Procedure for calculating turbine blade temperatures and comparison of calculated with observed values for two stationar...

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Material Information

Title:
Procedure for calculating turbine blade temperatures and comparison of calculated with observed values for two stationary air-cooled blades
Series Title:
NACA RM
Physical Description:
38 p. : ill. ; 28 cm.
Language:
English
Creator:
Brown, W. Byron
Slone, Henry O
Richards, Hadley T
Lewis Research Center
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington, D.C
Publication Date:

Subjects

Subjects / Keywords:
Aircraft gas-turbines -- Blades   ( lcsh )
Aerodynamics -- Research   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
Abstract: Local and average blade temperatures were calculated for two stationary air-cooled turbine blades with 10 tubes and 13 fins forming the internal heat-transfer surfaces. These temperatures were calculated using previously published NACA temperature-distribution equations and the most recent theories for determining heat-transfer coefficients, including for the first time the allowance for effects of variable wall temperature on gas-to-blade heat-transfer coefficients at the leading and trailing sections of turbine blades. Comparison of calculated and experimental blade temperatures, for gas temperatures of 300° and 1000°F, resulted in good agreement.
Bibliography:
Includes bibliographic references (p. 26-29).
Statement of Responsibility:
by W. Byron Brown, Henry O. Slone, and Hadley T. Richards.
General Note:
"Report date September 29, 1952."
General Note:
"Classification changed to unclassified Authority: Mr. J.W. Crowley Change #3065 Date Aug. 17, 1955 E.L.B."--stamped on cover

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003810529
oclc - 133466154
sobekcm - AA00006183_00001
System ID:
AA00006183:00001

Full Text

,f-A m'. 534
SI 1:DENTIAL cow
RM E52H07


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ir :FOR CALCULATING TURBINE BLADE TEMPERATURES

PRISON OF CALCULATED WITH OBSERVED VALUES FOR

TWO STATIONARY AIR-COOLED BLADES

:.Byrn Brown, Henry 0. Slone, and Hadley T. Richards

Lewis Flight Propulsion Laboratory
Cleveland, Ohio
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DATE AUG. 17, 1955
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NACA RM E52H07 CONFIDEIITIAL


NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

RESEARCH MEMORANDUM

PROCEDURE FOR CALCULATING TURBINE BLADE TEMPERATURES AND

COMPARISON OF CALCULATED WITH OBSERVED VALUES FOR TWO

STATIONARY AIR-COOLED BLADES

By W. Byron Brown, Henry 0. Slone, and Hadley T. Richards


SUMMARY

The accurate prediction of local turbine blade temperatures is
necessary for the design of cooled turbines. When current methods of
predicting blade temperatures are applied to cooled turbine blades,
discrepancies between calculated and measured temperatures in some
blade leading and trailing sections result. In an effort to reduce
these discrepancies and hence to improve blade-temperature predictions,
an investigation was conducted for stationary turbine blades with 10 tubes
and 13 fins forming the internal heat-transfer surfaces. Local blade
temperatures were calculated using previously published NACA temperature-
distribution equations and the most recent theories for determining heat-
transfer coefficients, including for the first time the allowance for
effects of variable wall temperature on gas-to-blade heat-transfer
coefficients at the leading and trailing sections of turbine blades.
The calculated temperatures were compared with measured temperatures.

Results indicate that calculated trailing-section temperatures can
be greatly reduced and leading-section temperatures increased in blades
which have an appreciable temperature gradient in these sections when
gas-to-blade heat-transfer coefficients based oh variable wall tempera-
ture rather than coefficients based on constant wall temperature are
used in the temperature-distribution equations. Applications of gas-
to-blade coefficients based on variable wall temperature and an average
blade-to-coolant coefficient for the coolant passage nearest the trail-
ing edge for the 10-tube and 13-fin blades resulted in probable errors
for a point near the trailing edge of 60 and 40 F, respectively, for a
300 F gas temperature and of 360 and 300 F, respectively, for a
10000 F gas temperature. For a point near the leading edge, a similar
procedure resulted in probable errors of 50 and 60 F, respectively, for
a 3000 F gas temperature and of 8 and 18 F, respectively, for a
10000 F gas temperature. In the blade midchord region, where variable-
wall-temperature effects are negligible, maximum probable errors of 80
and 13 F for gas temperatures of 3000 F and 10000 F were obtained for
the two blades.


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NACA RM E52H07


niTR'.T'DUrT iii

A knowledge of cooled-blade temperatures for a turbine of known
design and operatingg conditions is extremely important in the evaluation
of cooled turbines. Reliably calculated blade temperatures enable the
turbine designer to (1) determine accurately the required coolant flow
necessary for the turbine design characteristics considered, (2) examine
the thermal gradients in the blade, because large thermal gradients may
cause the blade to fail, and (3) determine the strength characteristics
of the turbine blade on the basis of stress-to-rupture data. Blade
strength decreases rapidly as the blade temperature increases; thus an
accurate calculation of blade tempieratLures is required. Also, experi-
mental and reliably calculated blade temperatures afford a check on the
blade fabrication techniques; that is, a c.rmparison between experimental
and calculated temperatures may indicate whether or not the thermal bond
between the internal heat-transfer surfaces and the blade shell is
satisfactory.

As early as 1945, equations for calculating blade temperature dis-
tributions were developed and published by the NACA. These investiga-
tions are summarized in references 1 and 2. Because the accuracy of the
calculated temperatures depends primarily on the heat-transfer coeffi-
cients inserted into the equations, it is quite important that appro-
priate values for these be determinable.

For forced-convection blade-to-coolant heat-transfer coefficients,
hereinafter called inside coefficients, pipe correlations agree with
experimental correlations for air in a stationary cascade (reference 3)
and heated liquids in a rotating cascade (reference 4).

The gas-to-blade heat-transfer coefficients, hereinafter called
outside coefficients, offer a more complex problem because turbine blade
shapes differ widely among themselves and from round tubes, so that a
great variety of pressure and velocity distributions occurs. For laminar
flow, methods have been published for computing both average and local
outside coefficients for a wedge-type flow which are applicable to tur-
bine blades (reference 5). The conditions covered by that investigation
for a constant wall temperature include an Euler number range (a measure
of the pressure gradient) from -0.09 to 2.0, Mach numbers approximating
zero, a Prandtl number range from 0.6 to 1.0, and a temperature ratio
(gas to wall) equal to 1.0. Additional analyses of the laminar region
which can also be applied to cooled blades with impermeable walls were
made for the case of transpiration cooling. These analyses are pre-
sented in references 6 to 9.

An approximate method of solving the laminar-boundary-layer equa-
tions for cylinders of arbitrary cross section is presented in refer-
ence 6. This method requires that the velocity and temperature profiles


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NACA RM E52H07


in the boundary layer be assumed. The conditions covered by this anal-
ysis include a constant wall temperature, a temperature ratio equal to
1.0, and a range of Prandtl number. An exact method of solving the
laminar-boundary-layer equations for a wedge-type flow presented in ref-
erence 7 includes the conditions of a constant wall temperature, Euler
number range from separation values to 1.0, Mach numbers approximating
zero, a Prandtl number of 0.7, and temperature ratios from 1.0 to 4.0.
Reference 8 presents a tabulation of these exact solutions of the
laminar-boundary-layer equations for most of the conditions of refer-
ence 7 and includes results from two additional temperature ratios of
1/4 and 1/2. Another approximate method which utilizes prepared charts
to reduce calculation procedures is presented in reference 9 for the
calculation of heat transfer in the laminar region around cylinders of
arbitrary cross section. This method is based on the exact boundary-
layer solutions for wedge-type flow presented in references 7 and 8. The
boundary-layer equations for cylinders of arbitrary cross section were
used in reference 9 to compare the solutions obtained in references 7
and 8 for wedge-type flow. For impermeable blade walls, it can be shown
(reference 9) that the effect of temperature ratios from 1.0 to 2.0 on
heat transfer is negligible in the laminar region. At present, a tem-
perature ratio of 2.0 is probably the limit for air-cooled blades having
impermeable walls. Also, it is pointed out in reference 9 that heat-
transfer coefficients obtained for elliptical cylinders compared favorably
with those determined from wedge-type-flow solutions. On the basis of
the foregoing results, the simplified methods reported in reference 5 for
the determination of heat-transfer coefficients in the laminar region are
currently adequate for application to air-cooled turbine blades having
impermeable walls.

