Survey of advantages and problems associated with transpiration cooling and film cooling of gas-turbine blades


Material Information

Survey of advantages and problems associated with transpiration cooling and film cooling of gas-turbine blades
Series Title:
Physical Description:
39 p. : ill. ; 28 cm.
Eckert, E. R. G ( Ernst Rudolf Georg ), 1904-
Esgar, Jack B
Lewis Research Center
United States -- National Advisory Committee for Aeronautics
Place of Publication:
Washington, D.C
Publication Date:


Subjects / Keywords:
Turbines -- Blades -- Research   ( lcsh )
Airplanes -- Motors -- Cooling   ( lcsh )
Aerodynamics -- Research   ( lcsh )
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )


Summary: Transpiration and film cooling promise to be effective methods of cooling gas-turbine blades; consequently, analytical and experimental investigations are being conducted to obtain a better understanding of these processes. This report serves as an introduction to these cooling methods, explains the physical processes, and surveys the information available for predicting blade temperatures and heat-transfer rates. In addition, the difficulties encountered in obtaining a uniform blade temperature are discussed, and the possibilities of correcting these difficulties are indicated. Air is the only coolant considered in the application of these cooling methods.
Includes bibliographic references (p. 24-26).
Statement of Responsibility:
by E.R.G. Eckert and Jack B. Esgar.
General Note:
"Report date February 12, 1951."
General Note:
"Restricted"--stamped on front cover

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003810653
oclc - 135022835
sobekcm - AA00006207_00001
System ID:

Full Text
Nk,91A-6 15'"'







By E. R. G. Eckert and Jack B. Esgar


Transpiration and film cooling promise to be effective methods of
cooling gas-turbine blades; consequently, analytical and experimental
investigations are being conducted to obtain a better understanding of
these processes. This report serves as an introduction to these cool-
ing methods, explains the physical processes, and surveys the informa-
tion available for predicting blade temperatures and heat-transfer
rates. In addition, the difficulties encountered in obtaining a uni-
form blade temperature are discussed, and the possibilities of cor-
recting these difficulties are indicated. Air is the only coolant
considered in the application of these cooling methods.


Gas-turbine blades are usually cooled by forcing air through the
hollow interior of the blade. The possibilities of this cooling method
are limited by the heat-transfer rates from the blade to the coolant.
Because of this limitation, other blade cooling methods are being
investigated. Transpiration cooling, also called sweat or porous-wall
cooling, promises to be a very effective means of cooling objects such
as rocket nozzles (reference 1) that must be in contact with high-
temperature, high-velocity gas streams. This method of cooling in
which the coolant is forced through a porous wall to form an insulating
layer of fluid between the wall and the gas stream is now being con-
sidered for use in gas-turbine stator and rotor blades. Film cooling
is another means of cooling surfaces that is similar to transpiration
cooling, although somewhat less effective. With film cooling the
insulating fluid layer or film is formed by allowing a coolant to flow
through slots or holes and then over the surface. References 2 and 3
have shown that film cooling with air as the coolant can be used effec-
tively to decrease the temperatures of gas-turbine blades.



Since transpiration and film cooling show considerable promise as
methods of turbine-blade cooling, experimental and analytical investi-
gations are being conducted at the NACA Lewis laboratory to gain more
knowledge and to expand the application of these cooling methods. This
report serves as an introduction to the basic principles of transpira-
tion and film cooling and reviews briefly the status of methods for
calculating heat transfer and temperatures in blades cooled by these
means. It deals only with the thermodynamic aspect of the cooling
methods. Neither stresses, a very important item for both methods,
nor fabrication techniques are considered, but problems that have to
be solved in the design and fabrication of transpiration and film-
cooled blades are discussed.


The following symbols are used in this report:

A area, sq ft

Cl to C4 constants

Cp specific heat at constant pressure, Btu/(lb)(oF)
d average width of pore in porous metal, ft

F factor in equation (1l)

g acceleration due to gravity, ft/sec2

H gas-to-surface heat-transfer coefficient, for porous or film-
cooled surface, Btu/(OF)(sq ft)(sec)

H' gas-to-surface heat-transfer coefficient for solid surface,
Btu/(F)(sq ft)(sec)

k thermal conductivity, Btu/(OF)(ft)(sec)

L length, ft

m exponent
Nu Nusselt number, --

p pressure, Ib/sq ft


Pr Prandtl number, k

Q heat flow rate, Btu/sec

r ratio of velocities in boundary layer (See equation (17))
Re Reynolds number, --

s slot width, ft

Nu H
St Stanton number, N or
RePr pcV

T temperature, OF

v flow volume per unit area and time, ft/sec

V velocity, ft/sec

w flow rate per unit area, lb/(sec)(sq ft)

W flow rate, lb/see

x distance, ft

y distance from center line, ft (See fig. 3)

Y distance from center line where = 0.5 (See fig. 33

8 boundary-layer thickness, ft

e temperature difference, oF

p viscosity, (lb)(sec)/sq ft

g cinematic viscosity
Skinematic viscosity, -, sq ft/sec

density, Ib/cu ft

4 NACA RM E50K15

T thickness, ft



ad adiabatic

c coolant

e effective

g gas

L laminar sublayer

m mean

max maximum

w wall

1,2 designate locations


Internal Cooling
The conventional method of cooling turbine blades is to make the
blades hollow and force cooling air through the blade interior. Usually,
the cooling air enters the blade at the root and discharges into the gas
stream at the tip. To make this cooling method more effective, inserts
can be provided within the blade to direct the cooling air along the
inner wall, or the inner surface area can be increased by the addition
of fins (fig. l(a)). An inherent disadvantage of this method is that
the gas-to-blade heat-transfer coefficient is very high because of the
high gas velocities. It is difficult to increase the blade-to-coolant
heat-transfer coefficient to a value higher than the gas-to-blade coeffi-
cient. When both heat-transfer coefficients are equal on a hollow or an
insert blade, the blade wall temperature lies half way between the gas
and the coolant temperatures. It would be possible to decrease effec-
tively the blade wall temperature by placing some thermally insulating
material on the outside surface of the blade. Such an insulation has
been tried in the form of ceramic coatings, but coatings of sufficient


thickness for insulating purposes lack the required structural strength
for gas-turbine operation. Some method of coating the blade with a
layer of air would insulate the blade better than any ceramic coating
because of the much lower thermal conductivity of air.

