Sonderforschungsbereich/Transregio 40 – Annual Report 2015
197
Modelling the heat transfer coefficient in
reacting cooling films
By G. Frank and M. Pfitzner
Institut für Thermodynamik LRT10, Universität der Bundeswehr München
Werner-Heisenberg-Weg 39, 85577 Neubiberg
In the literature the surface heat flux of a reactive boundary layer is calculated using the
inert heat transfer coefficient and the difference between maximum fluid temperature
and wall temperature. However experimental data showed, that the heat transfer coefficient in a reactive cooling film differs from its nonreactive counterpart. The objective of
this work is to investigate the heat transfer coefficient in a reactive boundary layer. In the
first part of this work the surface heatflux for a reactive laminar boundary layer on a flat
plate is analytically derived using boundary layer theory. The heat transfer coefficient of
a reactive cooling film emanating with Z = 1 from a porous surface is enhanced by a
factor of 1/(1 − Zst ) and by two correction constants Cf 0 and CC , which represent the
boundary layer thickening due to mass injection and changing fluid properties. To validate the analytical results a reference experiment of a diluted hydrogen boundary layer
flame over a porous flat plate has been investigated numerically using a finite rate combustion model and detailed chemistry. The surface heatflux of the numerical reference
data is compared to the newly derived analytical heat flux correlation.
1. Introduction
In combustion chambers of modern liquid rocket engines pressures up to 100 bar
prevail and with a LOx/LH2 fuel, temperatures up to 3500 K are reached. Combustion
chamber walls have to be cooled to counter those high thermal loads and assure safe
operation. Modern cooling systems of liquid rocket engines are based on regenerative
cooling, where coolant flows through cooling channels in the wall. On the hot gas side
this regenerative cooling system can be supported by the injection of a cooling film.
Those supplementary cooling methods, like film cooling and transpiration film cooling
are in the focus of present research to cope with future challenges. On a film cooled
wall a film of liquid or gaseous coolant forms a protective layer between the cooled wall
and the hot combustion products in the chamber. The coolant is injected through orifices
like slots or holes in the wall, or, in case of transpiration film cooling, through the porous
wall material.
In classical film cooling applications the heat flux to the wall can be obtained by
qw = h · (Tw − Taw )
(1.1)
with Tw being the wall temperature and Taw the adiabatic wall temperature for the film
cooled geometry. For most engineering applications the heat transfer coefficient h in
the above equation can be approximated by the heat transfer coefficient of the nonfilm cooled geometry, or, more accurately, by the heat transfer coefficient obtained in a
198
G. Frank and M. Pfitzner
situation where film and main flow have the same temperature. One is left with determing
the unknown adiabatic wall temperature Taw . A dimensionless film cooling effectiveness
can be defined
Taw − T∞
,
(1.2)
η=
Tc − T∞
using the adiabatic wall temperature Taw , the freestream temperature T∞ and the temperature of the injected cooling fluid Tc . For a few cooling film configurations the film
cooling effectiveness can be derived from boundary layer theory as a function of Reynolds and Prandtl number and a relation of coolant properties to mainstream properties [1]. For the large variety of cooling configurations, where an analytical solution is
not possible, the film cooling effectiveness has been experimentally investigated and
correlated in terms of Reynolds and Prandtl number, as well as blowing or momentum
ratio.
Turbulent fluctuations of the main flow can transport a reactive mixture into the cooling
film, for example radicals or even pockets of unburned oxidizer. In film cooled gas turbine
combustors it has been obeserved that pockets of incompletely burned fuel are transported into the oxidizer cooling film. With the additional heat release within the boundary
layer the theory of the inert measured film cooling effectiveness cannot be related to the
temperature field anymore. It has been shown by the authors, that instead of a temperature based film cooling effectiveness (Eq. (1.2)) a modified mixing effectiveness based
on a mixture fraction or mass fraction of an inert tracer gas can be used [2]
η⋆ =
Zw − Z∞
.
Zc − Z∞
(1.3)
For inert cooling films with unity Lewis number the temperature based (Eq. (1.2)) and
the mixture based cooling film efficiency (Eq. (1.3)) are identical due to the analogy of
heat- and mass transfer.