Local and average heat-transfer coefficients for turbulent flow are
computed in references 5, 10, and 11 for the cases of zero pressure
gradient (flat plate), constant wall temperature, low subsonic Mach
numbers, temperature ratio equal to 1.0, and a Prandtl number range from
0.5 to 10.

An equation for the average outside heat-transfer coefficient for
turbine blades including both laminar and tL.rbu1,lent flow is derived in
reference 5 for constant wall temperature, low subsonic Mach numbers,
temperature ratio equal to 1.0, and including the Euler number and
laminar to turbulent transition-ratio effects. Calculated average outside
coefficients using the equations of reference 5 compared favorably with
observed data for stationary blades in references 12 and 13 and with data
for a water-cooled turbine in reference 14.

The temperature equations reported in references 1 and 2 have been
used recently to compute cooled-blade temperatures in stationary and
rotating turbine blade cascades in which temperatures were measured
experimie..t.ali. Reference 15 gives a comparison between calculated and


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NACA RM E52H07


experimental blade temperatures for two stationary air-cooled blade con-
figurations. One of the configurations investigated was a hollow alumi-
num blade with long leading and trailing sections. The other configura-
tion was an air-cooled blade of low thermal conductivity with a long
trailing section, a short leading section, and 10 tubes forming the
internal heat-transfer surfaces. Herei:lafter, the terms long leading
and trailing sections will be used when referring to blades having a
sufficiently large area of low thermal conductivity metal in the uncooled
section to cause an appreciable temperature gradient. The methods
reported in reference 5 were used to compute constant wall temperature
outside coefficients for the calculations of reference 15; a stagnation
point coefficient was used to calculate local leading section tempera-
tures, and average outside cefficients were employed to calculate an
average midchord and local trailing section temperatures. Good agreement
was obtained for both blades at low and high gas temperatures except at
the trailing section of the 10-tube blade where the calculated tempera-
ture was at least 1000 F higher than the experimental value at a gas
temperature of 10000 F.

A rotating cascade of air-cooled thin-shelled blades (blades having
a mean wall thickness of approximately 0.040 in.) was used for a compar-
ison of average calculated and experimental blade temperatures in ref-
erence 16. An average constant-wall-temperature outside coefficient was
computed from reference 5 for the entire blade periphery, and the aver-
age calculated blade temperature was found to be approximately 300 F
less than the experimental value for a gas temperature of approximately
15000 F. Reference 17 contains a comparison of calculated and experi-
mental blade temperatures for the leading and trailing sections and the
midchord region of an aluminum water-cooled turbine. Once again, the
outside coefficients for constant wall temperature as computed from
reference 5 were used in these calculations. Generally, good agreement
resulted except at the leading section, where the maximum deviation was
470 F for a gas temperature range from 4000 to 16000 F.

On the basis of the foregoing results, the methods reported in
references 15 to 17 are considered accurate enough for calculating
cooled-blade temperatures at the midchord region of most blades and at
the leading and trailing sections of thin-shelled air-cooled blades.
Also, reliable temperatures can probably be calculated for blades of
high thermal conductivity even if the leading and trailing sections are
physically long. The methods are not accurate enough for long leading
and trailing sections of low thermal conductivity as indicated by the
results des-cribed for the trailing section of the 10-tube air-cooled
blade (reference 15). In future cooled-turbine-blade applications,
liquid-cooled blades and possibly some cast air-cooled blades may be
made of steel (low thermal conductivity) and have long leading and
trailing sections. Since low conductivity in a long blade section is


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NACA RM E52H07 CONFIDENTIAL 5


conducive to appreciable temperature gradients in that section, a
method of calculating blade temperatures more accurate than methods
obtainable to date is desired.

As was pointed out in the preceding discussion, local trailing-
section temperatures for a blade of low thermal conductivity and a long
trailing section resulted in calculated temperatures at least 1000 F
higher than the experimental values. Such trends were noticed as long
ago as 1945 by E. Schmidt (reference 18) who observed a much lower trail-
ing-section temperature than he calculated. This result was attributed
by E. R. G. Eckert to the shielding effect of the boundary layer, which
is strongly cooled in the forward portion of the blade; that is, the
metal temperature increases quite rapidly from the cooled midchord
region to the edge of the long trailing section, especially where metals
of low thermal conductivity are used. At the time, no method was given
for evaluating this effect numerically. Recently, however, some attempts
have been made to include these effects by consideration of variations
in wall temperature on outside coefficients. References 19 to 21 con-
sider laminar flow along flat plates and wedges, and reference 22 con-
siders the case for turbulent flow of an incompressible fluid along a
flat plate. Constant property values and pressure gradient effects are
included in references 19 and 21, whereas reference 20 allows for certain
variations in property values but does not include any pressure gradient
effect. On the basis of the results of references 19 to 22, it can be
concluded that the calculated trailing-section temperatures of the
10-tube blade (reference 15), which were considerably higher than the
experimental values, may be partially attributed to the effects of an
appreciable temperature gradient in the long trailing section.

Consequently, an investigation was conducted at the NACA Lewis
laboratory on two air-cooled turbine blade configurations (the 10-tube
blade of reference 15 and a 13-fin blade which has long leading and
trailing sections) in order to (1) obtain local experimental blade tem-
peratures around the blade periphery for two gas temperatures and a
range of cooling-air flow, and (2) compare the data with calculated
blade temperatures, wherein an attempt was made to eliminate the out-
standing discrepancies obtained in previous investigations. The experi-
mental investigation was conducted in a static cascade because the
instrumentation could be more complete and accurate than on a rotating
cascade.

The purposes of this report are to (1) apply variable-wall-
temperature corrections to the outside coefficients for sections of the
blade which require such corrections, and use these corrected coeffi-
cients to calculate blade temperatures, (2) compare the calculated tem-
peratures with experimental temperatures obtained in the investigation
just described in order to determine the adequacy of the variable-wall-
temperature correction in reducing the differences between experimental


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IJACA PR E52HOC'


and calculated temperatures for long leading and trailing sections, and
(3) present a detailed method for calculating local cooled-blade tem-
peratures by use of the best available theories for obtaining outside
and inside coefficients. The method of calculation is of such nature
that it can be carried out entirely from design data without any test
measurements.

The results of this investigation are presented for gas temperatures
of 3000 and 10000 F, a range of gas Mach number from 0.4 to 0.6, a mean
temperature ratio (gas to average wall) of 1.40, a range of coolicg-air
flow for Reynolds numbers from 5000 to 40,000 for each gas temperature,
and a range of Euler number in the laminar region from 1.0 to 0.


APPARATUS AND FRPCEDURE

Test Facility

A sectional view of the blade test section used in this investiga-
tion is shown in figure 1. Combustion air passed successively through
a flat-plate orifice, a combustor, and a plenum chamber prior to entering
the test section, and then into the exhaust system. The gasoline com-
bustor used in these investigations limited the gas temperature range to
between 3000 and 10000 F. The inlet duct to the test section was equipped
with a bellmouth to insure a uniform velocity profile at the entrance to
the cascade. The setup was insulated against heat loss from just down-
stream of the combustor to just downstream of the test section.