Film Cooling

The method of film cooling uses some gas or liquid to build an
insulating layer between the blade wall and the hot gases as shown in
figure l(b). The coolant is discharged by slots to the outside of the
surface and is carried downstream by the outside gas flow. In this way
an insulating film is built along the surface. This film is gradually
destroyed on its way downstream by turbulent mixing, which allows the
wall temperature to rise as shown in figure 2. The coolant film has to
be renewed at a certain distance by blowing new coolant through addi-
tional slots, or the wall temperature would eventually approach gas
temperature. The insulating effect of a gas layer is very good because
gases have a lower thermal conductivity than any known insulating mate-
rial, but a liquid layer is even better because the liquid coolant is
evaporated by the heat flowing from the hot gases into the coolant film.
The heat required for evaporation keeps the temperatures of the liquid
film and the blade surface down to the evaporation temperature until a
point downstream from the slot is reached where the liquid is completely
evaporated (reference 4). Inasmuch as the heat absorption by the
evaporation is the determining factor in this cooling method, it may be
called evaporative film cooling. For aircraft gas turbines, air is the
most readily available coolant; other gases or liquids would have to be
carried as extra weight in the airplane. The combustion gases flowing
around the turbine blades have physical properties that differ only
slightly from the properties of air; therefore, film cooling will be
considered for the case where the coolant is the same gas as the com-
bustion gases in this report.

A simple physical process that is similar to film cooling and
therefore helps to explain it is shown in figure 3. A line source of
heat S is placed in a uniform gas stream. This heat source may be
realized, for instance, by a heated wire. The heat that is transferred
from the wire to the gas is carried downstream by convection and, at the
same time, diffused by turbulent mixing. The temperature profiles in
the gas flow downstream of the heat source have the shapes indicated in
the figure for two distances from the heat source. An investigation by
Corrsin and Uberoi (reference 5) shows that the temperature profiles at
each location are similar in shape. They can be superimposed into a
single curve by an appropriate change in the scale of the coordinates,
as shown in the lower part of figure 3. The aBscissa of this curve is


the ratio y/Y, where y is the distance from the center line I-I and
Y is the distance where the temperature difference as measured against
the upstream temperature.reaches the value of 0.5 of the temperature
difference on the center line I-I. The ordinate is the ratio e/Omax
of the temperature difference 0 between the upstream temperature and
the temperature at y, and emax, the same temperature difference meas-
ured on the center line I-I (for y = 0). The investigations in refer-
ence 5 show that Y increases linearly with the distance x from-the
heat source. The area under all the temperature profiles at different
distances has to be the same for continuity reasons when the loss in
momentum of the fluid by the heat source can be neglected. In this case
the temperature difference Omax on the center line decreased inversely
proportional to the distance x. Conditions in the flow on a film-cooled
surface would be identical to the conditions considered in figure 3
when there are no frictional forces within the flowing gas and when the
neighborhood of the slot is excluded. In this case the center line I-I
represents the wall surface and the line source S represents the slot.
In reality the flow is retarded by viscous 'forces in the neighborhood
of the wall, which causes the convective heat flow and the intensity of
turbulence to be somewhat changed. The temperature distribution meas-
ured by Wieghardt (reference 6) on a film-cooled surface is also shown
in figure 3 in order to give an indication of the order of magnitude of
these changes. The same investigation shows that the difference between
the wall temperature and the gas temperature decreases inversely propor-
tional to x0.8 as opposed to the decrease of max inversely propor-
tional to x for the line heat source. The investigations of Wieghardt
will be subsequently discussed in more detail.

Transpiration Cooling

It was previously stated that, in the method of film cooling, the
air film is gradually destroyed by turbulent mixing with the hot gases.
In this way, the effectiveness of the film decreases in the downstream
direction from the point where it leaves the slot. This disadvantage is
avoided in the transpiration-cooling method where the cooling film is
continuously renewed along the blade surface. The blade is fabricated
out of some porous material and the coolant is forced through the porous
blade wall as shown in figure l(c). A liquid coolant is again more effec-
tive than a gas coolant because considerable heat is absorbed by the
evaporation process. Cooling with a liquid may be distinguished from
cooling with a gas by calling it evaporative-transpiration or sweat cool-
ing. The most important coolant for aircraft power plants is air for
the reasons mentioned previously and, therefore, only transpiration


cooling with air will be considered in this report. Because the cooling
film is continuously renewed in the transpiration-cooling method, the
blade temperatures will be more uniform along the blade circumference
than they will with film cooling. An additional advantage of transpira-
tion cooling that tends to decrease the heat transfer from the hot
gase- to the blade surface is the fact that there is a continuous slow
movement of the cooling air away from the blade surface because new
cooling air is continuously forced through the pores and then leaves
the surface. A counterflow is thus created between the heat flowing
from the hot gases towards the blade surface and the cooling air flowing
away from the blade surface. The cooling air continuously carries heat
away from the surface by convection and in this way decreases the over-
all heat transfer from the hot gas to the surface.

This principle is explained in more detail in the simple configura-
tion in figure 4 for a duct with the cross-sectional area constant along
its length and with air at a constant, uniform velocity flowing through
it. Two screens separated by the distance L are placed in the duct.
These screens are kept at two different temperatures T1 and T2.
First, the air velocity is considered to be zero. In this case when
free-convection currents are neglected, heat is conducted from the
screen with the higher temperature T2 through the stagnant air between
the screens to the screen with the temperature T1. This heat-conduction
process is described by the familiar equation

Q = -Ak (1)

in which Q is the heat flow per unit time, k is the thermal con-
ductivity of the air that will be assumed independent of temperature,
and T is the air temperature at the distance x from the screen
having the temperature Tl. In a steady state, the heat flow Q has
to be constant along x when heat losses through the duct walls are
neglected. An integration of equation (1) gives the well-known relation

T Tj = ClX (2)

stating that the temperature increases linearly from one screen to the
other. The heat flow is then determined by the equation

T2 T1

0 NACA RM E50K15

If the air moves with a finite velocity producing a weight flow w per
unit time and cross-sectional area, the heat flow through any cross
section in the duct is now composed of two parts: (1) the heat that is
conducted in the direction opposite to the air stream and (2) the heat
that is carried along by the air stream. Therefore, the heat flow is
represented by

