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Investigation on the influence of nonuniform initial temperature on the transient heat transfer measurement of film cooling
Experimental Thermal and Fluid Science 35 (2011) 1151–1161
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Investigation on the influence of nonuniform initial temperature on the transient heat transfer measurement of film cooling Cun-liang Liu ⇑, Hui-ren Zhu, Jiang-tao Bai, Zong-wei Zhang, Xia Zhang School of Power and Energy, Northwestern Polytechnical Univ., Xi’an 710072, China
a r t i c l e
i n f o
Article history: Received 7 May 2010 Received in revised form 30 January 2011 Accepted 31 January 2011 Available online 3 April 2011 Keywords: Transient heat transfer measurement Nonuniform initial temperature Film cooling Measurement deviation
a b s t r a c t Numerical and experimental investigations on the influence of nonuniform initial temperature on the transient heat transfer measurements are presented in this paper. The case of film cooling is investigated. When the initial wall temperature is nonuniform, the results of heat transfer coefficient and film cooling effectiveness, which are calculated by the equations derived with constant initial temperature, could deviate from the true values badly, especially in the condition of short test duration. Using initial wall temperature which is higher than the real values causes the results of heat transfer coefficient and film cooling effectiveness lower than the true values. And lower initial wall temperature produces higher results of heat transfer coefficient and film cooling effectiveness. However, when the initial temperature distribution in the region where conduction plays more influence on the wall surface temperature than the convection is well fitted by the cubic polynomial, accurate results can be obtained by the new equation which is derived from 1-D unsteady conduction model with nonuniform initial wall temperature. Some suggestions are also introduced to reduce the influence of nonuniform initial temperature when the initial temperature distribution is difficult to obtain and the equation derived from constant initial temperature has to be employed. Crown Copyright Ó 2011 Published by Elsevier Inc. All rights reserved.
1. Introduction Nowadays there are mainly two types of heat transfer measurement technique: steady state measurement technique and transient measurement technique. Transient heat transfer measurements are more and more employed by researchers because of short experiment duration which is able to reduce the experiment cost and workload, and is required in short-duration test facilities. At the beginning, it was used at high temperature in shock tunnels for the measurement of surface heat flux [1]. The use of transient technique at lower temperature started from Russell et al. [2] and Clifford et al. [3]. They employed phase change paints to study heat transfer within gas turbine blade cooling passages. Ireland and Jones [4,5] were the first to present a transient heat transfer measurement technique using thermochronic liquid crystal (TLC) coating where they tracked the movement of a single band of liquid crystal during a transient experiment. In Ireland and Jones [4,5], the basic principles and the data reduction method for the transient heat transfer measurements were also described in detail. The normal assumptions are that the test plate is a semiinfinite solid and the transient temperature response is governed by the one-dimensional heat conduction into the model. When the material of the model has a low thermal diffusivity (e.g. Per⇑ Corresponding author. E-mail address: [email protected] (C.-l. Liu).
spex) a one-dimensional (l-D) assumption is often a good approximation, since the surface temperature response is limited to a thin layer near the surface and the lateral conduction is small [6]. Following the step of Ireland and Jones, Jones and Hippensteele [7] measured the heat transfer coefficient on a compound-curve surface in a transient wind tunnel with TLC. Metzeger et al. [8] employed the transient liquid crystal method for local heat transfer measurements on a rotating disk with jet impingement. Ekkad and Han [9] and Zehnder et al. [10] measured the heat transfer characteristics in a square two-pass channel through the transient technique with TLC. The experiments mentioned in [2–10] are classified as a two-temperature system because the thermal boundary conditions are set by a single gas temperature and the wall temperature. Transient heat transfer measurement techniques are also widely used in three-temperature systems where boundary conditions are set by the freestream temperature, wall temperature, and injection temperature, such as film cooling experiments. Because both of the heat transfer coefficient and the film cooling effectiveness g are unknown at every measurement location, at least two equations are needed to obtain the two parameters. Vedula and Metzger [11] proposed a two-test strategy for film cooling transient measurement in which two experimental tests were performed with different local fluid temperatures to determine both the heat transfer coefficient and the film cooling effectiveness. Chambers et al. [12] further developed this idea to a three-test strategy for film cooling transient measurement. And Drost et al.