A cascade of seven blades was installed in the test section accord-
ing to the dimensions in figure 2. The test blade, installed as the
center blade, was the only blade through which cooling air was passed.
The other six blades had the same profile as the test blade. The cooling
air that was supplied to the test blade was obtained from the laboratory
refrigerated air system. The air passed successively riuLough a plenum
chamber, the blade entrance extension, the test blade, the blade exit
extension, another plenum chamber, a flat-plate orifice, and then into
the laboratory exhaust system (see fig. 1). Because it was impossible
to connect the plenum chambers directly to the blade, blade entrance
and exit extensions of the same shape as the blades were used to conduct
the air from the entrance plenum chamber to the test blade and from the
test blade to the exit plenum chamber. ITie blade entrance extension
had a span of 6 inches and the blade exit extension, a span of 3 inches.
Both extensions had an internal free-flow area of 0.043 square inch and
a hydraulic diameter of 0.396 inch. In order to reduce the amount of
heat conducted from the ends Li the hot test bide, metal was removed
in the form of 1/8 inch wide chordwise slots cut thr-.ugl the wills of
these extensions near the ends a.ajacent to the test blade. These slots
were then sealed with a low conductivity material to prevent coolant
leakage.


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NACA RM E5-'27


Blade Description

The air-cooled turbine blade configurations used in this investi-
gation were a 10-tube blade and a 13-fin blade. The reason for selection,
these two blade configurations is that the 13-fin blade has both .1 .,r7
leading and trailing sections and the 10-tube blade has a l c-in truailini,
section but a thin-shell leading section. Figure 3 shows the end views
of the two blades. The geometry factors pertinent to the two blade
configuraticns are given in the following table:

Geometry factor 10-tube 13-fin
blade blade

Blade chord, in. 2.00 2.00
Outside perimeter, in. 4.35 4.53
Span, in. 3.92 3.50
Portion of blade span exposed
to gas stream, in. 3.00 3.00
T.tas free-flow area of internal .181 .0962
cooling-air passage, sq in.
Hydraulic diameter of internal .103 .0670
cooling-air passage, in.

10-tube blade. The 10-tube blade used in this investigation was
similar to the 10-tube blades used in the investigations reported in
references 15 and 23. The outside wall of the blade tapered linearly
from the root to the tip for reduction of stresses during engine opera-
tion. The nominal thickness of the wall at the tip was 0.040 inch and
at the base, 0.070 inch. The blade shell was cast of high-temperature
alloy (X-40) in such a manner that the core area was constant over the
length of the blade. In order to increase the internal heat-transfer
surface, 10 tubes were inserted in the hollow blade (fig. 3). They
extended through the blade from tip to base. These tubes were brazed to
each other and to the inside surface of the hollow blade by Nicrobraze.
Of the 10 tubes, four were made of stainless-steel tubing having a
0.125 inch outside diameter and a wall thickness of 0.010 inch; and six
were made of low-carbon steel with a 0.156 inch outside diameter and a
wall thickness of 0.0155 inch. Availability at the time of fabrication
accounted for the difference in tubing materials.

13-fin blade. The 13-fin blade used in this investigation was
designed for heat-transfer investigations in a static cascade; therefore
there was no taper in the blade wall, but the outside blade profile was
essentially the same as for the 10-tube blade. The blade was machined
in two parts divided essentially at the mean camber line, and upon
assembly the parts were welded together at the leading and trailing
edges; therefore the 13 fins were not continuous (see fig. 3). The fins
had an average thickness of 0.036 inch, and the average fin spaci:,n was
0.046 inch. The blade was machined from high-temperature alloy S-816.


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NACA RM E52H07


Instrumentation

In order to calculate blade temperatures for the conditions of
cascade operation, it was necessary that the cooling-air weight flow,
the blade entrance cooling-air temperature, and the gas temperature be
known. Also, for purposes of comparison, blade temperatures were meas-
ured for a range of cooling-air weight flows.

The cooling-air weight flow was measured by means of a flat-plate
orifice downstream of the test section, and the cooling-air temperatures
were measured by means of thermocouples in the entrance and exit plenum
chambers (fig. 1). The exit cooling-air temperature was measured to
check the calculated values of this temperature. The total temperature
of the combustion gas was measured by means of thermocouples upstream of
the test section, and the combustion-gas weight flow was measured by
means of a flat-plate orifice upstream of the test section. The blade
temperature distribution was measured by 11 thermocouples in the blade
wall around the blade perimeter at the midspan (see fig. 3).

In order to obtain the cooling-air total temperatures at the blade
entrance and exit, thermocouples were placed in the walls of the blade
extensions to correct the values of cooling-air total temperatures meas-
ured in the plenum chambers for any heat picked up in the blade entrance
and exit extensions. The method of correcting the cooling-air tempera-
tures for the heat picked up in the blade entrance and exit extensions
is reported in reference 24.


Cascade Operation

Blade temperatures were measured for the 10-tube and 13-fin blades
for the following conditions:

Blade Gas Gas Gas Gas Mean Euler Cooling- Cooling- Average
tem- Prandtl Mach weight temper- number, air air blade
pera- number number flow ature laminar weight Reynoildcs inlet
ture Ib \ ratio region flow per number cooling-
(OF) sec) (gas to blade air ter-
average lsb perature
wall) (OF)

300 0.672 0.40 3.5 1.32 1.0 0.010 5000
10-tube and and to to and to to to -50
1000 0.654 0.60 4.5 1.61 0 0.070 40,000
300 0.672 0.40 5.0 1.28 1.0 0.005 5000
13-fin and and to to and to to to 8
1000 0.654 0.60 6.0 1.38 0 0.022 17,000


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After the cooling-air weight flow was set, frequent checks were made on
the blade temperatures in order to ascertain when steady-state conditions
were reached. When the blade temperatures and the combustion-gas weight
flow remained steady for approximately 15 minutes, the necessary readings
were taken.


METHODS OF CALCULATION

The present investigation was limited for simplicity to one-
dimensional blade temperature-distribution equations. In the central
portion of the blade section, where the blade temperatures at different
stations along the blade periphery do not differ very much, average
values of chordwise blade temperatures were calculated for the entire
region. At the two extreme sections, near the leading and trailing
edges, large temperature gradients occurred between the surface swept by
the coolant and the surface swept by the hot gas at a given spanwise
distance from the blade root. Here, chordwise temperature distributions
were calculated under the assumption that the heat conducted from the
section in the spanwise direction was negligible when compared with that
conducted chordwise from the outer blade surface to the coolant.

Blade temperatures were calculated from a design basis; that is, it
was assumed that only the blade geometry, effective gas temperature,
cooling-air weight flow, combustion-gas weight flow, blade velocity
distribution, and blade entrance effective cooling-air temperature were
known. The effective gas temperature was obtained from the definition
of recovery coefficient, which is a function of the total, static, and
effective gas temperatures. From the measured total gas temperature
and the calculated velocity distribution, the static gas temperature was
determined and used to calculate the effective gas temperature by assuming
a recovery coefficient of 0.89 (reference 25). Inasmuch as the cal-
culated temperature distributions are for the blade midspan position so
that a comparison could be made between calculated and experimental
blade temperatures, an average effective cooling-air temperature is
required in the temperature-distribution equations. It is sufficiently
accurate, because of the linear distribution of the cooling-air tem-
perature, to use an arithmetic average of the cooling-air total tem-
peratures at the blade entrance and blade exit for the average effective
cooling-air temperature. Since the blade entrance cooling-air total
temperature was known, the blade exit cooling-air temperature was cal-
culated from the equations presented in reference 26.