Q = -Ak + Cpw A (T TI)

where the specific heat of the air Cp will be assumed independent of
temperature. The second term of this equation contains the temperature
difference T T1 because it is usual to disregard the heat that is
carried along by an, unheated air stream (of temperature T1). Again,
the heat flow Q has to be constant along x in a steady-state process.
An integration of equation (4) together with the boundary condition
T = T1 when x = O yields

T T1 = C2 (e k

- 1)

S (5)

With the boundary condition T = T2 for x = L, the equation becomes

-- x
T T = (T2 T) e 1
e 1

The resultant heat flow Q from the screen with
to the screen with temperature T1 is

Q= -Ak(
x =0

the temperature T2

T2 T1
= -cpw A -


For w = 0 the numerator and denominator in this equation are zero.
However, it can be determined that the value for this indeterminate
form is the same as given by equation (3). In figure 4 the air tem-
peratures in the space between the two screens are plotted with the
value Cpw/k that characterizes the air velocity as the parameter.
For zero air velocity a straight line represents the air temperatures.
For finite velocity the temperature profiles are curved. A negative
value of the parameter Cpw/k describes a flow of the air from the
screen T2 towards the screen T1. The gradient of the lines at x = 0
determines the amount of heat that flows from one screen to the other
because, for x = 0, the second term on the right side of equation (4)
is zero. It can be seen that the heat flow decreases with increasing
velocities in the positive direction and increases for velocities in the
negative direction. The decrease in heat flow for positive velocities
is a consequence of the fact that part of the heat which leaves the
screen with the temperature T2 by conduction in a direction opposite
to the air velocity is picked up by the air stream and carried back
towards the screen with temperature T2 and only part reaches the
screen with the temperature T1.

This process by which the heat conduction flow is continuously
decreased on its way by the counterflowing air takes place within a
transpiration-cooled wall as well as within the boundary layers that are
built up on both surfaces of the wall. Figure 5(a) shows how the tem-
perature varies throughout the wall and through the boundary layers.
Figure 5(b) indicates the temperature variation within the wall in an
enlarged scale. The transpiration-cooled wall is represented in cross
section. The cooling air flows through the wall from left to right.
Hot gases with a temperature of 10000 F flow along the right surface
and build up a boundary layer that is usually turbulent. Within this
boundary layer, the temperature drops to the value on the outside wall
surface. Because the surface area in contact with the cooling air is
very great in a porous material, there are practically no temperature
differences between the cooling air on its flow through the pores and
the wall. The temperature difference between the air and the wall
shown in figure 5(b) is exaggerated. The cooling air therefore.leaves
the porous wall with the wall surface temperature Tw, g. It leaves the
wall with a small velocity normal to the surface. In passing away from
the surface, the cooling air picks up momentum from the gas flow until
it finally reaches the outside gas velocity. At the same time its tem-
perature increases either by conduction or by turbulent mixing until at
some distance the gas temperature Tg is reached. The condition within
the coolant-air film in the immediate vicinity of the wall surface cor-
responds principally to the condition in the air flow discussed in


connection with figure 4. The temperature TI corresponds to the wall
surface temperature Tw,g and the temperature T2 to the gas tempera-
ture Tg. The only difference is that the location of the plane in the
cooling-air film that corresponds to the location of the screen with
the temperature T2 is somewhat indeterminate because the turbulent
mixing process occurs over some finite distance. Essentially, the same'
process also takes place within the wall itself, the only difference
being that the heat is conducted not only within the cooling air but
also within the porous wall. The thermal conductivity k in equa-
tions (4) to (7) must be adjusted to take this process into account.
In addition, a heat-transfer process from the porous wall material to
the cooling air takes place as the air flows through the pores. This
process determines the temperature difference between the cooling air
and the solid material within the wall. A mathematical investigation of
this process by Weinbaum and Wheeler (reference 7) shows that this tem-
perature difference is very small except within a very narrow range
near that part of the wall surface where the cooling air enters. This
temperature difference may therefore be neglected in calculating the
wall temperatures. There is a boundary layer on the coolant entrance
side of the wall within which the air temperature increases from the
initial value Tc to the temperature with which the air enters the
pores. The thickness of this boundary layer and the temperature increase
within it, however, are much smaller than on the hot side of-the wall.
The shape of the temperature curve within this boundary layer corresponds
principally to the shape of the temperature profiles in figure 4 near
the screen with the temperature T2. The temperatures Tc, Tw,m, and
Tg shown in figure 5 were measured on a transpiration-cooled turbine
blade. The temperatures within the boundary layers are only approximate
shapes deduced from boundary-layer calculations. The temperatures
within the wall were determined using equation (6) and measured values
for the thermal conductivity of porous steel (fig. 6). The notation
SL in figure 5 will be explained later.

A relation between the outside gas-to-surface heat-transfer coeffi-
cient and the wall temperature on the outside of the turbine blade can
be easily derived in connection with figure 5. If there is no heat flow
along the wall within the boundary layer on the cooling-air side, which
quite often is the case, then the total heat conducted from the inner
wall surface into the cooling-air boundary layer is picked up by the
cooling air and is carried back into the wall by convection. In other
words the resultant heat flow Q in equation (4) at any place on the
wall is zero. In a steady state, the heat flow is zero at any cross
section throughout the wall. On the surface of the wall facing the gas
stream, the temperature of the cooling air is Tw,g. The heat that is


carried with the cooling air through the element dA of this surface
is therefore

wcp dA (Tw,g Tc) (8)

where w denotes the weight flow per unit area and time through dA
and Cp is the specific heat of the cooling air at constant pressure.
On the other hand, the heat that is transported through the gas boundary
layer to the surface may be defined with a heat-transfer coefficient by
the expression

H dA (Tg T,g) (9)

At high velocities in the gas stream, the effective temperature has to
be introduced as gas temperature into this expression. Because the
total heat flow through the area dA is zero, expressions (8) and (9)
have to be equal. Therefore,

H (Tg,- T,g) = Wcp (Tw,g Tc) (10)

This relation determines the gas-to-surface heat-transfer coefficient as
soon as the effective gas temperature, the coolant temperature, and the
wall temperature on the surface facing the gas stream are known from
measurements; or inversely, it allows for the calculation of the blade-
surface temperature from the gas-to-surface heat-transfer coefficient.