In a reactive cooling film the thermal boundary layer differs signifficantly from the
thermal boundary layer of inert cooling films. Although earlier work [2] suggests, that the
mixture based cooling film efficiency η ⋆ for reactive cooling films can be approximated
by empirical correlations for the temperature based cooling film effiency η of the inert
case, no connection between η ⋆ and the wall heat flux qw has been defined so far. In the
literature the driving temperature difference for the wall heat flux is naturally chosen as
the difference between maximum (flame) temperature in the boundary layer with heat
release inside the cooling film, and the wall temperature [3, 4]. An upper limit for this
maximum temperature is the adiabatic flame temperature derived from the composition
in the freestream Z∞ and at the wall Zw . The mixture based film cooling efficiency η ⋆
and the driving temperature difference are parameters determining the surface heat flux
for reactive cooling films.
Since different driving temperatures contribute mainly to the increase of the surface
heat flux, the heat transfer coefficient is often assumed to be the same as in the inert
case [3, 4]. Experimental data shows however, that the heat transfer coefficient in the
reactive boundary layer increases from −5% up to 50% of the inert value [3]. That means,
that the surface heat flux in the investigated gas turbine is increased up to a factor of 1.5
solely by the increase of the heat transfer coefficient.
This work aims to study the change of the heat transfer coefficient in a reactive boundary layer. The first part of this paper (section 2) is dedicated to an analytical analysis of
a reactive boundary layer over a porous flat plate. A similar laminar flow over a vapor-
heat transfer in reacting cooling films
199
F IGURE 1. Laminar flow over a porous flat plate
liquid interface was investigated using boundary layer theory [5] by Emmons [6] for unity
Schmidt number and constant fluid properties and by Andreussi [7] for variable fluid
properties. In those two works, the surface heat flux was directly coupled to the injected
mass into the boundary layer, since the heat transferred to the liquid surface was needed
to vaporize the fuel. In this work, surface heat flux and injection are independent and
the surface heat flux is derived for this configuration with variable fluid properties. The
convective heat transfer coefficient of the reactive boundary layer is compared to its
nonreactive counterpart. A reference experiment conducted by Volchkov [8], which is
similar to the generic setup in section 2, is described in section 3. This experiment has
been investigated numerically in [9] using OpenFOAM [10] and a finite rate combustion
model. Using experimental and numerical data, the surface heat flux of this boundary
layer diffusion flame is compared to the newly developed theoretical heat flux correlation
in section 5.
2. Theoretical background
In this section the surface heat flux is derived for a reactive boundary layer over a
porous flat plate. Fig. 1 depicts the investigated setup schematically. Fluid with a composition Z∞ and temperature T∞ flows with the uniform velocity u∞ over a porous flat
plate. Coolant with the composition Zw is injected through the porous material with the
velocity v0 (x). The wall and the coolant fluid are assumed to be in thermal equilibrium,
so the coolant is injected at wall temperature Tw . The porous plate is assumed to be
ideally smooth, meaning the pores of the material do not present obstacles for the flow
and a laminar boundary layer develops on the plate. Without injection (v0 = 0 m/s) the
velocity profile corresponds to the typical Blasius solution. The reference point for the
coordinate system is the upstream edge of the flat plate. The x-coordinate represents
the downstream position, the streamwise velocity component is u. The y-coordinate describes the distance to the plate surface and v is the velocity component perpendicular
to the plate. This setup represents an ideal transpiration film cooling situation. Film cooling efficiencies η and η ⋆ are either zero (Zw = Z∞ ) or unity for this setup.
For this flow the conservation equations of mass, momentum and energy can be
200
G. Frank and M. Pfitzner
simplified using boundary layer theory [5].