For simplicity in presentation, the temperature-distribution equa-
tions required in this investigation and the methods for determining the
outside and inside heat-transfer coefficients will be considered sepa-
rately in the following sections.


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NACA RM E52H07


Midchord-Blade-Temperature Equation

Average blade temperatures for the midchord section of the blade
(see fig. 3) were obtained from equation (22) reported in reference 2.
This equation, in the notation of this report, is

Tge B 1
= (1)
T T- + A
g, e a

where


A= (2)
hf Zi

(All symbols are defined in the appendix.) Blade temperatures were
determined by inserting into the equations the appropriate perimeters of
the blades in question and heat-transfer coefficients determined by
methods to be discussed in subsequent sections of this report. The
methods for determining Tg e and T" which are required in equa-
tion (2) have been discussed in the preceding section.


Chordwise-Temperature-Distribution Equation Through

Blade Leading and Trailing Sections

The leading and trailing sections of the 10-tube and 15-fin blades
were approximated by trapezoids. The one-dimensional blade chordwise-
temperature-distribution equation for a trapezoidal approximation to the
blade leading- and trailing-edge sections (equation (20), reference 1)
is, in the notation of this report,


T 2 [ (il1) Jo(it) + iJ1(il)iHo(i)]
ge B 2K (3)
T i 2t2
ge a i (it,) H,(i)2] [H(i(1)iJ1(i)] 2K kr


The geometry factors and the effect of variable wall temperature on out-
side heat-transfer coefficients incorporated in the terms K, 1, and
t2 were evaluated from their definitions given in reference 1 for a
wedge approximation of the blade leading and trailing sections.


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NACA RM E52H07


The temperature distributions in the leading and trailing sections
were calculated by insertion of appropriate blade dimensions and by use
of heat-transfer coefficients determined by the following methods.


Outside Heat-Transfer Coefficients

The use of different equations for the calculation of blade temper-
atures in various parts of an air-cooled turbine blade is accompanied by
the use of different outside heat-transfer coefficients in these equa-
tions. Consequently, outside coefficients will be discussed for the
midchord region, the trailing section, and the leading section in that
order.

Midchord region of blade. The following procedure was used to
determine an average outside coefficient for the blade midchord region
(fig. 3 shows blade midchord region). Local outside coefficients at
three points near the beginning, middle, and end of the pressure surface
of the blade midchord region and local coefficients at three similar
points on the suction surface were computed. The equation (equation (19),
reference 5)

1
3 _
Nug/Prg = Flam Reg laminarr flow) (4)

was used to calculate the local coefficients in the laminar region
because, as pointed out in the INTRODUCTION, the simplified procedure of
reference 5 is currently adequate for application to air-cooled
impermeable-wall blades. For the calculation of local coefficients in
the turbulent region, it is assumed that the effect of temperature ratios
from 1.0 to 2.0 on heat-transfer coefficients is negligible, as was the
case in the laminar region (reference 9). Consequently, the equation
(equation (26), reference 5)

1
Nug/Pr3 = 0.0296 Re 08 (turbulent flow) (5)

was used to calculate the local coefficients in the turbulent region.
An arithmetic average of the six local coefficients was used for the
average outside coefficient for the blade midchord region. The fluid
properties of equations (4) and (5) are based on an assumed wall tem-
perature. Therefore, the calculation of an average blade midchord tem-
perature is essentially an iteration process an average blade midchord
temperature is assumed to calculate an average outside coefficient, this
coefficient is used in equation (1) to calculate an average blade midchord


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NACA RM E52H07


temperature, and then the two values are compared. It can be shown from
equation (1) that a 25 percent change in the assumed blade temperature
will result in a change of approximately 3 percent in the calculated
blade temperature.

The location of the transition point on each blade surface is
required in order to determine whether laminar or turbulent flow pre-
vails at each of the six local midchord points in question, and hence
whether equation (4) or equation (5) is applicable at each of these
points. Moreover, if equation (4) is applicable, a knowledge of the
local Euler number is necessary for the calculation of Flam. The value
of the local velocity, the location of the transition point (considered
herein as the minimum pressure point), and the calculation of the
required local Euler number
2
Eu = -(dp/dx)/(pgUg,/x) (6)

were all determined from the velocity distribution about the blade in
question; the velocity distribution was calculated by the method of
reference 12 and checked by measurements made on a Lucite blade of sim-
ilar shape. From this calculated velocity distribution, the pressure
distribution and hence the density were obtained. The Reynolds number,
with properties based on wall temperature, was then determined.

Trailing section of blade. The trailing sections of the blades
considered in this investigation were approximated by trapezoids, to
which equation (3) is applicable. In these blades, which have a long
trailing section and are made of a high-temperature alloy (low thermal
conductivity), the wall temperature varied rapidly beyond the coolant
passage, and according to reference 22 the outside heat-transfer coeffi-
cient could consequently be quite different from the coefficients for a
constant wall temperature. This difference was provided for by use of
a curve for the turbulent boundary layer (fig. 4) which is based on
approximate analytical results of reference 22. The analysis of ref-
erence 22 is subject to the assumptions of a flat plate (no pressure
gradient), constant property values, and no frictional dissipation of
energy within the boundary layer. Since it can be assumed that trailing
sections of turbine blades have a negligible pressure gradient and the
effects of the other two assumptions are quite-small, figure 4 should be
applicable for trailing sections of turbine blades which have an appre-
ciable temperature gradient. Figure 4 presents the ratio of the outside
heat-transfer coefficient for variable wall temperature to that for con-
stant wall temperature as a function of the exponent n, where n is
given by the relation
*
Tg,e (T)
T -T
g,e B n (7)
Tg,e TB


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NACA RM E52H07 CONFIDENTIAL 13


and X is measured along the profile from a reference point where the
midchord region with uniform temperature TB ends and the region of
rapid change in wall temperature TB begins. In the investigations
reported herein, this reference point was taken at the end of the tube
or fin section (see fig. 3).

Equation (67) of reference 22, derived for a flat plate and tur-
bulent flow, expresses the relation between the heat-transfer coeffi-
cients based on variable and constant wall temperatures. In the nota-
tion employed herein, equation (67) is

h An Yn (X/j)n
S= (8)
hoX An (X/j)n

where n An (X/j)n is a power series expansion of the temperature ratio
f(TB,X/Tge)-I and Yn is a relation among certain gamma functions
defined as

(40n +2
r + r
rY ( 9)
Yn = 40n 32
r + --
39 39

In the calculations necessary for the construction of figure 4, only one
term in the series expansion was considered so that equation (8) sim-
plified to
h
S= Yn (10)

Values of n were inserted into equation (9) to obtain a set of ratios
of the variable- to constant-wall-temperature heat-transfer coefficients.

When an attempt is made to apply figure 4 to calculations for a
blade trailing section, the value of n is unknown. However, a value
of ho (constant-wall-temperature outside coefficient) can be deter-
mined by the method employed in calculating the midchord coefficient.
In this instance, two local outside coefficients were calculated on each
surface of the trailing section by application of equation (5) (tur-
bulent flow prevails over the blade trailing section). The average of
these four local coefficients To was used along with an assumed value
of n, say -0.6, and ho (variable-wall-temperature average outside
coefficient) was then obtained by use of figure 4. The temperature
distribution was then calculated by use of equation (3). Since the


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NACA RM E52H07


value of Tg was known, a logarithmic plot of Tg T against X
determined n. If this value equaled the assumed n, the temperature
distribution was the required one; if the two values of n differed, n
was adjusted until the initial and final values agreed, and the desired
value of ho was obtained from figure 4.