Extent of Laminar and Turbulent Boundary Layers

It is well known that gases flowing through a turbine blade grid
build boundary layers along the surface of the blades. The action of the
viscous forces within the flowing gases is practically confined to this
region and to the wake downstream of the blades. Sometimes the flow
separates from the blade surface on the convex or suction side of the
blade near the trailing edge. The boundary layers may be laminar or


When the blade is cooled, a temperature boundary layer is built up
around the blade, which, for gases, has a thickness of the same order of
magnitude as the flow boundary layer. The gas temperature is influenced
by the cooling process only within this boundary layer and within the
wake behind the turbine blade. If separation occurs, then the region
behind this separation has to be included. The remainder of the gas
flow is not influenced by the cooling process. The heat transfer from
the blade surface into this boundary layer depends very much on whether
the flow within this boundary layer is laminar or turbulent. Before a
calculation of the heat transfer can be started, it must therefore be
known in which places along the blade surface the boundary layer is-
laminar or turbulent. Consequently, a short examination as to what kind
of boundary layer is to be expected on a porous blade is required.

At some point on the nose of the turbine blade, the gas flow divides
into the part that flows along the suction side and the part that flows
along the pressure side of the blade. This point is called the stagna-
tion point and in this region the boundary layer is always laminar. At
some distance downstream of this point, the flow within the boundary
layer becomes turbulent. The location of this transition point mainly
depends on the local pressure gradient and the thickness of the boundary
layer at that point. The pressure distribution around a turbine blade
on the pressure and suction side is determined by the outside gas flow.
The pressure decreases at first with increasing distance from the stagna-
tion point. Usually, it reaches a minimum somewhere along the blade on
both sides and-then increases. The transition to turbulent flow within
the boundary layer probably occurs on turbine blades near the point where
the pressure minimum is reached. The exact location of the point of
transition cannot be predicted yet by calculation although great advances
have been made in recent years in understanding the transition process.
The factors influencing transition are now fairly well known. Some
information also exists on their mutual importance. The investigations
in this field that are of special importance for the prediction of the
transition point on turbine blades will be mentioned briefly.

Tollmien and Schlichting (references 8 and 9) show that on a solid
surface the transition is caused by the fact that the boundary layer
becomes unstable against small fluctuations under the influences of the
outside pressure distribution along the surface and of the boundary-layer
thickness. A pressure decrease in flow direction stabilizes, and a
pressure increase destabilizes, the boundary layer. Outside disturbances
that may be present to a considerable degree in the flow in a gas turbine
move the transition point forward. A theory of this process is given by
Taylor in reference 10. It was found that increasing curvature decreases
the stability on a concave surface (reference 11). Temperature differ-
ences within the boundary layer increase the stability when the blade is


cooler than the gas (reference 12). A comparison of average gas-to-blade
heat-transfer coefficients calculated by Brown and Donoughe (refer-
ence 13) with measured data indicates that, in internally-cooled turbine
blades, the transition usually occurs near the point where the pressure
minimum is reached along the surface.

All these data referred to boundary layers on a solid blade. By
the transpiration-cooling process, the stability of the boundary layer
is decreased. Some information on this process is found in reference 14.
No experimental data concerning the point of transition on turbine blades
are available; however, it may be expected that laminar boundary layers
will not be present downstream of the minimum-pressure point. Laminar
boundary layers, therefore, will usually be confined to a region near
the blade nose, and along the rest of the surface the boundary layer
will be turbulent. It will be discussed later why it is well worth
while to study the question of how-the laminar part can be extended
over a larger part of the blade surface.

Heat Transfer in Transpiration-Cooled Laminar Boundary Layer

The heat transfer in the laminar boundary layer can be analytically
investigated. All the available information stems from calculations.
The essential features of this type of heat transfer can be learned by
a discussion of the conditions on a flat plate.

Figure 7 shows the sketch of such a porous plate in a gas stream
of uniform velocity Vg. The plate is fabricated from porous material
and cooling air is blown through it from a box attached to the back of
the plate. The cooling air in this box has a temperature Tc and the
outside gas temperature is Tg. The distribution of the cooling air
along the plate surface depends on the local distribution of the porosity.
Two important conditions may be considered. The first one is to find a
distribution of the cooling air that keeps the outside surface tempera-
ture of the plate Tw g constant over the whole plate length. The
amount of coolant passing through the porous plate may be characterized
by a velocity vc with which the cooling air moves away from the plate
surface. This velocity vc is not the actual velocity of the cooling
air coming out of the pores, but rather it is the volume flow through
a unit surface of the plate. The cooling air has this velocity when all
the jets coming out of the pores combine into a continuous flow and the
velocity differences are equalized. It is assumed in all the calcula-
tions for transpiration cooling that these conditions occur very close
to the plate surface. The distribution of the velocity vc along the
plate that gives a constant wall temperature when equation (10) is valid


is vc6 where x denotes the distance along the plate from the
leading edge (reference 15). In this case the partial differential
equations describing the flow and the heat transfer within the boundary
layer can be transformed into total differential equations and solved
exactly. Such a calculation shows that the boundary-layer thickness
along the wall increases proportional to the square root of x and
the local heat-transfer coefficient H decreases inversely proportional
to the square root of x. The heat-transfer coefficient is obtained
from the dimensionless equation

Nu = FA/ (1)


Hx Vg
Nu = --, Re = g
k g

The factor F is a function of the coolant-flow velocity vc and the
Prandtl number Pr of the gas. The factor F is presented in fig-
ure 8 for a gas with a Prandtl number of one with the assumptions that
the property values are constant through the boundary layer and the
Mach number is small. Negative values of the abscissa indicate suction
of the gas into the wall. The flow boundary layer for this case was
calculated by Schlichting (reference 16). The temperature boundary
layer and therefore the heat-transfer coefficient for Pr = 1 follow
immediately from these calculations considering the similarity between
velocity and temperature profile. For the case investigated, the
porosity of the plate has to be varied in such a way that the velocity
vc decreases inversely proportional to the square root of x.