∂(ρu) ∂(ρv)
+
∂x
∂y
∂u
∂u
ρu
+ ρv
∂x
∂y
∂h
∂h
+ ρv
ρu
∂x
∂y
∂Z
∂Z
ρu
+ ρv
∂x
∂y
=0
∂u
∂
µ
∂y
∂y
µ ∂h
∂
=
∂y P r ∂y
µ ∂Z
∂
=
∂y Sc ∂y
=
(2.1)
(2.2)
(2.3)
(2.4)
The energy conservation equation is written in terms of enthalpy including the heats of
formation. The Eckert number is assumed to be very small and viscous heating therefore
negligible. The mixture can be described by a conserved scalar, differential diffusion is
neglected and the diffusion of the mixture fraction can be described by a global Schmidt
number Sc. The fluid properties are assumed to vary, but to be always in local thermal
equilibrium, so that ρ, µ, P r, Sc = f (p, h). With variable fluid properties, velocity and
temperature fields are coupled and cannot be solved independently like in the classic
Blasius solution. With unity Lewis Number Le = Sc/P r = 1, energy (Eq. (2.3)) and
mixture fraction(Eq. (2.4)) transport equations become formally identical, so mixture
fraction and energy transport are analogous.
The continuity equation (2.1) can be satisfied by introducing a streamfunction Ψ with
ρu = ∂Ψ/∂y and ρv = −∂Ψ/∂x. The remaining partial differential equations are transformed into ordinary differential equations using the Dorotnitzyn-Stewartson transformation.
r
Z
ρ∞ u ∞ y ρ
Ψ
dy,
f (η) = √
(2.5)
η=
2µ∞ x 0 ρ∞
2ρ∞ µ∞ u∞ x
With the introduction of the Chapman-Rubesin parameter C
ρµ
C=
ρ ∞ µ∞
(2.6)
the boundary layer equations are written as
′
(Cf ′′ ) + f f ′′ = 0
′
C ′
+ f h′ = 0
h
Pr
′
C ′
Z
+ f Z′ = 0
Sc
(2.7)
(2.8)
(2.9)
with the prime (′ ) indicating differentiation with respect to η. The boundary conditions to
the above equations become
r
2x
η=0
f = −ρv0 (x) ·
f ′ = 0 h = h(Tw , Zw ) Z = Zw
ρ ∞ u ∞ µ∞
η→∞
f ′ = 1 h = h(T∞ , Z∞ ) Z = Z∞ .
A similarity solution for the velocity, temperature and mixture profiles in the boundary
layer is only√possible, if f (η = 0) = const. The injected massflow per area ρv0 (x) has to
vary like 1/ x to ensure self similar profiles. The injection is characterised by the pa-
heat transfer in reacting cooling films
201
rameter f0 = f (η = 0), with f0 > 0 corresponding to suction and f0 < 0 corresponding
to injection into the boundary layer.
Now the wall heat flux for this flow shall be derived. The wall heat flux can be generally
written as
∂T q̇ = −λ
,
(2.10)
∂y y=0
or transformed into the new coordinate system
r
ρ∞ u∞ ∂T ρ
.
q̇ = −λ
ρ∞ 2µ∞ x ∂η η=0
(2.11)
The temperature T can be related to the mixture fraction Z in a non-reacting system
with negligible variations of cp as Tu (Z) = TO + Z(TF − TO ). In a system with chemical
reactions T (Z) shall be described by the Burke Schumann solution. The chemical reactions in the system are described as νF [F ] + νO [O] → [P ], with [F ], [O], [P ] indicating
fuel, oxidizer and products respectively. The Burke Schumann solution is then written as
(
if Z ≤ Zst
Tu (Z) + ZZst (Tad,F l − Tu (Zst )),
T (Z) =
1−Z
Tu (Z) + 1−Z
(T
−
T
(Z
)),
if
Z > Zst
ad,F l
u
st
st
with
s · YF − YO + YO,0
s · YF,0 + YO,0
1
=
.
YF,0
1 + s · YO,0
Z=
and Zst
s=
WO νO
WF νF
(2.12)
Here Tad,F l is the adiabatic flame temperature at stoichiometric mixture fraction Zst and
Wi are the molar masses of fuel and oxidizer respectively. A Burke Schumann solution of
a diluted hydrogen air flame is depicted in Fig. 8 as the reference solid line in the graph.
In rocket combustion systems, the fuel is used as coolant for film cooling, therefore
Zw = 1, TF = Tw and the wall heat flux can be written now
r
ρ∞ u∞ Tw − Tad,F l ∂Z ρ
.