Leading section of blade. The effect of variable wall temperature
on outside heat-transfer coefficient was also taken into consideration
for the blade leading section. Here laminar flow prevails, and the
analysis of reference 21 for wedge flows, flat-plate flow, and stagna-
tion flow, assuming constant property values, was used to account for
the effect of variable wall temperature. Figure 5 gives the relation
between the outside heat-transfer coefficients for variable and constant
wall temperatures as a function of the exponent n (see equation (7)
wherein X is replaced by x for this case). The values used in con-
structing figure 5 were obtained from table 1 (Euler number of zero)
and table 2 (Euler number of 1.0) of reference 21 for a Prandti number
of 0.7 by dividing the outside coefficients for various values of n
(variable wall temperature) by the value for n equal to zero (constant
wall temperature). Since the leading sections of turbine blades are in
the laminar region and have Euler numbers ranging from 1.0 (stagnation
point) to zero (transition from laminar to turbulent), figure 5 is
applicable to blade leading sections. For this investigation, the
constant-wall-temperature outside coefficient was determined in a way
similar to that for the midchord coefficient, that is, as an average of
12 local outside coefficients (six on each blade surface located between
the coolant passage nearest the blade leading section and the stagnation
point) determined from equation (4) for constant property values. Also,
local Euler numbers were calculated for the 12 points and averaged.
Therefore, figure 5 and values of ho and Eu were used to determine
ho for the leading section in the manner used for the trailing-section
turbulent-flow outside coefficient. For the leading section, however,
x was measured from the stagnation point, as in reference 21.


Inside Heat-Transfer Coefficient

Different inside heat-transfer coefficients were required for use
in the two temperature-distribution equations (equations (1) and (3)).
In the midchord region of the blades, the heat-transfer surface area was
greatly augmented by the use of tubes and fins. In order to make
allowance for this, a so-called effective inside coefficient was
required. On the other hand, heat leaving either the leading or trail-
ing blade section is picked up by the cooling air adjacent to each sec-
tion, and the presence of tubes or fins in the midchord region had little
or no effect. Consequently, an average inside coefficient for the
coolant which swept the leading and trailing sections was required for


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NACA RM E52H07


the calculation of temperatures in these regions. The methods of deter-
mining the effective and the average blade-to-coolant heat-transfer
coefficients that were required follow.

Effective coefficient. For the midchord region of the blades,
where the tubes or fins substantially increased the heat-transfer sur,
face area, an effective coefficient was determined. An average heat-
transfer coefficient for fully developed turbulent flow in pipes was
obtained from equation (90) of appendix F, reference 26,

Nua = 0.019 Rea0.8 (11)

The fluid property values in equation (ll) were based on wall temperature
in references 16 and 26, which was considered adequate for these investi-
gations. It can be shown (reference 27) that for air flowing in a smooth
round tube, a satisfactory correlation of average inside heat-transfer
:oefficients was obtained for a range of wall temperatures from approxi-
mately 1500 to 16000 F and an inlet-air temperature near 750 F when the
fluid properties were based on wall temperatures. When the range of wall
temperatures was extended to cover a more complete range from approxi-
mately 750 to 26000 F and the inlet-air temperature varied from 750 to
10000 F, a good correlation of the heat-transfer data was obtained only
when the fluid properties were based on film temperature (an average of
the air and wall temperatures); it was assumed in reference 27 that the
thermal conductivity of the air varied as the square root of the tempera-
ture. Since the best correlation of heat-transfer data over an extended
range of wall temperature and inlet-air temperature was obtained when the
fluid properties were based on film temperature, the fluid property values
in equation (11) were based on film temperature (an average of the cooling-
air and wall temperatures) in this investigation.

The required effective inside coefficient, based on inner wall sur-
face area only, was determined from equation (6) of reference 16, which
in the notation employed herein is


h = hi m(2) tanh (LrcPr) + Z T (12)
-r= (mr + Tr) LrLpr r=l
r=l r=l


where ih is obtained from equation (11) and is based on film tem-
perature.

.For application to air-cooled turbine blades with internal surfaces
other than fins, the 10-tube blade in this instance required replacement
of the internal surface by equivalent fins. The following general method
of reference 16 was used:


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(1) By inspection, the blade was divided into s sections of equi-
valent fins which appeared to have equal fin lengths, spacings, and
thicknesses, and the number of fins a. in each section was determined.

(2) For each section, the inside blade shell perimeter Ii.r

li,r = a.(mrn + Tr and the total wetted perimeter li,W,r were
determined.

(3) The quantities required for equation (12) were determined by
the following definitions:

armr= ir aT (13)

and

L i,W,r .rmr
Lr2 (14)


The relation between bh and hi for the 10-tube and 15-fin
blades is shown in figure 6. Examples showing the application of equa-
tion (12) to tube- and fin-type blades are presented in appendix B of
reference 16.

Average coefficient. Average inside coefficients for computing
leading- and trailing-section temperatures were calculated from equa-
tion (11). In each case, the coefficient was determined for the coolant
which swept the one end of the blade section in question, and the dimen-
sions of this particular passage were used in equation (11). The cool-
ant flow for each of these passages was taken as the average coolant
flow per unit of flow area.

Since hi required for both leading- and trailing-section temper-
ature calculations and for determining hf is based on film tempera-
ture, which necessitates the assumption of a wall temperature, an
iterative process similar to that used in calculating midchord outside
coefficients was used here.


RESULTS AND DISCUSSION

The comparisons of calculated and measured blade temperatures for
the 10-tube and 13-fin blades for the conditions of this investigation
(listed previously in APPARATUS AND PROCEDURE section) are shown in fig-
ures 7 to 9 for the midchord, trailing, and leading sections of the
blades, respectively. In each case the calculated temperatures are
plotted as the ordinates and the measured temperatures, as the abscissas,
with the 450 line representing perfect agreement.


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Comparison of Calculated and Measured Midchord Blade Temperatures

Calculated midchord temperatures are compared with measured midchord
temperatures in figure 7. In this region, temperatures were observed on
each of the blade surfaces and the arithmetic average of these measure-
ments was used to represent the measured midchord temperature. For the
calculation of the midchord temperatures, the average effective inside
heat-transfer coefficient was obtained by considering three groups of
fins (or effective fins) for each of the blades. It must be remembered
that in some cases more fin groups might be necessary since it is not
advisable to average fins (or tubes) which differ very much in dimen-
sions.

Figure 7 shows that the calculated and measured midchord tempera-
tures agree very closely for both the 10-tube and 13-fin blades; that
is, the points for both blades are very close to the 450 perfect agree-
ment line. The quantity usually selected to compare the magnitudes of
errors is the probable error (the number of errors greater than the
probable error is the same as the number less than the probable error).
For the midchord temperatures of the blades investigated, the probable
error was found to be a maximum of 80 F for a gas temperature of 3000 F
and a maximum of 130 F for the 10000 F gas temperature. The good agree-
ment between calculated and experimental temperatures clearly shows that
departures from Mach numbers close to zero and a temperature ratio of
1.0 (the conditions used in obtaining equations (4) and (5)) are not
enough to require corrections in the present investigation. Also, the
assumption that the effect of temperature ratios from 1.0 to 2.0 on
heat-transfer coefficients in the turbulent region is negligible appears
to be justified. The applicability of equations (4) and (5) to other
blade configurations and conditions will be discussed in a subsequent
section of this report.


Comparison of Calculated and Measured Temperatures

Near Blade Trailing Edge

In figure 8, calculated blade temperatures are plotted against
measured temperatures for the thermocouple located nearest the trailing
edge of each blade (see fig. 3). For the 3000 F gas temperature the
agreement is again very good; the probable error for the 10-tube blade
was found to be 60 F and for the 13-fin blade, 40 F.