The second important case is a plate where the velocity vc is
constant. This case can be obtained with a constant wall thickness and
porosity. Only. an approximate solution of the differential equations is
possible for this case (reference 17). The local behavior of the
boundary-layer thickness, the wall temperature, and the heat-transfer
coefficient is different from the case previously discussed. As in the
other case, the boundary-layer thickness starts with the value zero at
the leading edge of the plate but it increases linearly in the down-
stream direction for great values of x. The local heat-transfer
coefficient starts with the value infinity at the leading edge and
approaches asymptotically the value zero with increasing x. The wall
temperature is equal to the gas temperature at the leading edge and
approaches the cooling-air temperature asymptotically with increasing
x. It is interesting to note that the behavior of these values is


different when the air is sucked into the plate instead of blown from
it. In this case the boundary-layer thickness, the heat-transfer
coefficient, and the wall temperature reach a constant value some
distance from the leading edge.

Conditions near the leading edge of a turbine blade where the
boundary layer is expected to be laminar differ from those on a flat
plate by the fact that the pressure and, therefore, the velocity Vg
outside the boundary layer are not constant but vary along the blade
surface. Exact calculations are possible only for certain types of
this velocity variation. Solutions were presented for constant
property values, small Mach numbers, and for the case that Vg varies
proportional to x (references 15 and 16). The solutions can be
extended to include the effect of the variation of property values and
a variation of the gas velocity Vg proportional to xm. The resulting
formula for the heat-transfer coefficient will have the same form as
equation (11). The factor F will then also depend on the ratio of
the gas temperature Tg to the wall temperature T, g and on the
exponent m. This exponent is related to the local pressure gradient
dp/dx by the equation

dp mpgVg2
dx x

Normally, the variation of the velocity Vg proportional to xm does
not correspond too well to the velocity variation encountered on turbine
blades. For such a variation only approximate calculations of the heat-
transfer coefficient are possible. A method using the integrated
momentum and heat-flow equations of the boundary layer, similar to the
procedure proposed by von rirmin (reference 18), sees to be best suited
for this purpose.

Heat Transfer in Transpiration-Cooled Turbulent Boundary Layer

No exact calculations are possible for the turbulent boundary
layer with or without transpiration cooling. Experimental information
obtained on transpiration-cooled turbulent boundary layers is also very
meager. The present knowledge of the heat transfer in the transpiration-
cooled turbulent boundary layer is therefore much less reliable than
that of the laminar boundary layer. A first, approximate theory for
the turbulent case was presented by Rannie (reference 19). The essen-
tial features of this theory may-be explained with the help of figure 5.
The turbulent fluctuations within the boundary layer on the gas side
of the wall decrease in a direction towards the wall. On the wall


itself they must be zero. Very little detailed information exists on
this decrease of turbulence. However, to simplify the real conditions
for heat-transfer investigations the boundary layer can be divided into
two parts, a very thin sublayer near the wall (indicated by BL in
fig. 5) within which the flow is assumed to be laminar, and a turbulent
part where the heat transfer by turbulent mixing is assumed to exceed
by far the heat transfer by conduction. This idealization, which gave
useful results for the boundary layer on a solid wall, is used by Rannie
in his theory. In addition he assumes that the transpiration-cooling
process does not alter the value of 8L or the conditions in the
turbulent part of the boundary layer; only the laminar sublayer is
influenced. The temperature drop through the laminar sublayer on a
solid wall is practically linear in accordance with equation (2). For
the porous wall it is non-linear and can be described by equation (6).
In this way a formula was derived in reference 19 for the heat-transfer
coefficient and by use of equation (10) the wall temperature could also
be calculated. Comparisons of this formula with experimental data
obtained in the Jet Propulsion Laboratory, California Institute of-..
Technology showed fair agreement (reference 19). The formula was
simplified by Friedman (reference 20) in restricting it to fluids with
a Prandtl number near one. The formula for the wall temperature Twg
derived by him is

Twg- T r (12)
(2= )(2
Tg -Tc er+r- 1

where r is the ratio of the velocity parallel to the surface at the
border between the laminar sublayer and the turbulent part of-the
boundary layer (at the distance 8L in fig. 5) to the stream velocity
Vg outside the boundary layer. The value CP is given by

pc v
H' (13)

with p the density; c the specific heat; vc, the velocity of
the cooling air; and H the heat-transfer coefficient that would be
present on a solid surface under the same outside flow conditions.

The formula was used by Friedman to study heat transfer in a tube,
but it can be applied to the flat plate as well. Reference 21 gives
the equation for heat transfer over a flat plate with turbulent flow as

Nu = 0.0296 (Re)08 (Pr)1/3 (4)


If both sides of equation (14) are divided by (Re)(Pr), the
equation takes the form

Nu H 0.0296
(Re)(Pr) = St = PpVg (Re)1/5 (pr)2/3 (15)

where the Reynolds number is based on the length x. Substituting the
value of H' from equation/(15) into equation (13) results in

(Re)1/5 (pr)2/3 Pc (16
0.0296 pgVg

The value Cp therefore depends mainly on the ratio of density times
velocity of the coolant to density times velocity of the outside gas
flow and, to a smaller degree, on the Reynolds number for the outside
flow and on the Prandtl number. The value 0.5 was used by Friedman
for the velocity ratio r. For the flat plate it is better to use the
relation in reference 22

r = (17)

Friedman also compared equation (12) with test data and found good
agreement. For the specific values Re = 105 and Pr = 0.7, the tem-
perature ratio given by equation (12) is plotted in figure 9 as a
function of the mass-flow ratio. The temperature ratio calculated from
equations (10) and (11) for the laminar boundary layer on a flat plate
is also included. The figure reveals that the amount of coolant flow
necessary to obtain the same wall temperature is very much lower for
the laminar boundary layer than for the turbulent one. This fact means
that very big gains can be obtained if the laminar portion of the
boundary layer on the surface of the turbine blade can be increased.

The conditions on a turbine blade in the region of the turbulent
boundary layer differ again from the conditions on a flat plate by the
fact that the velocity varies along the blade surface. Practically no
information on the influence of this velocity variation is available,
neither on the solid surface nor on a transpiration-cooled wall. It
seems, however, that the influence of a pressure gradient is less on a
turbulent boundary layer than on a laminar one. Nothing is known of the
heat transfer in the separated region that often exists on the downstream
part of the suction surface of the blade.


Heat Transfer in Film Cooling

It was pointed out that in film cooling thecooling-air film is
diffused by turbulent mixing with the hot gases and, in this way, is
gradually destroyed on its downstream path after leaving the slot.
When the wall surface does not receive heat by radiation or by con-
duction into or from the interior of the wall, it assumes a temperature
that is called the adiabatic wall temperature.