0894-1777/$ - see front matter Crown Copyright Ó 2011 Published by Elsevier Inc. All rights reserved. doi:10.1016/j.expthermflusci.2011.01.021
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Nomenclature a d c D h U I q ReD T t X Y Z
square root of thermal diffusivity depth of the heat penetration (m) specific heat of wall (J/kg K) film hole inlet diameter (m) convective heat transfer coefficient (W/m2 K) velocity (m/s) momentum flux ratio ¼ qc U 2c =qg U 2g heat flux (W/m2) Reynolds number based on film hole inlet diameter (= qgUgD/lg) temperature (K) time (s) streamwise coordinate originating at the center of cooling hole exit (m) normal coordinate (m) spanwise coordinate originating at the center of cooling hole exit (m)
d k
range of the nonuniform initial wall temperature (m) thermal conductivity of wall (W/m K)
Subscripts aw adiabatic wall c jet g mainstream i initial t = 0 s surface 0 without injection 1 infinite ld long duration sd short duration TC1 measured by 1# TC10 thermocouple measured by 10# thermocouple
Greek symbols g film cooling effectiveness q density (kg/m3)
[13] performed 6–8 tests by varying the injection flow temperatures to reduce the measurement error of the results. In [11–13] a single narrow-band TLC was used. Ireland and Jones [14] further developed a double-TLC transient measurement technique which used a mixture of two carefully chosen liquid crystals to obtain the heat transfer coefficient and the film cooling effectiveness by only one test. Wide band TLC and infrared thermography method can also be used in transient heat transfer measurements to get full surface distributions of the heat transfer coefficient and the film cooling effectiveness through a single test [15,16]. Some basic methods used in the transient heat transfer measurements and the applications of the transient heat transfer measurements were reviewed by Baughn [17], Ireland and Jones [18], Ekkad and Han [19]. And the recent developments in the technique of the transient test can be found in Ireland et al. [20], Buttsworth et al. [21], Jenkins et al. [22] and Jonsson et al. [23]. The mathematical model of the transient heat transfer measurements mentioned above can be described as:
qffiffiffiffi 8 2 @Tðy;tÞ k 2 @ Tðy;tÞ > ¼ a ; a ¼ 2 > qc; y P 0; t P 0; @t @y < ; Tð0; tÞ ¼ T s ðtÞ; y ¼ 0; t ! 1 : qðtÞ ¼ h½T s ðtÞ T g ðtÞ k @Tðy¼0;tÞ > @y > : t ¼ 0; y ! 1 : T i ðyÞ ¼ T i : ð1Þ For three temperature systems, Tg(t) is the adiabatic wall temperature Taw(t). And for transient experiments without heat foil: q(t) = 0. There are four implicit assumptions in that mathematical model: h and other unknown parameters, such as film cooling effectiveness, are invariant with time; the test plate is a semi-infinite solid; the conduction process in the wall is one-dimensional; the initial wall temperature is uniform. Although h can depend on the level and the distribution of the surface temperature, these effects are always insignificant for force convection, so the first assumption is acceptable. In many cases, the second assumption is easy to achieve by making the duration of one test be sufficiently short that the thermal pulse is not affected by the thermal boundary conditions at remote surface of the test plate. Recently researchers have paid much attention to the third assumption and made great effort to consider the influence of the lateral conduction. Lin and Wang
[24] and Ling et al. [25] solved the 3D unsteady heat conduction equation to avoid the lateral conduction error. To reduce the large computation cost introduced by solving 3D unsteady heat conduction equation, Kingsley-Rowe et al. [26] introduced an approximate analysis and derived a correction parameter to consider the lateral conduction error. Von Wolfersdorf [27] further presented an analytical model to address the effect of lateral conduction in the transient film cooling measurements. For the fourth assumption, researchers tried many methods [17] to achieve a uniform initial wall temperature, such as preheating the model to a uniform temperature and exposing the model to the flow field quickly or introducing the flow by switching fast valves. Those methods made the test facilities very complicated and expensive. However, very few published works try to consider and solve the problem in the presence of nonuniform initial wall temperature except Jones and Hippensteele [7] and Liu et al. [28]. Jones and Hippensteele [7] regarded the initial wall temperature as an unknown parameter and used the double-TLC method to get the initial wall temperature and h simultaneously. However, they just considered the nonuniformity normal to the conduction direction and assumed that the wall temperature at every location was still a constant in the conduction direction. Liu et al. [28] further developed a new transient technique to consider the temperature nonuniformity in the conduction direction. The core of this new technique is an equation obtained by Laplace transformation from 1-D unsteady conduction equation with nonuniform initial wall temperature. However, Liu et al. [28] did not present how much influence the nonuniform initial temperature has on the transient measurement results. In the present paper, the authors investigate the influence of the nonuniform initial temperature in the conduction direction on the results of the transient heat transfer measurement through both the numerical and experimental methods. 2. Transient measurement theory in the presence of nonuniform initial wall temperature for film cooling measurements To give a general equation, the case of film cooling is considered. When considering the case of no secondary injection, it just needs to make the cooling effectiveness g be 0. Although some
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descriptions about the transient measurement theory in the presence of nonuniform initial wall temperature can be found in Liu et al. [28], a relatively detailed description is introduced in this section. The mathematical model of the transient film cooling measurement in the presence of nonuniform initial wall temperature is:
qffiffiffiffi 8 @Tðy;tÞ 2 Tðy;tÞ > ¼ a2 @ @y ; a ¼ qkc; y P 0; t P 0; 2 > < @t y ¼ 0; t ! 1 : h½T s ðtÞ T aw ðtÞ ¼ > > : t ¼ 0; y ! 1 : T i ðyÞ ¼ f ðyÞ:
k @Tðy¼0;tÞ ; @y
Tð0; tÞ ¼ T s ðtÞ;
where the Taw(t) stands for the adiabatic wall temperature. This value can be replaced by the film cooling effectiveness, which is defined by:
T aw ðtÞ T g ðtÞ : T c ðtÞ T g ðtÞ
ð3Þ
Usually, it is difficult to create a step change in the air temperature. To be more general, the function of Tg(t) and Tc(t) are in the form of power series in the present study,
T g ðtÞ ¼
M X m¼0
Gm
tm ; Cðm þ 1Þ
T c ðtÞ ¼
N X n¼0
Cn
tn : Cðn þ 1Þ
ð4Þ
When step change in the temperature is needed, it just needs to make Gm(m P 1) and Cn(n P 1) be 0. The key point for our analysis is the initial condition. In the former work including [11–13,15,16], the initial wall temperature was always considered as constant. The temperature profile is similar with the curve shown in Fig. 1a. But in the real experiment, the initial temperature in the region near the test surface is usually nonuniform due to the temperature difference between the test plate and the air in the test duct. The temperature profile is similar to the curve shown in Fig. 1b. In Fig. 1b, d is the extent of the nonuniform initial temperature in the wall. The inner boundary of d is defined as the location at which |(T T1)/(Ts T1)| = 0.01 . In the present paper, the nonuniform initial wall temperature is approximated by the polynomial of third order:
e þ By e 2 þ Dy e þ Cy e 3: T i ðyÞ ¼ f ðyÞ ¼ A
ð5Þ
For transient heat transfer measurements, semi-infinite test plate is assumed. But the polynomial (5) cannot describe the temperature distribution properly in the whole semi-infinite region due to its mathematical character. In most cases, including the experiment introduced in the following section, the nonuniform initial temperature usually exists in a limited extent in the test plate, and its distribution is similar with the temperature profile as shown in Fig. 1b. The polynomial (5) is sufficient to express this
0
Ti ( y )
0
Ti ( y )
δ
" # pffiffik 2n X b t E0 T s ðtÞ ¼ C n gb C 2k þ 1 n¼0 k¼0 " # pffiffik M 2m X X b t k E0 Gm ð1 gÞb2m þ C 2þ1 m¼0 k¼0 ( pffiffi 2e e e 0 þ a B ½1 E0 þ 2a C 2pffiffiffitffi 1 þ E0 þ AE 2 b p b b ) pffiffi e 6a3 D 2 t 1 E0 ; t pffiffiffiffi þ þ b b p b2 N X
ð2Þ
g¼
kind of initial wall temperature distribution in the region [0, d]. And it is also sufficient to get accurate results according to the analysis in the following sections. Eqs. (2)–(5) can be solved analytically using the Laplace transform technique. The solution is the function of wall surface temperature Ts with time t. It is also the equation to calculate h and g: 2n
ð6Þ
pffiffiffiffiffiffiffiffi pffiffi pffiffi 2 qck; E0 ¼ eb t erfc b t and erfc b t is the compffiffi e and D e C e are equal with plementary error function of b t . When B; 0, Eq. (6) regresses to expression derived from constant initial wall e temperature with the value A. where b ¼ h=
3. Numerical investigation on the influence of nonuniform initial temperature 3.1. Method For the case of film cooling, step changes in Tg(t) and Tc(t) are assumed. If the values of the following parameters are specified: the property of the wall material: q; c; k; the initial wall temperature distribution; the air temperatures: Tg and Tc; the convective heat transfer coefficient h; the film cooling effectiveness g and the time t, the corresponding wall surface temperature history Ts(t) can be calculated accurately from (2) by numerical method, such as finite difference method. So the first step of investigation is to specify the values of q; c; k, Ti(y), Tg, Tc, h, g and t to calculate Ts(t) using the numerical method. Then fit the Ti(y) with the e B; e and D. e C e At last substitute the polynomial (5) to obtain A; e B; e ; D, e C e Tg, Tc, t and Ts(t) into the Eq. (6) to values of q; c; k; A; calculate h and g. Because two equations, at least, are needed to calculate h and g, t and Tg and Tc are assigned two different values respectively to obtain two Ts(t) s with the same h and g. To show the influence of nonuniform initial wall temperature, calculations using the Eq. (7) which is derived from (2) with constant initial temperature Ti:
T s ðtÞ ¼ ð1 gÞT g ½1 E0 þ gT c ½1 E0 þ T i E0 ;
ð7Þ
are also carried out with the same q; c; k, Tg, Tc, t and Ts(t). The differences between the calculated h and g and the imposed values can show the influence of the nonuniform initial temperature. The sketch of the computational model is shown in Fig. 2. For all the calculations, wall material property q; c; k were set as 1200 Kg/ m3, 1430 J/kg K, 0.17 W/m K which are equal with perspex’s q; c; k. The extent of computation domain L is 0.04 m. And the penetration time computed from L2/16a2 is about 1000 s.
Δy Δy
y
(a)
y
Fig. 1. Two kinds of initial wall temperature.
(b)
m-1 B1
m
m+1
y B2
Fig. 2. Sketch of the computational model.
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Table 1 Combinations of h and g for the present study. (h, g) (60, 0.1) (120, 0.1) (200, 0.1)
(60, 0.4) (120, 0.4) (200, 0.4)
(60, 0.8) (120, 0.8) (200, 0.8)
Table 2 Combinations of t and Tg and Tc for the present study. (t, Tg, Tc)
The nonuniform initial wall temperature was produced by computation with the method mention above from 0 s to a certain time at a constant initial wall temperature Ti which was also the initial temperature used in Eq. (7) and a specified Taw which was carefully chosen to obtain representative initial wall temperature distributions. In the calculations, Ti was set as 295 K. And two initial wall temperature distributions which are shown in Fig. 3 were produced and employed in the calculations to show the influence of different nonuniformity degrees. Fig. 3 only shows the fitting curves in the region [0, d] where d = 0.013 and the fitting curves agree well with the initial wall temperature distributions in this region.