For the 10000 F gas temperature, the probable errors were 360 F and
300 F for the 10-tube and 13-fin blades, respectively. Trailing-section
temperatures were also calculated with no allowance for the effect of
variation in wall temperature on the outside heat-transfer coefficient,
and the calculated temperatures were at least 1000 to 1500 F higher than


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NACA RM E52B07


the measured temperatures (see reference 15 for 10-tube blade results).
It thus appears that outside heat-transfer coefficients which include
wall-temperature-variation effects should be used in equation (3) for
calculating accurate temperatures in cooled blades with long trailing
sections.

The relatively large probable errors obtained for the trailing
section of the blades at the 10000 F gas temperature may be due to
separation of flow along this section of each blade. Calculated tem-
peratures are dependent on the use of appropriate outside heat-transfer
coefficients; since no method for the calculation of such coefficients
is as yet known for separated regions, special corrections for separa-
tion of flow cannot be made at the present time the separated region
is simply included in the turbulent region. However, the conditions of
equation (5) negligible pressure gradient, low subsonic Mach numbers,
and a temperature ratio equal to 1.0 were fulfilled closely enough in
the present investigation that the use of equation (5) precludes serious
differences between experimental and calculated trailing-section tem-
peratures. The justification of the assumption that the temperature
ratio has a negligible effect on heat transfer in the turbulent region
is not so apparent from these results as for the midchord temperature
calculations.


Comparison of Calculated and Measured Temperatures

Near Blade Leading Edge

A plot of calculated against measured blade temperatures for the
thermocouple located nearest the blade leading edge is shown in figure 9.
Agreement at this point is also good. The maximum deviation between
calculated and measured temperatures was found in the case of the 13-fin
blade when the gas temperature was 10000 F. The probable errors were
found to be 50 and 80 F for the 10-tube blade and 60 and 180 F for the
13-fin blade for gas temperatures of 3000 and 10000 F, respectively.

Temperatures were also calculated for this blade location by use
of outside coefficients based on constant wall temperature for the
10000 F gas temperature. Calculated temperatures were found to be of
the order of 500 to 700 F below the measured values for the 13-fin
blade, which has a long leading section. A negligible difference
between calculated and measured temperatures for the 10-tube blade
which has a short leading section (thin shell) resulted. Thus, in order
to be able to calculate leading-section temperatures accurately, outside
coefficients, which allow for variations in wall temperature, should be
used in equation (3) for blades with long leading sections, whereas
constant-wall-temperature outside coefficients may be used in equa-
tion (3) for blades with short leading sections (thin shell).


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For the present investigation the fact that equation (4) and fig-
ure 5, used to determine the outside coefficients, are based on constant
property values has little effect on the coefficients, because for the
laminar region a change from constant property values results in a change
of less than 1.0 percent in the coefficient (references 7 to 9).


Influence of Errors in Calculated Temperatures

on Design Considerations

The agreement between calculated and measured temperatures at all
locations on the blades investigated is considered to be well within the
accuracy required for turbine design. Errors in prediction of blade
temperatures in the trailing or leading sections are less serious than
in the midchord region because evidence from endurance tests (refer-
ence 28) and calculations based on reference 29 indicate that the midchord
region of the blade is the main stress carrying member. The effect of
errors in calculated average blade midchord temperatures on blade mate-
rial life will therefore be discussed.

At gas temperatures higher than the 10000 F temperature investigated
herein, the probable errors in the calculated midchord temperatures would
be somewhat higher. An increase in gas temperature from 5000 to 10000 F
resulted in an increase in the maximum probable error from 80 to 130 F
for the midchord-temperature calculations. With stress-rupture char-
acteristics as a basis, for most low-alloy steels suitable for non-
strategic turbine blades 250 F is about the maximum error permissible for
design practice. For example, consider a turbine rotor blade made from
Timken 17-22A(S) steel, centrifugally stressed to 25,000 pounds per
square inch at the critical section, and designed for a stress-ratio
factor of 2.0. (The stress-ratio factor is an indication of the blade
stress carrying capacity and is defined in reference 29 as the ratio of
the integrated allowable stress to the integrated centrifugal stress at
the blade critical section.) An error of 250 F in the average midchord
temperature would result in stress-ratio factors of 2.36 and 1.56,
respectively. On the basis of limited results, it is indicated in ref-
erences 28 and 29 that turbine design calculations should be based on
stress-ratio factors above 1.5. It is believed that for the higher gas
temperatures encountered in current turbine engines, approximately
17000 F, the probable error in the calculated average midchord tempera-
ture would not exceed the permissible value of 250 F. Modifications to
this calculation method for other blade configurations and conditions
are given in the following section.


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2IACA RM E52H07


APPLICATION OF ANALYSIS TO OTHER BLADE CONFIGURATIONS AND COrDITIONS

The 10-tube and 13-fin blades were selected for this investigation
because they had long leading and trailing sections (sufficient amount
of low thermal conductivity metal area in section to cause an appreciable
temperature gradient), except the leading section of the '10-tube blade,
and large differences in coolant passage geometry. The results already
presented indicate that calculated trailing-section temperatures are
greatly reduced and leading-section temperatures increased in blades
which have an appreciable temperature gradient in these sections when
outside heat-transfer coefficients are corrected by newly published
formulas for variable wall temperatures. Also, the results show that
the same calculation procedures applied equally well to both blades and
hence that passage configuration effects are negligible. However, the
use of'equation (1) yielded only an average midchord temperature at the
blade milsparn position. In order to calculate an average blade midchord
temperature at a position other than midspan for a stationary blade,
equation (20) of reference 2 must be applied. This equation requires
that the effective cooling-air temperature at the blade root be known,
whereas equation (1) of this investigation utilizes an average cooling-
air temperature. For rotating blades, equation (18) of reference 2 is
applicable; in this case, too, the blade root effective cooling-air
temperature must be known. However, for calculating local temperatures
at the leading and trailing sections, equation (3) is still applicable.

The procedure just described is applicable to other configurations
with long leading and trailing sections (long metal heat-conduction
paths). On the other hand, blades such as those reported in refer-
ences 29 and 30 have thin blade shells so that the cooling-air passages
are extended well into the leading and trailing sections and the wall-
temperature variations in these regions are greatly reduced. For such
blades the effect of wall-temperature variation can often be neglected
and the calculation of blade temperatures is simplified by using the
outside heat-transfer coefficients calculated by means of equations (4)
and (5) without making a correction for wall-temperature variation. This
was verified in the present investigation by the leading-section cal-
culations for the 10-tube blade which has a thin-shell leading section.
That is, the correction in the outside heat-transfer coefficient for
the 10-tube blade leading section was only about 2 percent, whereas that
for the 13-fin blade leading section was 20 percent. A comparison of
calculated and measured average blade temperatures for two thin-shell
blades investigated in a turbojet engine using the simplified method
(no variable-wall-temperature correction) resulted in errors in calcu-
lated temperatures of about 300 F (reference 16). Some of this error
may have been due to poor contact between the internal heat-transfer
surface and the shell of the blades, although it may be possible that
the theories verified in this investigation are not entirely adequate
for rotating air-cooled blades.


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NACA RM E52H07 CONFDErNTIAL 1i


For forced-convection liquid-cooled turbine blades with long lead-
ing and trailing sections, exactly the procedure described herein can
be used to account for the variable-wall-temperature effect when cal-
culating outside heat-transfer coefficients. The inside heat-transfer
coefficient should be calculated by means of an appropriate pipe cor-
relation, as shown in reference 4.

The effect of temperature ratio may in some cases be an Irriprct.ant
condition influencing outside heat-transfer coefficients. Nevertheless,
the effect seems to be negligible for temperature ratios from 1.0 to
2.0. As pointed out earlier, a temperature ratio (gas-to-wall) of 2.0
is the probable limit for current air-cooled blades with impermeable
walls. Also, even though turbine inlet temperatures may exceed 17000 F
in the near future, it is believed that the temperature ratio will still-
be close to 2.0 for air-cooled blades with impermeable walls. Conse-
quently, the methods employed in this investigation may be utilized for
such high-temperature applications. For future cooled-turbine designs,
probably those utilizing transpiration or liquid cooling, where tempera-
ture ratios higher than 2.0 may be encountered, the methods used herein
must be modified. A detailed discussion of these modifications is beyond
the scope of this report.


