined time constant being substantially equal to the mean time constant at said leading edge of said blade.

3. A hollow turbine part as claimed in claim 1, further comprising means for supplying fluid cooling to at least one of said leading edge and said trailing edge to thereby lower the heat transfer coefficient thereof and modify the thickness thereof required to provide said temperature response therefor.

4. A hollow turbine blade for use in a fluid medium, said blade comprising a pressure surface wall and a suction surface wall and having a leading and trailing edge, said walls having a thickness distribution in the direction from the leading edge to the trailing edge of said blade such that the temperature response at each portion of said walls in substantially the same as the other portions of said walls with changes of the temperature of said fluid and such that the mean temperature time constant is substantially equal to the mean temperature time constant at said leading edge of said blade, said mean time constant at each portion of said walls being the mean of the temperature time constants at the outer surface thereof, at the inner surface thereof and at an intermediate point between the surfaces thereof, each of said outer surface, inner surface and intermediate point time constants being determined by replacing the temperature response at said each point to a step change of said fluid temperature with approximately a first order response thereto, said temperature response at each point being calculated from the following equations:

.delta.T/.delta.t = a(.delta..sup.2 T/.delta..sub.y.sup.2)

where T(y,t) represents a temperature at an arbitrary position and an arbitrary time in a small element of said wall, t represents elapsed time after a sudden temperature change of said motive fluid, a represents thermal diffusivity of the blade material, and y represents the axis oriented in the blade wall thickness direction with its origin located at the blade surface in the main air flow side;

in the case of heating, the boundary conditions and initial conditions are:

at y = 0,

.alpha..sub.gx (T.sub.g - T(o,t)) = (- .lambda..delta.T/.delta.y) y = o

at y = l,

.alpha..sub.cx (T(l,t) - T.sub.c) = (- .lambda..delta.T/.delta.y) y = l

at t = 0,

T(y,O) = T.sub.o

at t = .infin.,

T(y,.infin.) = T.sub.A - (T.sub.A - T.sub.B) y/l

in the case of cooling, the boundary conditions and initial conditions are:

at y = 0,

.alpha..sub.gx (T(o,t) - To) = (.lambda..delta.T/.delta.y) y = o

at y = l,

.alpha..sub.cx (T(l,t) - T) = (-.lambda..delta.T/.delta. y) y = l

at t = 0,

T(y,0) = T.sub.A - (T.sub.A - T.sub.B) y/l

at t = .infin.,

T(y,.infin.) = T.sub.o

where in the boundary and initial condition equations T.sub.g represents the recovery temperature of the fluid, T.sub.c represents the cooling air flow temperature, To represents the temperature of the entire region kept in an equilibrium state that is realized before heating or after cooling, l represents the wall thickness and .lambda. represents the conductivity of the blade material, and T.sub.A and T.sub.B are:

T.sub.A = T.sub.g (1 + .alpha..sub.cx l/.lambda. + .alpha..sub.cx /.alpha..sub.gx T.sub.c /T.sub.g)/(1+ .alpha..sub.cx l/.lambda. + .alpha..sub.cx /.alpha..sub.gx)

T.sub.B = T.sub.g (1 + .alpha..sub.cx l/.lambda. T.sub.c /T.sub.g + .alpha..sub.cx /.alpha..sub.gx T.sub.c /T.sub.g)/(1+ .alpha..sub.cx l/.lambda. + .alpha..sub.cx /.alpha..sub.gx)

where .alpha..sub.gx represents an effective local heat transfer coefficient on main air flow side, .alpha..sub.cx represents an effective local heat transfer coefficient on the cooling air flow side;

and where, in the case of heating:

T(y,t) = T.sub.N + T.sub.A - (T.sub.A - T.sub.B)y/l

and in the case of cooling:

T(y,t) = T.sub.o - T.sub.N

where T.sub.N represents the non-steady state term of the temperature, and is expressed as follows: ##SPC2##

X{ (C.sub.1 + C.sub.2 l+ C.sub.2 K.sub.g /.alpha..sub.n.sup.2) sin (.alpha..sub.n l) - (C.sub.1 K.sub.g /.alpha..sub.n - C.sub.2 /.alpha..sub.n + C.sub.2 K.sub.g l/.alpha..sub.n cos (.alpha..sub.n l)+ C.sub.1 K.sub.g /.alpha..sub.n - C.sub.2 /.alpha..sub.n }

where C.sub.1, C.sub.2, K.sub.g and K.sub.c are constants defined by the following equations:

C.sub.1 = T.sub.o - T.sub.A

C.sub.2 = (T.sub.A - T.sub.B)/ l

K.sub.g = .alpha..sub.gx /.lambda., and

K.sub.c = .alpha..sub.cx /.lambda.

and where .alpha..sub.n is a positive root in the equation

tan(.alpha..sub.n l) = (K.sub.c + K.sub.g).alpha..sub.n /.alpha..sub.n.sup.2 - K.sub.c K.sub.g .