U.S. patent number 3,836,283 [Application Number 05/313,843] was granted by the patent office on 1974-09-17 for construction of axial-flow turbine blades.
This patent grant is currently assigned to The Director of National Aerospace Laboratory of Science and Technology. Invention is credited to Masakatsu Matsuki, Toyoaki Yoshida.
United States Patent |
3,836,283 |
Matsuki , et al. |
September 17, 1974 |
**Please see images for:
( Certificate of Correction ) ** |
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.
Inventors: |
Matsuki; Masakatsu (Tokyo,
JA), Yoshida; Toyoaki (Tokyo, JA) |
Assignee: |
The Director of National Aerospace
Laboratory of Science and Technology (Masao Yamanouchi, Tokyo,
JA)
|
Family
ID: |
12697673 |
Appl.
No.: |
05/313,843 |
Filed: |
December 11, 1972 |
Foreign Application Priority Data
|
|
|
|
|
May 8, 1972 [JA] |
|
|
47-44663 |
|
Current U.S.
Class: |
416/96R; 416/97A;
416/97R; 415/115 |
Current CPC
Class: |
F01D
5/189 (20130101); F05D 2260/201 (20130101); Y02T
50/60 (20130101); Y02T 50/676 (20130101); Y02T
50/673 (20130101) |
Current International
Class: |
F01D
5/18 (20060101); F01d 005/18 () |
Field of
Search: |
;416/92,95-97
;415/115-116 |
References Cited
[Referenced By]
U.S. Patent Documents
Foreign Patent Documents
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|
|
|
|
|
|
924,012 |
|
Mar 1947 |
|
FR |
|
892,698 |
|
Oct 1953 |
|
DT |
|
910,400 |
|
Nov 1962 |
|
GB |
|
Primary Examiner: Powell, Jr.; Everette A.
Attorney, Agent or Firm: Brooks Haidt & Haffner
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 .
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.
* * * * *