SUMMARY OF RESULTS

Temperatures calculated from previously published NACA equations
and measured blade temperatures were compared for a 10-tube and a 13-fin
air-cooled turbine blade tested in a static cascade. The results of
this investigation for gas temperatures of 3000 and 10000 F, subsonic
gas'Mach numbers, mean temperature ratios (gas-to-average wall) of 1.3
to 1.6, and a range of cooling-air weight flow are summarized as follows:

1. For those blades having long leading and trailing sections
(sufficient amount of low thermal conductivity metal area in section to
cause an appreciable temperature gradient), calculated trailing-section
temperatures were greatly reduced and leading-section temperatures
increased when outside heat-transfer coefficients were corrected by newly
published formulas for variable wall temperatures.

2. For the midchord region of the blades investigated, the maximum
probable error between calculated and measured blade temperatures was
80 F at a gas temperature of 3000 and 130 F at a 10000 F gas temperature.

3. For the trailing section of the blade, the probable errors were
60 and 40 F at the 3000 F gas temperature and 360 and 300 F at the
10000 F gas temperature for the 10-tube and 13-fin blades, respectively.


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NACA RM E52H07


4. For the leading section of the blade, the probable errors were
50 and 60 F at the 300 F gas temperature and 80 and 180 F at the
10000 F gas temperature for the 10-tube and 13-fin blades, respectively.

5. The short (thin shell) leading section of the 10-tube blade
required only a 2 percent correction of the outside heat-transfer
coefficient for the variable-wall-temperature effect, whereas the long
leading section of the 13-fin blade required a 20 percent correction.


Lewis Flight Propulsion Laboratory
National Advisory Committee for Aeronautics
Cleveland, Ohio


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NACA RM E52H07


APPENDIX SYMBOLS

The following symbols are used in this report:

A flow area of coolant passage, sq ft

An coefficients of power series

cp specific heat at constant pressure, Btu/(lb)(oF)

Dh hydraulic diameter, 4 times flow area, ft
wetted perimeter
Eu Euler nl.riber, -(dpg/dx)/(pgcoUTg c/x)

Flam variable, evaluated in figure 8, reference 5

h local heat-transfer coefficient (constant wall temperature),
Btu/(sec)(sq ft)(F)

h average heat-transfer coefficient, Btu/(sec)(sq ft)(oF)

hf average effective inside heat-transfer coefficient,
Btu/(sec)(sq ft)(F)

h local outside heat-transfer coefficient (variable wall
temperature), Btu/(sec)(sq ft)(F)

ho* average outside heat-transfer coefficient (variable wall
temperature), Btu/(sec)(sq ft)(OF)

iHo,H1 Bessel functions

j chordwise distance from blade trailing or leading edge to
coolant passage, ft

Jo,iJ1 Bessel functions

1

K h _
kB sin

k thermal conductivity, Btu/(sec)(ft)(0F)

Lr fin length of fins in group r, ft (see equation (14))


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2 perimeter, ft

mr spacing of fins in group r, ft

U.u Nusselt number of cooling air, (hi Dh)/ka,F

Nug Nusselt number of gas, hox/kg,w

n exponent of X in relation (Tg,e Ti)/(Tg,e TB) Xn

Prg Prandtl number of gas, (cp,g,wlg,w)/kg,w

p static pressure, lb/sq ft absolute

Reg Reynolds number for cooling air, (wa Dh)/(A 'a,F)

Reg Reynolds number for gas, (Ug, a g,w x)/',w

r index of summation

s number of fin groups

T temperature, OF

T average temperature, OF

T average cooling-air total temperature, OF
a

T blade temperature for case with variable wall temperature, OF

U velocity, ft/sec

w air weight flow, lb/sec

X distance along blade surface as shown in figure 3, ft

x distance along blade surface from stagnation point, ft

Yn function defined by equation (8)

y distance from blade trailing or leading edge to blade
element, ft (see reference 1)

a number of fins in group r
r


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p Hl(i1l) Jo(iS2) + iJl(i~1) iHo(i2) except in.equation (9),
where r denotes the usual gamma function

1 1
TI(l tan #) 2 TI(l tan 4)
'1t2 2K y' + 2 tan 2 2K 2 tan 9 '
1

2K + -----
2 tan f



hfl i

p viscosity, lb/(ft)(sec)

p density, lb/(cu ft)

T ,T2 trapezoidal thicknesses at leading or trailing edge and
coolant passage, respectively, ft

Tr thickness of fins in group r, ft
1
2 Ei 22
(r B2 1 where ih is based on film temperature
(7B r1

tan-1 T 2j 1

Subscripts:

a air

B blade

e effective

F evaluated at film temperature

g gas

i inside


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NACA RM E52H07


o outside

r any internal section of blade formed by fins or equivalent
fins

W wetted

w evaluated at wall temperature

X local point

o0 free stream

Superscripts:

denotes linear dimension increased by T1/2

refers to variable wall temperature


REFERENCES

1. Livingood, John N. B., and Brown, W. Byron: Analysis of Temperature
Distribution in Liquid-Cooled Turbine Blades. NACA Rep. 1066,
1952. (Supersedes NACA TN 2321.)

2. Livingood, John N. B., and Brown, W. Byron: Analysis of Spanwise
Temperature Distribution in Three Types of Air-Cooled Turbine
Blade. NACA Rep. 994, 1950. (Supersedes NACA RM's E7Blle and
E7G30.)

3. Cohen, H.: Heat Transfer in Air-Cooled Gas-Turbine Blades.
Engineering, vol. 173, no. 4484, Jan. 4, 1952, pp. 21-23.

4. Freche, John C., and Schum, Eugene F.: Determination of Blade-to-
Coolant Heat-Transfer Coefficients on a Forced-Convection,
Water-Cooled, Single-Stage Turbine. NACA RM E51E18, 1951.

5. Brown, W. Byron, and Donoughe, Patrick L.: Extension of Boundary-
Layer Heat-Transfer Theory to Cooled Turbine Blades. NACA RM
E50F02, 1950.

6. Eckert, E. R. G., and Livingood, John N. B.: Calculations of Laminar
Heat Transfer Around Cylinders of Arbitrary Cross Section and
Transpiration-Cooled Walls with Application to Turbine Blade
Cooling. NACA RM E51F22, 1951.


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NACA RM E52H07


7. Brown, W. Byron: Exact Solution of the Laminar Boundary Layer
Equations for a Porous Plate with Variable Fluid Properties
and a Pressure Gradient in the Main Stream. Paper presented
at First U.S. National Congress of Applied Mechanics (Chicago),
June 11-16, 1951.

8. Brown, W. Byron, and Donoughe, Patrick L.: Tables of Exact Laminar-
Boundary-Layer Solutions When the Wall is Porous and Fluid Proper-
ties are Variable. NACA TN 2479, 1951.

9. Eckert, E. R. G., and Livingood, John N. B.: Method for Calculation
of Heat Transfer in Laminar Region of Air Flow Around Cylinders
Sof Arbitrary Cross Section (Including Large Temperature Differences
and Transpiration Cooling). NACA TN 2733, 1952.

10. Boelter, L. M. K., Grossman, L. M., Martinelli, R. C., and
Morrin, E. H.: An Investigation of Aircraft Heaters.
XXIX Comparison of Several Methods of Calculating Heat Losses
from Airfoils. NACA TN 1453, 1948.