(2.13)
q̇ = −λ
ρ∞ 2µ∞ x 1 − Zst
∂η η=0
The wall heat flux thus is dependent on the gradient of the mixture fraction at the
wall, which can be found by solving Eq. (2.9). To solve this equation, an assumption
on the Chapman Rubesin parameter needs to be made. Since (ρµ)w /(ρµ)∞ 6= 1, the
Chapman Rubesin parameter C is assumed to be constant, but not unity, for 0 < η < ηst .
Here ηst is the position of stoichiometric mixture fraction. The solution of Eq. (2.9)
Rη
Rη
exp − 0 Sc
Z − Zw
Z −1
C f dη dη
0
Rη
=
(2.14)
= R ηst
Zst − Zw
Zst − 1
exp − 0 Sc
0
C f dη dη
and the gradient of the mixture fraction at the wall can be expressed as
1
∂Z 1
.
R η Sc
= R ηst
·
∂η η=0 Zst − 1
f dη dη
exp
−
0
0 C
(2.15)
This expression is constant, as long as ηst is constant. With self similar mixture fraction
profiles, this condition is fulfilled.
202
G. Frank and M. Pfitzner
1.5
CC
Cf0
1.5
1
0.5
1
0.5
-0.2
-0.1
0
0
0.5
f0
1
1.5
C
F IGURE 2. Normalized mixture fraction gradient at the wall for different injection parameters and
Chapman Rubesin parameters
The wall heat transfer coefficient α can now be defined using
and
Tref = Tad,F l
to
r
ρ∞ u ∞
1
∂Z ρ
.
α = −λ
ρ∞ 2µ∞ x 1 − Zst ∂η η=0
q̇ = α(Tw − Tref )
(2.16)
This heat transfer coefficient is quite similar to the heat transfer coefficient of an inert
laminar boundary layer αnr , which can be found in the literature, see e.g. [1]. Equation (2.11) also describes the surface heat flux of an inert laminar boundary layer with
variable fluid properties. In a nonreactive flow, the temperature is often made dimensionless via θ = (T − T∞ )/(Tw − T∞ ) and the nonreactive wall heat transfer coefficient
can be described as
r
∂θ ρ∞ u ∞
ρ
(Tw − T∞ )
(2.17)
q̇nr = −λ
ρ∞ 2µ∞ x
∂η η=0
with
q̇nr = αnr (Tw − T∞ )
r
ρ
ρ∞ u∞ ∂θ αnr = −λ
.
ρ∞ 2µ∞ x ∂η η=0
(2.18)
For an inert, laminar flat plate boundary layer with unity Chapman Rubesin parameter,
the following Nusselt number correlation has been found as a good fit for the above
equation (2.18).
√ √
3
N unr (x) = 0.332 Re P r for P r > 0.6
(2.19)
Equations (2.16) and (2.18) are quite similar. The dimensionless temperature θ is
unity at the wall and zero in the free stream. In an inert flowfield its transport equation
can be written as (C/P r ·θ′ )′ +f θ′ = 0, which is for unity Lewis number formally identical
to the mixture fraction transport equation (2.9), therefore dθ/dη|η→0 = dZnr /dη|η→0 .
The nonreactive heat transfer coefficient defined in Nusselt correlations like Eq. (2.19)
is valid for unity Chapman Rubesin parameters and no injection. In a reactive case,
where C 6= 1 and f0 6= 0 the mixture fraction gradient at the wall differs (according to
equation (2.9)). It is dependent on the injection parameter f0 , the Chapman Rubesin
parameter C and the Schmidt number Sc. In classical Nusselt correlations the dependence on the Prandtl number, or in the Le = 1 case the analogous Schmidt number,
is included. For example the Nusselt
√ number in equation (2.19) is dependent on the
cubic root of the Prandtl number 3 P r. In Fig. 2 the mixture fraction gradient at the
heat transfer in reacting cooling films
203
wall is normalized by the value dZnr /dη|η→0 at no injection (f0 = 0), unity Chapman
Rubesin parameter and Schmidt number (C = 1, Sc = 1) and plotted against the injection parameter and the Chapman Rubesin parameter. This constant is called Cf 0 and
CC respectively. With increasing injection, the boundary layer thickens and the mixture
fraction gradient at the wall decreases linearly. The normalized √
mixture fraction gradient at the wall follows the Chapman Rubesin parameter with 1/ 3 C in the investigated
regime. A Chapman Rubesin parameter smaller than unity decreases the boundary
layer thickness, whereas a higher Chapman Rubesin parameter thickens the boundary
layer.