Short duration
(12, 315, 335) (10, 340, 320)
3.2. Results and discussion
Long duration
(220, 315, 335) (200, 340, 320)
Figs. 4–6 show the relative deviations of h and g: Dg = v|gcalculated greal|/greal and Dh = |hcalculated hreal|/hreal, which were calculated from different initial conditions and different test durations. ‘Ti-nonuni1’ or ‘Ti-nonuni2’ in the legend means the h and g were calculated using Eq. (6) with initial wall temperature distribution ‘Ti-nonuni1’ or ‘Ti-nonuni2’. ‘Ti’ in the legend means constant initial wall temperature Ti of value 295 K is employed to calculate h and g using Eq. (7) in the presence of real initial temperature distribution ‘Ti-nonuni1’ or ‘Ti-nonuni2’. ‘tld’ or ‘tsd’ in the legend means long test duration or short test duration in Table 2 is employed to calculate h and g using Eq. (7) or Eq. (6). All the three figures show that the relative deviations of h and g calculated by Eq. (6) are very small, no matter under which initial temperature distribution or whether long or short duration is employed. The maximal deviations of h and g calculated by Eq. (6) are about 4% and 2% respectively. This indicates that although polynomial (5) can only well fit the initial temperature distribution in the region [0, d], it is already sufficient to get accurate results. The reason can be obtained from the perspective of thermal resistance analysis. Change of wall surface temperature Ts with t is influenced by two heat transfer processes: convection at the wall surface and conduction in the wall. Biot number Bi ¼ hy=k is the ratio of the thermal resistance of conduction per unit area y=k to the thermal resistance of convection per unit area 1/h. The value of Bi determines which heat transfer process has more influence on the change of unsteady temperature field in the solid. When Bi < 1, conduction has more influence on the change of Ts with t. This
In the present work, Eq. (2) was discretized using fully implicit scheme: nþ1 n T nþ1 2T nþ1 T nþ1 m þ T mþ1 m Tm ; ¼ a2 m1 2 Dt Dy
ð8Þ
where Dt = 0.01 s and Dy = 0.05 mm. Discretization scheme (8) is of first-order accuracy in time and second-order accuracy in space and stable with time step. Gauss–Seidel iterative method was used to calculate the algebraic equations formed from (8) at each time step. The convergence of iteration was determined based on 7 n max T nþ1 6 10 T . m m The third kind of boundary condition, including the heat transfer coefficient h and the adiabatic wall temperature Taw which is computed from Taw = (1 g)Tg + gTc, was imposed at B1 position shown in Fig. 2. And adiabatic condition was imposed at B2 position. In the present work, h and g were assigned three typical values respectively according to the measurement results of h and g in Liu et al. [28]. So there are nine combinations of h and g (see Table 1). Two combinations of Tg and Tc were used for all the calculations. Moreover, to study the influence of the nonuniform initial temperature on the measurements with different durations, two kinds of duration: tld and tsd were studied here. The combinations of t and Tg and Tc are shown in Table 2.
300
300 calculated values and fitting curve
299
299
298
298
Ti
Ti
calculated values and fitting curve
297
297
296
296
295
295
294
0
0.01
0.02
y
Ti − nonuni1
0.03
0.04
294
0
0.01
0.02
y
Ti − nonuni 2
Fig. 3. Initial wall temperature distributions employed in the calculations.
0.03
0.04
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8
8 Ti-nonuni1 -tsd Ti-nonuni1 -tld Ti-tsd Ti-tld
6
6
5
5
4
4
3
3
2
2
1
1
0
60
80
100
120
h
140
160
180
Ti-nonuni2 -tsd Ti-nonuni2 -tld Ti-tsd Ti-tld
7
Δη×100%
Δη×100%
7
0
200
60
80
Δη = ηcalculated − 0.1 / 0.1
100
120
h
140
160
40 Ti-nonuni1 -tsd Ti-nonuni1 -tld Ti-tsd Ti-tld
Ti-nonuni2 -tsd Ti-nonuni2 -tld Ti-tsd Ti-tld 30
Δh×100%
30
Δh×100%
200
Δη = ηcalculated − 0.1 / 0.1
40
20
10
0 60
180
20
10
80
100
120
h
140
160
180
200
Δh = hcalculated − hreal / hreal
0 60
80
100
120
h
140
160
180
200
Δh = hcalculated − hreal / hreal
Fig. 4. Relative deviations of g and h calculated by different initial conditions at g = 0.1.