More information on the value of this adiabatic wall temperature
Tw,ad may be obtained from an experimental investigation by Wieghardt
(reference 6), which is concerned primarily with the de-icing of air-
plane wings. He therefore investigated a hot-air jet blown into a cold
air stream by a slot in a flat plate. As a first approximation,
however, his results should be applicable to the film-cooling process
as well. He derived the expression

/ \0.8
Tg ,a ad c_ s (1 8
S-a 21.8 (18)
T-T P Vg
6 c gg

in which x is the distance in downstream direction from the slot and
s, the slot width. The formula is applicable for x/s values greater
than 100 and for a mass-flow ratio smaller than or equal to one. It
gives a decreasing adiabatic wall temperature at a certain distance x
with increasing mass-flow ratio. More research is required for x/s
values smaller than 100, because this range will find most application
in gas-turbine blades. The experiments (reference 6) showed that, for
mass-flow ratios greater than one, the adiabatic wall temperature
increased again with increasing mass-flow ratio. A value of one for
the mass-flow ratio therefore gives the best cooling effectiveness.

When there is heat flow through the wall by internal conduction
from a hotter location in the wall, or when heat is radiated from the
gas to the wall surface, a finite heat transfer by convection from the
wall to the air film has to balance this heat exchange and the tempera-
ture profile on a normal to the wall must have a finite gradient at
the wall surface. A heat-transfer coefficient H describing this heat
flow dQ has to be defined by the equation

dQ= HdA (Te T,g) (19)

in analogy to the corresponding equation for the heat transfer in high-
velocity flow. The effective temperature Te has to be chosen in


such a way that the temperature difference Te Tw,g is zero where
the heat flow dQ is zero. Only this difference avoids a change of
the heat-transfer coefficient to values infinite and zero as the
temperature difference is varied. The exact evaluation of Te is
difficult owing to wall temperature gradients in the direction of flow,
which influence the air-film temperature profile normal to the wall
surface. The temperature 'e affecting heat transfer will therefore
probably be slightly different from the adiabatic wall temperature
Tw,ad in equation (18). No information is presently available on
the magnitude of the heat-transfer coefficients in film cooling or on
an exact method of evaluating Te in equation (19).

The. information obtained by Wieghardt's investigations may be
used to compare the effectiveness of film and transpiration cooling
for the case when the heat flow dQ in equation (19) is zero. If
a wall is assumed to be cooled by a continuous succession of slots
having the distance x from each other, equation (18) gives the
highest values for the effective wall temperature. The amount of
coolant flow per length x of the surface is spcVc. In a
transpiration-cooled wall of the same dimensions,'the coolant flow
through the length x is xpcvc. In order to compare both walls at
the same amount of coolant flow per unit area, the coolant velocity
Vc on the transpiration-cooled wall has to be compared with a coolant

velocity x Vc on the film-cooled wall. Therefore the expression

s cc

x pgVg

in equation (18) is equivalent to the expression


in equation (16). In this way the curve for film cooling is inserted
in figure 9. It can be seen that the transpiration cooling is more
efficient than the film cooling. In this comparison it must be kept
in mind that equation (18) describing the temperatures obtained in film
cooling is valid only for a ratio x/s greater than 100 and that the
slot width is of the same order of magnitude as the boundary-layer



Required Flow Distribution

The information in the previous section can be used to determine
the wall-temperature distribution around the surface of a transpiration-
cooled porous blade when the velocity of the cooling air vc is
prescribed, or it can be used to determine the distribution of the
cooling-air velocity around the blade surface that is necessary to keep
the wall temperature uniform. A first approximate answer to this
-question may be obtained with the equations presented and the accuracy
of such a determination can be increased when more information on the
heat-transfer coefficients under the circumstances prevailing on blade
surfaces becomes available.

The local distribution of the cooling-air velocity vc necessary
to keep the surface temperature of a plane wall at a constant value is
shown in figure 10. The necessary coolant velocity begins with an
infinite value at the leading edge and decreases rapidly in the region
of the laminar boundary layer with increasing distance from the leading
edge. It rises again when the boundary layer becomes turbulent in
accordance with figure 9 and decreases in the turbulent region. This
decrease, however, is much slower than the decrease in the laminar

Basically, the conditions around a turbine blade for constant
surface temperature will be similar. They differ mostly in the region
near the stagnation point where the finite thickness of the blade
causes the cooling-air velocity to start out with a finite value.
The required velocity vc decreases from this value along both the
suction and pressure side of the surface to the transition point, where
a rise in the necessary velocity vc has to be anticipated followed by
a more gradual decrease of its value in the downstream direction. The
leading edge region where high coolant velocities vc are required
probably extends a greater distance downstream on a blade than on a
flat plate.

The problem that has to be solved in the design of a turbine blade
is providing the necessary distribution of the cooling air by proper
choice of the local porosity of the blade wall and its thickness.
Probably the variation of vc determined theoretically along both sides
of the blade can be approximated well enough by a constant value;
however, the high vc values necessary near the leading edge have to
be provided by some means. Figure 11 shows the temperature distribution
that was measured on a porous blade in a static cascade at the


NACA Lewis laboratory. The manufacturer tried to maintain a uniform
porosity around the blade perimeter. It can be seen that the tempera-
tures are very low on both sides of the blade. For comparison, blade
temperatures are also given which were measured in a static cascade
on two blades cooled internally with air; namely, a hollow blade and
a blade where the internal area was increased by brazing 10 tubes into
the hollow blade shell. It can be seen that the temperatures obtained
in the central part of the porous transpiration-cooled blade are
extremely low. Even with liquid cooling such low temperatures have not
been obtained at comparable coolant flows.

The reason for the high leading-edge temperatures is apparent
from a comparison of the blade cross section in figure 11 and the
velocity distribution necessary to maintain a constant wall tempera-
ture as shown in figure 10. Figure 11 shows that the wall thickness
near the leading edge was appreciably higher than at the sides of the
blades. Owing to a high pressure drop through the thicker wall,the
velocity was therefore small in the region where, according to fig-
ure 10, a high value of this velocity is necessary to maintain a con-
stant wall temperature. Near the trailing edge, no especially high
values of the coolant velocity are necessary according to figure 10;
however, the Coolant velocity in this region in the turbine blade was
extremely low because of the long path length of the coolant through
the material in this region. This fact is evidenced by a measured
local coolant-flow distribution shown in figure 12. Special attention
must therefore be directed to the leading-edge and the trailing-edge
regions of such a transpiration-cooled turbine blade. Further inves-
tigations are necessary to determine whether a local change in the
porosity can make the temperature sufficiently uniform around the
periphery of the blade or whether other special means have to be used
for cooling the leading- and trailing-edge regions.