11. Johnson, H. A., and Rubesin, M. W.: Aerodynamic Heating and
Convective Heat Transfer Summary of Literature Survey.
Trans. A.S.M.E., vol. 71, no. 5, July 1949, pp. 447-456.

12. Hubbartt, James E., and Schum, Eugene F.: Average Outside-Surface
Heat-Transfer Coefficients and Velocity Distributions for Heated
and Cooled Impulse Turbine Blades in Static Cascades.
NACA RM E50L20, 1951.

13. Donoughe, Patrick L.: Outside Heat Transfer of Bodies in Flow-A
Comparison of Theory and Experiment. Thesis submitted to Case
Inst. Tech., June 1951.

14. Freche, John C., and Schum, Eugene F.: Determination of Gas-to-
Blade Convection Heat-Transfer Coefficients on a Forced-Convection,
Water-Cooled Single-Stage Aluminum Turbine. NACA RM E50J23, 1951.

15. Ellerbrock, Herman H., Jr.: Some NACA Investigations of Heat-
Transfer of Cooled Gas-Turbine Blades. Paper presented at the
General Discussion on Heat Transfer. Inst. Mech. Eng. (London)
and A.S.M.E. (New York) Conference (London), Sept. 11-13, 1951.

16. Ziemer, Robert R., and Slone, Henry 0.: Analytical Procedures for
Rapid Selection of Coolant Passage Configurations for Air-Cooled
Turbine Rotor Blades and for Evaluation of Heat-Transfer, Strength,
and Pressure-Loss Characteristics. NACA RM E52G18, 1952.


CONFIDENTIAL


CONFIDENTIAL









NACA RM E52H07


17. Schum, Eugene F., Freche, John C., and Stelpflug, William J.:
Comparison of Calculated and Experimental Temperatures of
Water-Cooled Turbine Blades. NACA RM E52D21, 1952.

18. Petrick, E. N.: A Survey of German Hollow Turbine Blade Develop-
ment. Pt. I Initial Investigations and Developments. Purdue
Univ., Purdue Res. Foundation, pub. by USAF-AMC, Wright-Patterson
Air Force Base, Dayton (Ohio) Oct. 1949.

19. Schuh, H.: Laminar Heat Transfer in Boundary Layers at High
Velocities. Rep. and Trans. 810, British M.A.P., April 15,
1947.

20. Chapman, Dean R., and Rubesin, Morris W.: Temperature and Velocity
Profiles in the Compressible Laminar Boundary Layer with Arbitrary
Distribution of Surface Temperature. Jour. Aero. Sci., vol. 16,
no. 9, Sept. 1949, pp. 547-565.

21. Levy, Solomon: Heat Transfer to Constant-Property Laminar Boundary-
Layer Flows with Power-Function Free-Stream Velocity and
Wall-Temperature Variation. Jour. Aero. Sci., vol. 19, no. 5,
May 1952, pp. 341-348.

22. Rubesin, Morris W.: The Effect of an Arbitrary Surface-Temperature
Variation Along a Flat Plate on the Convective Heat Transfer in
an Incompressible Turbulent Boundary Layer. NACA TN 2345, 1951.

23. Ellerbrock, Herman H. Jr., and Stepka, Francis S.: Experimental
Investigation of Air-Cooled Turbine Blades in Turbojet Engine.
I Rotor Blades with 10 Tubes in Cooling-Air Passages.
NACA RM E50104, 1950.

24. Brown, W. Byron, and Slone, Henry 0.: Pressure Drop in Coolant
Passages of Two Air-Cooled Turbine-Blade Configurations.
NACA RM E52D01, 1952.

25. Esgar, Jack B., and Lea, Alfred L.: Determination and Use of the
Recovery Factor for Calculating the Effective Gas Temperature
for Turbine Blades. NACA RM E51G10, 1951.

26. Brown, W. Byron, and Rossbach, Richard J.: Numerical Solution
of Equations for One-Dimensional Gas Flow in Rotating Coolant
Passages. NACA RM E50E04, 1950.


CONFIDENTIAL


CONFIDENTIAL









NACA RM E52807


27. Humble, Leroy V., Lowdermilk, Warren H., and Desmon, Leland G.:
Measurements of Heat-Transfer and Friction Coefficients for
Subsonic Flow of Air in Smooth Tubes at High Surface and Fluid
Temperatures. NACA Rep. 1020, 1951. (Supersedes NACA RM's
E7L31, E8L03, E50E23 and E50H23.)

28. Stepka, Francis S., and Hickel, Robert 0.: Experimental Investi-
gation of Air-Cooled Turbine Blades in Turbojet Engine.
IX Evaluation of the Durability of Noncritical Rotor Blades
in Engine Operation. NACA RM E51J10, 1951.

29. Esgar, Jack B., and Clure, John L.: Experimental Investigation
of Air-Cooled Turbine Blades in Turbojet Engine. X Endurance
Evaluation of Several Tube-Filled Rotor Blades. NACA RM E52B13,
1952.

30. Bartoo, Edward R., and Clure, John L.: Experimental Investigation
of Air-Cooled Turbine Blades in Turbojet Engine. XII Cooling
Effectiveness of a Blade with an Insert and with Fins Made of
a Continuous Corrugated Sheet. NACA RM E52F24, 1952.


CONFIDENTIAL


CONFIDENTIAL













CONFIDENTIAL


NACA RM E52H07






















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CONFIDENTIAL









NACA RM E52H07


Figure 2. Cascade geometry.


CONFIDENTIAL


CONFIDENTIAL









NACA RM E12HO7


o Thermocouple locations
//// Midchord section


(a) 10-tube blade.


Leading
edge


(b) 13-fin blade.


Figure 3. Sectional view of test blades showing
blade thermocouple locations.


CONFIDENTIAL


Trailing
edge


CONFIDENTIAL









NACA RM E52H07



1.2




1.1




1.0




.9




.8




- .7
0


CONFIDENTIAL


Figure 4. Correction factor for gas-to-blade heat-transfer coefficients
for variable wall temperature in turbulent boundary layer.

CONFIDENTIAL









NACA RM E52H07


Figure 5. Correction factor for gas-to-blade heat-transfer coefficients
for variable wall temperature in laminar boundary layer.


CONFIDENTIAL


OL
-1.0


CONFIDENTIAL









NACA RM E52H07


u0




Blade
07 --
10-tube
13-fin /



.06

S/


/
.05-
/




.04-



//

.03


I-I


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.01





0 .01 .02 .03 .O0

hi, Btu/(sec)(sq ft)(oF)

Figure 6. Chart for determination of effective
blade-to-coolant heat-transfer coefficients.
CONFIDENTIAL


CONFIDENTIAL








NACA RM E52H07


(a) 10-tube blade.


600 o




400,




i~f~i\ _____ _____ A .-__ ____ _____ ___ ____ _____


j/ -
200 400 600 81
Experimental temperature, OF


(b) 13-fin blade.

Figure 7. Comparison of calculated average and
experimental average midchord blade temperatures
for two air-cooled blades in a static cascade.


CONFIDENTIAL


CLuu


CONFIDENTIAL









NACA RM E52H07


Combustion gas 0o
800- temperature -- o
(OF) c
1000 --
A 300

600




400





200_






(a) 10-tube blade.


Experimental temperature, OF

(b) 13-fin blade.

Figure 8. Comparison of calculated and experimental local
temperatures near blade trailing edge for two air-cooled
blades in a static cascade.

CONFIDENTIAL


CONFIDENTIAL










CONFIDENTIAL


NACA RM E52H07


400 60U
Experimental temperature,


1UUU


(b) 13-fin blade.


Figure 9. Comparison of calculated and experimental local
temperatures near blade leading edge for two air-cooled
blades in a static cascade.


CONFIDENTIAL
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