means only the initial wall temperature in the region [0, y] where Bi < 1 has great influence on the change of Ts with t. In the region (y, 1) where Bi > 1, the influence of initial temperature on Ts(t) is very small. So accurate results can be calculated by Eq. (6) when the polynomial (5) fits the initial temperature distribution well in the region [0, y] where Bi < 1. In the present calculations, the maximum of 1/h is 1/60, and the fitting region is [0, 0.013], and k = 0.17 W/m K. The region [0, y] where Bi < 1 is in the fitting region. So the results calculated by Eq. (6) are very accurate. Figs. 4–6 also show that the relative deviations of h and g are very large when Eq. (7) is used to calculate h and g in the condition of short test duration tsd. The maximal relative deviation of h is about 12% at the presence of Ti-nonuni1 and 36% at the presence of Ti-nonuni2. However, the relative deviations of h and g calculated by Eq. (7) in the condition of long test duration tld are very small. Especially in the case of Ti-nonuni1, they are as small as the deviations of h and g calculated by Eq. (6). That is because long test duration produced a larger heat penetration depth dld than d, and in the region (d, dld) the initial wall temperature is just Ti. Moreover, when the nonuniformity of the initial temperature is not very severe, the absolute difference between Ti and the real initial tem-
perature in the region [0, d] is not very large. So the relative deviations of h and g calculated by Eq. (7) in the condition of long duration are very small, especially at the presence of Ti-nonuni1. In the condition of short test duration, dsd is smaller than d. In the region [0, dsd], the temperature difference between Ti and the real initial temperature is large, especially when the nonuniformity degree of the initial temperature is large like Ti-nonuni2. So the h and g calculated by Eq. (7) in the condition of short test duration are inaccurate. From the above discussion we know that if the initial wall temperature distribution could be measured, accurate results can be calculated by equations which contain the information of the nonuniform initial wall temperature, such as Eq. (6). When the initial wall temperature distribution is difficult to obtain and constant initial wall temperature has to be employed to calculated h and g, some methods have to be employed to reduce the influence. Two approaches are suggested here. One is to reduce the region [0, y] where Bi < 1 by using low heat conductivity material as the test plate or/and performing the transient measurements in a high h condition. All the figures in Figs. 4–6 show that the relative deviations of h and g calculated with the same initial condition and the
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8
8
Ti-nonuni1-tsd Ti-nonuni1-tld Ti-tsd Ti-tld
6
6
5
5
4
4
3
3
2
2
1
1
0 60
80
100
120
h
140
160
180
Ti-nonuni2-tsd Ti-nonuni2-tld Ti-tsd Ti-tld
7
Δη×100%
Δη×100%
7
0
200
60
80
100
120
h
140
160
40
40 Ti-nonuni1-tsd Ti-nonuni1-tld Ti-tsd Ti-tld
Ti-nonuni2-tsd Ti-nonuni2-tld Ti-tsd Ti-tld 30
Δh×100%
30
Δh×100%
200
Δη = ηcalculated − 0.4 | / 0.4
Δη = ηcalculated − 0.4 | / 0.4
20
10
0 60
180
20
10
80
100
120
h
140
160
180
200
Δh = hcalculated − hreal / hreal
0 60
80
100
120
h
140
160
180
200
Δh = hcalculated − hreal / hreal
Fig. 5. Relative deviations of g and h calculated by different initial conditions at g = 0.4.
same duration decrease as h increases, especially when constant initial temperature is used. Also, properly increasing the transient test duration is an effective and economical method to reduce the influence of the nonuniform initial temperature when the nonuniform initial temperature is unavoidable and difficult to measure. 4. Experimental investigation on the influence of nonuniform initial temperature Numerical investigation on the influence of nonuniform initial temperature has been introduced in the above section. However, to be more convictive, experimental validation on the influence of nonuniform initial temperature is also required. This section will present and analyze the measured h and g results of cylindrical hole film cooling with different initial wall temperatures which were measured in the real experiment tests. 4.1. Experimental apparatus and procedure The experimental apparatus and procedure in the present paper are the same with those in Liu et al. [28]. Only a brief description is
present. Fig. 7 shows a schematic of the overall test setup. Mainstream passed through valves, settling chamber, contraction section, mesh heater, second contraction and then entered the test section. The heated secondary flow discharged into environment through solenoid valve before starting a test. The test plate is a perspex plate with thickness of 40 mm in Y direction shown in Fig. 8. A single layer of narrow-band liquid crystal which was used to measure the surface temperature was sprayed on the test plate surface. Configuration of the cylindrical hole row in the present study is shown in Fig. 8. The hole diameter D is 10 mm. The lateral spacing and the inclined angle of the holes is 3D and 35° respectively. The mainstream velocity was maintained at 17 m/s, the corresponding Reynolds number based on film hole inlet diameter, ReD, was 10,000. The turbulence intensity of mainstream was of the order of 2%. Two momentum flux ratios I were tested: 1 and 4. Because the temperature difference between mainstream and jet is not very large, the density ratio qc/qg is nominally equal to 1. A transient test was initiated by switching the solenoid valve and the butterfly valve simultaneously to introduce the mainstream and the heated secondary flow injection into the test section. Different tests were distinguished by varying the heating
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8
8 Ti-nonuni1-tsd Ti-nonuni1-tld Ti-tsd Ti-tld
6
6
5
5
4
4
3
3
2
2
1
1
0
60
80
100
120
h
140
160
180
Ti-nonuni2-tsd Ti-nonuni2-tld Ti-tsd Ti-tld
7
Δη×100%
Δη×100%
7
0
200
60
80
Δη = ηcalculated − 0.8 | / 0.8
100
120
h
140
160
40
Ti-nonuni1-tsd Ti-nonuni1-tld Ti-tsd Ti-tld
Ti-nonuni2-tsd Ti-nonuni2-tld Ti-tsd Ti-tld
30
30
Δh×100%
Δh×100%
200
Δη = ηcalculated − 0.8 | / 0.8
40
20
20
10
10
0 60
180
80
100
120
140
160
180
200
h
Δh = hcalculated − hreal / hreal
0 60
80
100
120
140
160
180
200
h
Δh = hcalculated − hreal / hreal
Fig. 6. Relative deviations of g and h calculated by different initial conditions at g = 0.8.