Using Cf 0 and CC the mixture fraction gradient at the wall for the reactive case can
be related to the gradient of the normalized temperature at the wall in the inert case.
Combining now equations (2.16) and (2.18), the heat flux in a reactive boundary layer
can be finally defined with the nonreactive heat transfer coefficient to:
αnr
(Tw − Tad,F l ),
(2.20)
q̇ = Cf 0 · CC ·
1 − Zst
In contrast to the nonreactive case the driving temperature difference is between the
injection temperature Tw and the highest fluid temperature, which is the adiabatic flame
temperature Tad,F l when using the simplified Burke Schumann solution. The convective
heat transfer coefficient of the reactive case is enhanced by the factor 1/(1 − Zst ) compared to the nonreactive case. This factor 1/(1 − Zst ) represents the distance between
the wall and the high temperature reaction zone. This distance determines the temperature gradient to the wall, since between wall and the high temperature reaction zone
the temperature rises more steeply than in the inert case. The temperature gradient
becomes steeper when the reaction zone moves closer to the wall.
In the literature the surface heat flux of a reactive boundary layer is calculated using
the inert heat transfer coefficient and the difference between maximum fluid temperature and wall temperature. In this chapter the surface heatflux was analytically derived
for a reactive laminar boundary layer on a flat plate. Our analysis confirms the driving
temperature difference, but we found that the inert heat transfer coefficient is dependent on the distance between the high temperature zone and the wall, as well as on the
different fluid properties and the injection at the wall. A correction factor of 1/(1 − Zst )
has been identified Burke Schumann combustion. The correction factors for the different fluid properties at the wall CC and for the injection Cf 0 can be determined from the
graphs in Fig. 2.
3. Reference experiment
To validate the derived heat transfer correlation from the previous chapter, experimental data on a setup similar to the one sketched in Fig. 1 is desirable. Several investigators studied combustion phenomena in boundary layers, where fuel is injected into
the boundary layer via a porous surface or a liquid pool, e.g. [8, 11, 12]. For the current
study, we chose the experiment conducted by the group of Volchkov [8] as reference
experiment.
This experiment investigates a diluted hydrogen air flame in a laminar flat plate boundary layer. The experimental setup is sketched in Fig. 3. A porous plate with the length
L = 0.145 m is mounted flush with the bottom wall of a wind tunnel with a 0.105 m × 0.105 m
cross section. The top wall of the wind tunnel was removed during the experiment, so the
flow was not confined. Upstream of the porous plate a flame holder of height H = 3 mm
204
G. Frank and M. Pfitzner
F IGURE 3. Experimental and numerical setup of the Volchkov test case
was installed. Air at ambient temperature T∞ = 293 K and pressure p∞ = 1 bar flows
over the flat plate with a freestream velocity of U∞ = 2 m/s. The flow is laminar. A
hydrogen-nitrogen mixture with a hydrogen mass fraction of KH2 = 6.2 % is injected
through the porous material into the boundary layer at a uniform mass flow rate.
This real experimental setup differs from the ideal setup studied in the previous chapter. The injection of the fuel mixture
does not satisfy the boundary condition of the ideal
√
setup for f0 as ρv0 (x) ∼ 1/ x. So velocity, temperature and mixture fraction profiles
are not self similar. The injection parameter f0 on the other hand is quite small downstream of the leading edge, so the mass injection into the boundary layer is negligible for
the larger part of the plate. The ideal setup is semi inifinite in z-direction. Experimental
measurements were taken in the middle of the plate, so disturbances from the side-walls
are assumed to be negligible. The flame holder upstream of the plate induces a small
recirculation zone and disturbs the laminar boundary layer.