Another difficulty is present in turbine blades, which may be
explained with the help of figure 13. This figure shows the outside
pressure distribution around the turbine blade. The cooling-air
pressure inside the turbine blade is practically constant because of
the low cooling-air velocities. This constant value is also indicated
in the upper part of the figure. It can be seen that the pressure
difference available to force the cooling air through the porous wall
differs along the blade surface. Near the leading edge where high r.
coolant velocities ve are required, the pressure difference is
smallest. The higher pressure difference and higher coolant flow on
the suction side of the blade is responsible for the lower temperature


seen in figure 11 on this side of the blade. The differences in the
available pressure difference can be decreased by using a wall with
low porosity, which makes the required inside pressure high. A high
coolant pressure, however, should be avoided because this pressure
may not be available in the compressor of the gas-turbine engine. It
would be sufficient to provide a high coolant pressure only for the
region near the stagnation point and possibly near the trailing edge.
Another possibility is to let the variation of the local porosity take
care of the differences in the available pressure as well as the dif-
ferences in the necessary coolant flow.

Porosity and Permeability

The manufacturer of porous material is concerned with the fabri-
cation of porous structures with a certain predetermined and repro-
ducible porosity. On the other hand, the considerations in the previous
paragraphs determine what the permeability of a wall should be. The
porosity is defined as the value one minus the ratio of the density of
the porous material to the density of a solid piece made of the same
material. The permeability expresses the capacity of a porous material
to pass liquids or air when pressure differences are present. In order
to prescribe the required porosity to the manufacturer of transpiration-
cooled walls, the law relating the porosity and the mass flow through a
porous wall must be known. For low velocities Darcy's law

Ap cc c(2
T 5 2 (20)

gives the relation between the pressure difference Ap acting on the
two surfaces of a plain porous wall with the thickness T, the viscosity
p the velocity vc of the coolant flowing through the porous wall,
and the average width d of the pores. This law holds for Reynolds
numbers smaller than one when the Reynolds number is based on the
dimension d. It gives a linear relationship between pressure drop
and velocity analogous to laminar flow in a tube. For large Reynolds
numbers another law holds

Ap PCvc
= C4 (21)
T C4 d

according to which the pressure drop is proportional to the density
Pc of the coolant and the square of the velocity vc. Usually, there


is a large range of Reynolds numbers within which the law gradually
changes from the linear to the quadratic relationship. Usually, this
is the range that is of interest for the applications considered. In
addition the constants in both equations depend on the structure of
the porous wall and cannot be predicted exactly (reference 23). The
relation between the pressure drop and the velocity, therefore, has to
be experimentally determined for each material. When the density of
the coolant varies considerably on its way through the porous wall, it
is more expedient to calculate with the mass flow pcVc per unit area.
It was shown in references 23 and 24 that, when the change in density
is caused by pressure variation, the two foregoing equations become
relations between the differences Ap2 of the square of the pressure
and the mass flow rate PcVc.

In turbine blades the surfaces cannot always be considered as
plane walls in calculating the mass flow through the walls, especially
near the leading and the trailing edge of the blade. The mass flow
through curved walls may be determined by methods similar to the ones
used to study the heat flow by conduction through curved walls. In
the rangq of low Reynolds numbers where the relations between the
pressure square and the mass flow rate is linear, there is a complete
analogy between both physical processes. The lines of constant tempera-
ture then correspond to the lines of constant pressure and the direction
of the heat flow on any place within the wall coincides with the
direction of the mass flow. In the range where the relation between
pressure-square values and mass-flow rate is nonlinear this analogy
,ceases to exist. In this case an adaptation of the relaxation method
to the nonlinear character of the equations may be used to determine
local flow rates through a porous wall.


The results of a survey on transpiration and film cooling have
shown that:

1. Transpiration cooling is an extremely effective means of
cooling objects in high-temperature, high-velocity gas streams such
as gas-turbine blades; however, special care will be required to
provide sufficient cooling at the leading and trailing edges.

2. The laminar boundary layer on most turbine blades is appar-
ently limited to the portion of the blade near the leading edge.
Because transpiration cooling is much more effective for laminar
boundary layers than for turbulent layers, means of extending the
laminar layer require investigation.


3. Fabrication techniques must be developed for porous blades to
provide high strength with controlled, variable permeability.

4. Film cooling is also an effective method for the cooling of
turbine blades, although it is less effective than transpiration

5. Although considerable research has been conducted on transpi-
ration and film cooling, additional research is required for a more
thorough understanding of the heat transfer under the special con-
ditions prevailing for turbine blades.

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


1. Canright, Richard B. : Preliminary Experiments of Gaseous Transpi-
ration Cooling of Rocket Motors. Prog. Rep. No. 1-75, Power Plant
Lab. Proj. No. MX801, Jet Prop. Lab., C.I.T., Nov. 24, 1948.
(AMC Contract No. W-535-ac-20260, Ordnance Dept. Contract No.

2. Kuepper, K. H.: Temperature Measurement on Two Stationary Bucket
Profiles for Gas Turbines with Boundary-Layer Cooling. Trans. No.
F-TS-1543-RE, Air Materiel Command, U.S. Air Force, Jan. 1948.
(ATI No. 18576, CADO.)

3. Dempsey, W. W.: Turbine Blade Cooling (Final Hot Test Rep.).
Rep. No. 2037, Stalker Development Co., June 29, 1949.

4. Kinney, George R., and Sloop, John L.: Internal Film Cooling
Experiments in 4-Inch Duct with Gas Temperatures to 20000 F.
NACA RM E50F19, 1950.