power of the mainstream and the secondary flow to produce different Tg(t) and Tc(t) which were measured by K-type thermocouples. And the time between two tests was 40 min at least to guarantee the recovery of the test plate temperature. Eq. (6) was used as the calculation formula in the present measurements. And 6–8 tests were carried out at identical momentum flux ratios. This multiple-test method is according to Drost et al. [13]. The initial wall temperature distribution Ti(y) was measured by K-type thermocouples shown in Fig. 9a. The location of this crosssection which is perpendicular to the mainstream is 180 mm from the start of the test plate. The test plate had been in the environment for 40 min at least to ensure a uniform temperature in the plate before installation. But the temperature difference between the test plate and the air in the test duct produced nonuniform initial temperature distribution in the Y direction region [0, 10 mm] during the short installation process according to the measurement. Because the test region of interest is at least 10 mm from the edge of test plate in the X, Z directions and the air temperature in the test duct was uniform, the temperature in the X–Z plane under the test region surface were considered as uniform. So the nonuniform initial temperature distribution only existed in the Y
direction of the test region, and the present initial temperature measurement method is reasonable. In every test case, the measured initial wall temperature distribution was similar with the curve in Fig. 9b, and the biggest temperature difference in the distribution was between 1 °C and 2 °C.
4.2. Results and discussion Error analysis has been performed with the method proposed by Kline and McClintock [29]. In the present experiment, the measurement uncertainties include temperature uncertainties: DTg, DTc, DTi, DTs; and time uncertainty pffiffiffiffiffiffiffiffi Dt in Tg(t), Tc(t), Ts(t); and material property uncertainty D qck. The estimated uncertainty intervals for the present experiment are: DTg = DTc = DTi = DTs = ±0.2 °C; pffiffiffiffiffiffiffiffi Dt = ±0.1 s; D qck ¼ 20. The temperature and time uncertainties in Tg(t) and Tc(t) have been assessed to be the uncertainties of the fitting coefficients in Eq. (6). According to the uncertainty intervals given above, the relative uncertainties in heat transfer coefficient is about 6%, and in local film cooling effectiveness is about 15% at g = 0.1 and 3% at g = 0.7.
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Pressure measurement station
Temperature acquisition
Computer Blower
Air tank
valve
CCD camera
Mesh heater
-
Tungsten halogen lamp
TC Test section Butterfly valve
Test plate TC
Settling Contraction + Secondary chamber section contraction
controller
Solenoid valve
Blower Air tank Flow meter
-
valve
Solenoid valve
+ Air heater
Fig. 7. Sketch of the experiment system.
Z
Y X
3D D
30mm
0
D
40mm
X 35°
Fig. 8. Sketch of the test plate and the film-cooling hole row.
27
(a)
(b) 100mm
26.5
20mm T/ oC
100mm
40mm
Y
Ti Ti0
11 1 2 3 4 5 6 7 8 9 10
26
25.5
25
0
0.002
0.004
Y/m
0.006
Fig. 9. Sketch of thermocouples’ distribution in the test plate and the measured initial temperature distribution.
0.008
0.01
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I =1
I =4
I =1
I =4
Fig. 10. Laterally averaged g and h calculated with different initial wall temperatures.
Three kinds of initial temperature distributions were used to calculate the film cooling effectiveness and the heat transfer coefficient: one is the real nonuniform initial wall temperature marked as ‘‘Ti-real’’, one is the temperature measured by the 1# thermocouple marked as ‘‘Ti-TC1’’, and one is the temperature measured by the 10# thermocouple marked as ‘‘Ti-TC10’’. Fig. 10 shows the laterally averaged results of film cooling effectiveness g and heat transfer coefficient h. Relative to the results of ‘‘Ti-real’’, the cooling effectiveness results of ‘‘Ti-TC1’’ are lower by about 10% under I = 1 and about 20% under I = 4, and the heat transfer coefficient results of ‘‘Ti-TC1’’ are lower by about 10% under both momentum ratios. However, the cooling effectiveness results of ‘‘Ti-TC10’’ are higher by about 7% under I = 1 and about 10% under I = 4, and the heat transfer coefficient results of ‘‘Ti-TC10’’ are higher by about 10% under both momentum ratios. The above deviations are explained as the following. The temperature of ‘‘TiTC1’’ is higher than the real initial wall temperature. The liquid crystal color-change time should be shorter under ‘‘Ti-TC1’’. Using the real but relatively long color-change time in the calculation makes the calculated convective heat flux smaller than the real.
And thus the results of cooling effectiveness and heat transfer coefficient are lower accordingly. Similar explanation can be drawn to the results of ‘‘Ti-TC10’’ which are relatively higher. Actually, similar with the experimental results of ‘‘Ti-TC10’’, the numerical results of Dg = (gcalculated greal)/greal and Dh = (hcalculated hreal)/hreal calculated with constant initial temperature Ti in Section 3.2 are all positive because the value of Ti is lower than the real initial wall temperature. In Fig. 10, film cooling effectiveness results were compared with the data of Goldstein et al. [30]. In Goldstein et al. [30], the film cooling effectiveness of cylindrical hole row, which has the same or similar geometry parameters with the current hole row, has been measured using mass analogy method under similar operating conditions. Although the deviation between the ‘‘Ti-real’’ results and the Goldstein et al. [30] data is on the same order as the deviation between the ‘‘Ti-real’’ results and the ‘‘Ti-TC1’’ results, the ‘‘Ti-real’’ results have relatively better agreement in authors’ opinion. These results show that obtaining accurate initial temperature distribution is very important for transient heat transfer measurement of film cooling. Small overestimate or underesti-
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Fig. 11. Local g and h calculated with different initial wall temperatures under I = 1 at X/D = 20.