Wall temperatures were measured in this experiment with thermocouples at distinct
locations on the plate. Also temperature measurements in the boundary layer were
performed using a thermocouple probe. The temperature measurements were not corrected due to radiation heat loss. From the measured temperatures, the heat transfer is
given in terms of a Stanton number St for different flow conditions and is fitted to the
correlation St = 0.47Re−0.5
P r0.6 .
x
4. Numerical method
The authors investigated the experimental setup by Volchkov [8] numerically in Frank
[9]. The geometry is discretized with a 2D block structured grid with 239 × 86 cells,
as shown in Fig. 3. The grid is uniformly spaced up to an estimated boundary layer
thickness δ(L) and then coarsened towards the top freestream boundary. In streamwise
direction the grid is refined towards the leading edge of the porous plate. The grid spacing in the main mixing and reaction zone is ≈ 0.2 mm. The porous plate is modelled
as a velocity inlet with constant mass flow perpendicular to the plate. The injected fluid
and the porous wall are assumed to be in thermal equilibrium, therefore Tc = Tw . The
wall temperature distribution is prescribed and taken from experimental data. The numerical investigations are performed in the OpenFOAM framework [10] with a finite rate
combustion model and the Marinov mechanism [13] (10 species, 27 reactions). Since
heat transfer in reacting cooling films
2
2cm
8cm
12cm
y [cm]
1.5
CFD
205
exp.
1
0.5
0
500
1000
T [K]
1500
2000
F IGURE 4. Comparison of numerical and experimental temperature profiles
no multicomponent diffusion model is available in OpenFOAM, unity Schmidt number
Sc = 1 is assumed.
5. Results
5.1. Comparison with experimental results
Numerical and experimental data has been compared and are described in detail in [9].
Temperature profiles in the boundary at 2 cm, 8 cm and 12 cm downstream the leading edge of the plate are compared to experimental measurements in Fig. 4. Maximum
flame temperature as well as the distance between reaction zone and the wall were
overpredicted in the numerical simulation. Temperature measurements had been conducted with thermocouples, but are not corrected by radiation losses. We expect nearly
equilibrium temperatures in the flame, but also with radiative correction the experimentally measured temperatures are far lower than expected. The discrepancy in maximum
flame temperature is believed to be due to the very simple boundary condition for the
porous wall in the numerical simulation. The injection into the boundary layer is probably nonuniform in the experiment, since the flame heats the porous material at the
leading edge, thus changing the flow field inside the porous medium. Numerical results
can be certainly improved, when including multi component diffusion for the hydrogen
combustion. Nevertheless the numerical and experimental temperature gradients to the
wall agree well.
5.2. Comparison with boundary layer theory
Using the Dorotnitzyn Stewartson transformation Eq. (2.5) the numerical data has been
post processed. In Fig. 5 profiles of the Chapman Rubesin parameter are plotted for different locations downstream of the leading edge. These profiles seem quite self similar,
except for the first downstream location x = 1 cm, where probably the effect of the recirculation zone after the flame holder is visible. The Chapman Rubesin parameter at the
wall is Cw ≈ 0.4. The stoichiometric mixture fraction for this diluted hydrogen air flame
is Zst ≈ 0.32. In Fig. 6 mixture fraction profiles are plotted for different downstream locations and compared to boundary layer profile with no injection f0 = 0 indicated by the
solid line. Numerical and analytical data compare well. The numerical boundary layer is
slightly thicker than the theoretical one, which is probably due to the flame holder upstream the plate. Also the curvature of the mixture fraction profile changes from concave
for low mixture fractions to convex at Z → 1 in contrast to the undisturbed boundary layer
206
G. Frank and M. Pfitzner
5
5
1 cm
2 cm
3 cm
4 cm
5 cm
4
3
η
η
3
1 cm
2 cm
3 cm
4 cm
5 cm
Blasius
4
2
2
1
1
0
0
0.2
0.4
0.6
0.8
Chapman-Rubesin parameter C [-]
0
1
0
F IGURE 5. Profiles of the Chapman Rubesin
parameter of the Volchkov flame
0.2
0.4
0.6
mixture fraction [-]
0.8
1
F IGURE 6. Mixture fraction profiles of the
Volchkov flame
2000
5
4
η
3
temperature T [K]
1 cm
2 cm
3 cm
4 cm
5 cm
2
1
1500
B-S
1 cm
2 cm
3 cm
4 cm
5 cm
1000
500
0
500
1000
1500
temperature T [K]
2000
F IGURE 7. Temperature profiles of the
Volchkov flame
0
0.2
0.4
0.6
0.8
1
mixture fraction [-]
F IGURE 8. Temperature versus mixture fraction of the Volchkov flame with reference Burke
Schumann solution
profile. The distance of the stoichiometric mixture fraction to the wall is nearly constant
at ηst ≈ 1.1 in dimensionless coordinates. The increase in the Chapman Rubesin parameter is low for η < ηst and very steep in the oxygen side of the reaction zone η > ηst .