5. Corrsin, Stanley, and Uberoi, Mahinder S.: Spectrums and Diffusion
in a Round Turbulent Jet. NACA TN 2124, 1950.

6. Wieghardt, K.: Hot-Air Discharge for De-Icing. AAF Trans.
No. F-TS-919-RE, Air Materiel Command, Dec. 1946.


7. Weinbaum, S., and Wheeler, H. L., Jr.: Heat Transfer in Sweat-
Cooled Porous Metals. Prog. Rep. No. 1-58, Air Lab. Proj.
No. MX121, Jet Prop. Lab., C.I.T., April 8, 1947. (AMC Contract
No. W-535-ac-20260, Ordnance Dept. Contract No. W-04-200-ORD-455.)

8. Tollmien, W.: The Production of Turbulence. NACA TM 609, 1931.

9. Schlichting, H.: Uber die theoretische Berechnung der kritischen
Reynoldsschen Zahl einer Reibungsschicht in beschleunigter und
verzogerter Striomung. Jahrb. d. D. Luftfahrtforschung, 1940,
S. I 97-112.

10. Taylor, G. I.: Statistical Theory of Turbulence. V Effect of
Turbulence on Boundary Layer. Theoretical Discussion of Relation-
ship between Scale of Turbulence and Critical Resistance of
Spheres. Proc. Roy. Soc. London, vol. CLVI, no. A888, ser. A,
Aug. 17, 1936, pp. 307-317.

11. Liepmann, Hans W.: Investigations on Laminar Boundary-Layer Stability
and Transition on Curved Boundaries. ACA ACR 3H30, 1943.

12. Lees, Lester: The Stability of the Laminar Boundary Layer in a
Compressible Fluid. NACA Rep. 876, 1947. (Formerly NACA
TN 1360.)

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

14. Lees, Lester: Stability of the Laminar Boundary Layer with
Injection of Cool Gas at the Wall. Rep. No. 124, Aero. Eng.
Lab., Princeton Univ., May 20, 1948. Tech. Rep. No. 11,
Proj. Squid, under Navy Dept. Contract N6-ORI-105, T.O. III,
Phase I, NR 220-038.)

15. Eckert, E. R. G.: Heat Transfer and Temperature Profiles in
Laminar Boundary Layers on a Sweat-Cooled Wall. Tech. Rep.
No. 5646, Air Materiel Command, Nov. 3, 1947.

16. Schlichting, Hermann, und Bussmann, Karl: Exacte Lisungen fu{r die
laminare Grenzschicht mit Absaugung und Ausblasen. Schriften
d. D. Akad. Luftfahrtforschung, Bd. 7B, Heft 2, 1943.


17. Yuan, Shao Wen: Heat Transfer in Laminar Compressible' Boundary
Layer on a Porous Flat Plate with Fluid Injection. Jour. Aero.
Sci., vol. 16, no. 12, Dec. 1949, pp. 741-748.

18. von Karman, Th.: On Laminar and Turbulent Friction. NACA TM 1092,

19. Rannie, W. D.: A Simplified Theory of Porous Wall Cooling.
Prog. Rep. No. 4-50 Power Plant Lab. Proj. No. MX801, Jet Prop.
Lab., C.I.T., Nov. 24, 1947. (AMC Contract No. W-535-ac-20260,
Ordnance Dept. Contract No. W-04-200-ORD-455.)

20. Friedman, Joseph: A Theoretical and Experimental Investigation of
Rocket-Motor Sweat Cooling. Jour. Am. Rocket Soc., no. 79,
Dec. 1949, pp. 147-154.

21. Colburn, Allan P.: A Method of Correlating Forced Convection Heat
Transfer Data and a Comparison with Fluid Friction. Trans. Am.
Inst. of Chem. Eng., vol. XXIX, 1933, pp. 174-210.

22. Eckert, E. R. G.: Introduction to.the Transfer of Heat and Mass.
McGraw-Hill Book Co., Inc., 1950.

23. Green, Leon, Jr., and Duwez, Pol: The Permeability of Porous Iron.
Prog. Rep. No. 4-85, Jet Prop. Lab., C.I.T., Feb. 9, 1949.
(Ordnance Dept. Contract No. W-04-200-ORD-453.)

24. Grootenhuis, P.: The Flow of Gases through Porous Metal Compacts.
Engineering, vol. 167, April 1, 1949, pp. 291-292.



I \


A rI




-. NACA..

Gas temperature
T ----- --- --- ---- -- -- ---


Figure 2. Temperature distribution along film-cooled

NACA RM E50K15 29

S x
I a s I

Figure 3. Temperature profile downstream of line source of heat.




-.NACA /

0 x

Figure 4. Effect of air-flow rate on temperature
distribution between two locations having prescribed

0 0 0 0 0 0
o o C3 01 N o

SQ ainqrazadmal

ai z 'anmeadIma,




d S
r 4)

0 a
I 1

) 0
^> (U








,0 g<

,1 0 )



32 NACA RM E50K15



O 4




r i
0 m




\ '
\ 4
O .41
\ g

\ o *a
----- ---- -- ^ ---- ---- ---- ---- ----00 i4
\ .4- 4
---- ---- --- \ ---- ---- ---- --- ---- H

I- CD U)




X i


Figure 7. Coolant velocity, heat-transfer coefficient, and
boundary-layer thickness for transpiration-cooled plate at
constant temperature with laminar flow.




Figure 8. Factor F in laminar-fldw Nusselt equation Nu = F/Re
for transpiration cooling.





T 9-TJ


Film cooling

(transpiration cooling)

r aminsar on
(transpiration cooling)






Figure 9. Effect of mass-flow ratio on wall temperature for
transpiration and film cooling. Re = 105.

1_ ---- -

A 'jj3ooIaA AOtJ UHTooj0



j a
J Ds



















.I -
*- c a

B >
-a a i-


Chord, percent

Figure 11. Porous-blade temperature distribution for
vW/vg = 0.026.

Gas temperature

***---- ----- -- --*--/

Hollow blade

S l .O-Tube blade

Pressure surface

Suction surface

Coolant temperature









Suction surface

-Pressure surface

3 ( Pressure iurfac

S// ,

Suction surface-

o \


0 20 40 60 80 100
Chord, percent

Figure 12. Permeability variation around porous blade for
p&P 49.5 b2/ft5.


tlon surface


Coolant pressure

20 40 60 80
Chord, percent

Figure 13. Pressure distribution around turbine blade.

NACA-Lnley 2-12-51 580




II 16I 08I 16i iiii
3 1262 08106 573 1