mate of the initial temperature can make the experiment results deviate from the true values badly. Moreover, from Fig. 10 we also find that the deviation caused by the inaccurate initial wall temperature is large in the downstream region and small in the upstream region under I = 4 case. Under large momentum ratio case, the large-flux jet lifts off the wall immediately after ejecting and reattaches on the wall in the downstream region. Higher jet temperature in the experiment makes the change of liquid crystal color on the downstream wall more quickly than that on the upstream wall. So the test duration time in the downstream region is shorter than the upstream region. According to the analysis in Section 3.2, the influence of inaccurate initial temperature could be larger on the experiment results in the downstream region. This also provides experimental validation for the numerical investigation in Section 3. For the I = 1 case, because the jet flux is relatively small, the liquid crystal color-change time in most of downstream region is relatively long. So the deviations keep almost the same order in the downstream region. Figs. 11 and 12 also show the local spanwise distributions of g and h at X/D = 20 position for both momentum ratios, which were calculated by the three different initial wall temperature distributions respectively. In the streamwise midspan region where the liquid crystal colorchange time is relatively long and close, the deviations keep almost the same order in the whole spanwise region.
Fig. 12. Local g and h calculated with different initial wall temperatures under I = 4 at X/D = 20.
5. Conclusion A new equation used to calculate h and g in the transient heat transfer measurement was derived by Laplace transformation from 1-D unsteady conduction model with nonuniform initial wall temperature in the conduction direction. Both numerical and experimental methods were performed to investigate the influence of the nonuniform initial temperature on the transient measurement results of film cooling. The results show that accurate initial temperature distribution is very important for transient heat transfer measurement. Small overestimate or underestimate of the initial temperature by about 1–2 °C makes the experiment results deviate from the true values more than 10%. Using initial wall temperature higher than the real values causes the results of h and g lower than the true values. And using initial wall temperature lower than the real values causes the results higher than the true values. When the nonuniform initial temperature distribution is well fitted in the region where the conduction thermal resistance is smaller than the convection thermal resistance, accurate h and g can be calculated by the newly derived equation. If the initial temperature distribution is difficult to obtain and constant initial temperature has to be employed, the following methods are suggested to reduce the influence of nonuniform initial temperature: using material with
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low heat conductivity as the test plate, performing the transient measurements in a high h condition as well as properly increasing the transient test duration. Acknowledgments This work is sponsored by the National Basic Research Program of China (China 973 Program) under number of 2007CB707701. References [1] D.L. Schultz, T.V. Jones, Heat Transfer Measurements in Short Duration Hypersonic Facilities, AGARD NO. 165, 1985. [2] C.M.B. Russell, T.V. Jones, G.H. Lee, Heat transfer enhancement using vortex generators, in: Proceeding of International Heat Transfer Conference, Munich, 1982. [3] R.J. Clifford, T.V. Jones, S.T. Dunne, Techniques for Obtaining Detailed Heat Transfer Coefficient Measurements within Gas Turbine Blade Vane Cooling Passages, ASME Paper No. 83-GT-58, 1983. [4] P.T. Ireland, T.V. Jones, The measurement of local heat transfer coefficients in blade cooling geometries, in: AGARD Conf. on Heat Transfer and Cooling in Gas Turbines CP 390, paper 28, 1985. [5] P.T. Ireland, T.V. Jones, Detailed measurements of heat transfer on and around a pedestal in fully-developed channel flow, in: International Heat Transfer Conference, San Francisco, 1986. [6] R.P. Vedula, D.E. Metzger, W.B. Bickford, Effects of lateral and anisotropic conduction on determination of local convection heat transfer characteristics with transient tests and surface coatings, in: Winter Annual Meeting of ASME, Chicago, 1988, HTD-I, pp. 21–27. [7] T.V. Jones, S.A. Hippensteele, High-resolution heat-transfer-coefficient maps applicable to compound-curve surface using liquid crystal in a transient wind tunnel, NASA Technical Memorandum 89855, 1988. [8] D.E. Metzeger, R.S. Bunker, G. Bosch, Transient liquid crystal measurement of local heat transfer on a rotating disk with jet impingement, ASME Journal of Turbomachinery 113 (1991) 52–59. [9] S.V. Ekkad, J.-C. Han, Detailed heat transfer distributions in two-pass square channels with Rib turbulators, International Journal of Heat and Mass Transfer 40 (1997) 2525–2537. [10] F. Zehnder, M. Schüler, B. Weigand, J. von Wolfersdorf, S.O. Neumann, The Effect of Turning Vanes on Pressure Loss and Heat Transfer of a Ribbed Rectangular Two-Pass Internal Cooling Channel, ASME Paper No. GT200959482, 2009. [11] R.J. Vedula, D.E. Metzger, A Method for the Simultaneous Determination of Local Effectiveness and Heat Transfer Distributions in Three Temperature Convection Situations, ASME paper No. 91-GT-345, 1991. [12] A.C. Chambers, D.R.H. Gillespie, P.T. Ireland, A novel transient liquid crystal technique to determine heat transfer coefficient distributions and adiabatic wall temperature in a three temperature problem, ASME Journal of Turbomachinery 125 (2003) 538–546.