For the solution of Eq. (2.9) a constant Chapman Rubesin parameter for η < ηst was
assumed, which is approximately satisfied.
Temperature profiles at distinct downstream locations are plotted in Fig. 7. Here the
maximum flame temperature, as well as the distance between maximum flame temperature and wall (ηst ) is nearly constant downstream the plate. The wall temperature decreases from 1100 K to 900 K downstream the plate, the temperature profiles are therefore not similar in the near wall region, although the mixture fraction profiles are self
similar over the most part of the plate. Temperature versus the mixture fraction is plotted in Fig. 8 and compared to a reference Burke Schumann solution of a global one
step hydrogen combustion. A Burke Schumann solution exists for each boundary condition Tw and therefore at each downstream position. To simplify matters, only one Burke
Schumann solution with a mean injection temperature TF is plotted in Fig. 8.
Using diluted hydrogen fuel like in the experiment, flame temperatures are found to
be quite low around 1800 K. In the lean part of the flame the Burke Schumann solution
underpredicts the temperature, whereas in the rich part of the flame the Burke Schumann solution seems to be a good approximation. In Sec. 2 the gradient ∂(T )/∂(Z) at
heat transfer in reacting cooling films
207
heatflux [W/m2]
0
-10000
-20000
analytic
analytic, CC=1.3
analytic, αnr
CFD
experiment
-30000
-40000
-50000
0
0.05
0.1
x-axis [m]
0.15
F IGURE 9. Specific heat flux for the Volchkov test case
Z → 1 is estimated using the Burke Schumann solution. For this flame, this approach
seems justified.
The surface heatflux for this testcase is depicted in Fig. 9. The reference heatflux (
solid line ) is evaluated from the numerical data using the temperature gradient at the
wall and equation (2.10). To evaluate the experimental heatflux ( solid squares), the wall
temperature gradient has been extracted from the measured temperature profiles and
multiplied with the heat conductivity at the wall from the numerical data. Numerical and
experimental surface heat flux agree very well. The driving temperature difference for all
analytic heatflux calculations is taken as the difference of the wall temperature Tw and
the maximum fluid temperature Tmax in the boundary layer at each position downstream
the leading edge. The heat flux evaluated using the noreactive heat transfer coefficient
αnr from the Nusselt correlation (2.19) is depicted as dash dotted curve. This curve has
the best agreement with the quantitative value of the heat flux of all shown analytic heat
flux calculations. However the gradient of this curve is too low downstream compared to
the reference data set. In the upstream region the heat flux is slightly underpredicted.
Equation (2.20) with Cf 0 = CC = 1 is depicted as dashed curve. The heat transfer coefficient in this case is enhanced by the factor 1/(1 − Zst ). This equation overpredicts the
heatflux for all downstream positions, but fits the slope of the reference data better in
the upstream region, than the correlation using the nonreactive heat transfer coefficient.
The Chapman Rubesin parameter has been determined to Cw = 0.4 at the wall. Using
Fig. 2, the constant CC has been determined to CC = 1.3. Using this correction factor
as well, this correlation is depicted as dotted curve. With decreasing Chapman Rubesin
parameter an increase in the mixture fraction gradient and therefore an increase in the
wall heat flux is expected. As can be seen in Fig. 9 the heatflux is even more overpredicted using this correction factor. For this setup however the mixture fraction gradient at
the wall is lower than the Blasius reference solution, as was shown in Fig. 6. In fact the
boundary layer seems thicker in the Volchkov testcase than the reference undisturbed
flat plate boundary. This thickened boundary layer is possibly related to the flame holder
at the leading edge of the plate. The Nusselt correlation (2.19) is √
not strictly applicable
for this test case. Additionally the injection does not vary like 1/ x, so this boundary
layer does not show true self similar behaviour. Instead of using a Nusselt correlation
for a laminar flat plate boundary layer, the actual nonreactive heat transfer coefficient
can be evaluated from a numerical simulation of the inert Volchkov test case.
208
G. Frank and M. Pfitzner
6. Conclusions
In the literature the surface heat flux of a reactive boundary layer is calculated using
the inert heat transfer coefficient and the difference between maximum fluid temperature
and wall temperature. However experimental data showed, that the heat transfer coefficient in a reactive cooling film differs from its nonreactive counterpart. The objective of
this work is to investigate the heat transfer coefficient in a reactive boundary layer.
In the first chapter the surface heat flux for a reactive laminar boundary layer on a
flat plate was analytically derived using boundary layer theory. Our analysis confirms
the driving temperature difference, but we found that the inert heat transfer coefficient is
dependent on the distance between the high temperature zone and the wall, as well as
on the different fluid properties and the injection at the wall. The surface heat flux has
been described in terms of the nonreactive heat transfer coefficient in equation (2.20).
The temperature in the boundary layer rises from wall temperature to maximum flame
temperature, where the maximum heat release occurs. Therefore the closer the reaction zone is to the wall, the steeper the temperature gradient gets and the wall heat
flux increases. A correction factor of 1/(1 − Zst ) has been identified that represents this
effect. Injecting mass into the boundary layer thickens it. This reduces the temperature
gradient to the wall. This effect has been included in correlation (2.20) with a correction factor Cf 0 . Changing fluid properties are characterized by the Chapman Rubesin
parameter. Chapman Rubesin parameters lower than unity lead to thinner boundary
layers, whereas parameters greater than unity lead to a thickened boundary layer. This
effect has been included in correlation (2.20) with a correction factor CC .
To validate the correlation (2.20) a reference experiment conducted by Volchkov [8]
has been chosen, that has already been investigated numerically by the authors in [9].
This experiments investigates a diluted hydrogen air flame in a laminar boundary layer
over a porous flat plate. Numerical investigations are performed in the OpenFOAM
framework with a finite rate combustion model and the Marinov reaction mechanism
(10 species, 27 reactions). Although some discrepancies between numerical and experimental results had been found [9], temperature gradients to the wall and the wall
heat flux of the numerical data agree well with the experimental data set. The numerical
results have been postprocessed using the same coordinate transformation as in the
first chapter. Mixture fraction profiles through the boundary layer are mostly self similar,
although the experimental setup does not strictly fulfill the necessary boundary conditions. Assumptions made to derive equation (2.20) have been confirmed, namely a
nearly constant Chapman Rubesin parameter in the near wall section of the boundary
layer and temperature profile similar to a simple Burke Schumann solution.
The heatflux from the numerical data sat has been compared to three analytical correlations. Using the inert heat transfer coefficient, as suggested by literature, the quantitave value of the heatflux agrees well with the reference data. However the slope of the
heat flux curve is too low compared to the reference data. The nonreactive heat transfer
coefficient αnr used in this work was evaluated using a Nusselt correlation of a laminar flat plate boundary layer, which is a simplification. The actual boundary layer in the
Volchkov test case has been found to be thicker, due to the existence of the flame holder
and the constant mass injection over the plate length. The Nusselt correlation used here
is not strictly applicable in this test case. Probably due to this simplification of the nonreactive heat transfer coefficient, the heatflux using correlation (2.20) is overpredicted.
However the qualitative heatflux distribution on the surface matches the reference data
better.
heat transfer in reacting cooling films
209
Acknowledgments
Financial support has been provided by the German Research Foundation (Deutsche
Forschungsgemeinschaft – DFG) in the framework of the Sonderforschungsbereich
Transregio 40.
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