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[13] U. Drost, A. Bolcs, A. Hoffs, Utilization of the Transient Liquid Crystal Technique for Film Cooling Effectiveness and Heat Transfer Investigations on a Flat Plane and a Turbine Airfoil, ASME Paper No. 97-GT-026, 1997. [14] P.T. Ireland, T.V. Jones, Note on the Double Crystal Method of Measuring Heat Transfer Coefficient, OUEL Report 1710/87, 1987. [15] D.N. Licu, M.J. Findlay, I.S. Gartshore, M. Salcudean, Transient heat transfer measurements using a single wide-band liquid crystal test, ASME Journal of Turbomachinery 122 (2000) 546–552. [16] S.V. Ekkad, S. Ou, R.B. Rivir, A transient infrared thermography method for simultaneous film cooling effectiveness and heat transfer coefficient measurements from a single test, ASME Journal of Turbomachinery 126 (2004) 597–603. [17] J.W. Baughn, Liquid crystal methods for studying turbulent heat transfer, International Journal of Heat Fluid Flow 16 (1995) 365–375. [18] P.T. Ireland, T.V. Jones, Liquid crystal measurements of heat transfer and surface shear stress, Measurement Science Technology 11 (2000) 969–986. [19] S.V. Ekkad, J.C. Han, A transient liquid crystal thermography technique for gas turbine heat transfer measurements, Measurement Science Technology 11 (2000) 957–968. [20] P.T. Ireland, A.J. Neely, D.R.H. Gillespie, A.J. Robertson, Turbulent heat transfer measurements using liquid crystals, International Journal of Heat Fluid Flow 20 (1999) 355–367. [21] D.R. Buttsworth, R. Stevens, C.R. Stone, Eroding ribbon thermocouples: impulse response and transient heat flux analysis, Measurement Science Technology 16 (2005) 1487–1494. [22] S. Jenkins, J. von Wolfersdorf, B. Weigand, T. Roediger, H. Knauss, E. Kraemer, Time-resolved heat transfer measurements on the tip wall of a ribbed channel using a novel heat flux sensor—part II: heat transfer, ASME Journal of Turbomachinery 130 (2008) 011019. [23] M. Jonsson, D. Charbonnier, P. Ott, J. von Wolfersdorf, Application of the Transient Heater Foil Technique for Heat Transfer and Film Cooling Effectiveness Measurements on a Turbine Vane Endwall, ASME Paper No. GT2008-50451, 2008. [24] M. Lin, T. Wang, A transient liquid crystal method using a 3-D inverse transient conduction scheme, International Journal of Heat Mass Transfer 45 (2002) 3491–3501. [25] J.P.C.W. Ling, P.T. Ireland, L. Turner, A technique for processing transient heat transfer, liquid crystal experiments in the presence of lateral conduction, ASME Journal of Turbomachinery 126 (2004) 247–258. [26] J.R. Kingsley-Rowe, G.D. Lock, J.M. Owen, Transient heat transfer measurements using thermochromic liquid crystal: lateral-conduction error, International Journal of Heat Fluid Flow 26 (2005) 256–263. [27] J. von Wolfersdorf, Influence of lateral conduction due to flow temperature variations in transient heat transfer measurements, International Journal of Heat and Mass Transfer 50 (2007) 1122–1127. [28] C.L. Liu, H.R. Zhu, J.T. Bai, D.C. Xu, Experimental research on the thermal performance of converging slot holes with different divergence angles, Experimental Thermal and Fluid Science 33 (2009) 808–817. [29] S.J. Kline, F.A. McClintock, Describing uncertainties in single-sample experiments, Journal of Mechanical Engineering 75 (1953) 3–8. [30] R.J. Goldstein, P. Jin, R.L. Olson, Film cooling effectiveness and mass/heat transfer coefficient downstream of one row of discrete holes, ASME Journal of Turbomachinary 121 (1999) 225–232.
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Experimental Thermal and Fluid Science journal homepage: www.elsevier.com/locate/etfs
Investigation on the influence of nonuniform initial temperature on the transient heat transfer measurement of film cooling Cun-liang Liu ⇑, Hui-ren Zhu, Jiang-tao Bai, Zong-wei Zhang, Xia Zhang School of Power and Energy, Northwestern Polytechnical Univ., Xi’an 710072, China
a r t i c l e
i n f o
Article history: Received 7 May 2010 Received in revised form 30 January 2011 Accepted 31 January 2011 Available online 3 April 2011 Keywords: Transient heat transfer measurement Nonuniform initial temperature Film cooling Measurement deviation
a b s t r a c t Numerical and experimental investigations on the influence of nonuniform initial temperature on the transient heat transfer measurements are presented in this paper. The case of film cooling is investigated. When the initial wall temperature is nonuniform, the results of heat transfer coefficient and film cooling effectiveness, which are calculated by the equations derived with constant initial temperature, could deviate from the true values badly, especially in the condition of short test duration. Using initial wall temperature which is higher than the real values causes the results of heat transfer coefficient and film cooling effectiveness lower than the true values. And lower initial wall temperature produces higher results of heat transfer coefficient and film cooling effectiveness. However, when the initial temperature distribution in the region where conduction plays more influence on the wall surface temperature than the convection is well fitted by the cubic polynomial, accurate results can be obtained by the new equation which is derived from 1-D unsteady conduction model with nonuniform initial wall temperature. Some suggestions are also introduced to reduce the influence of nonuniform initial temperature when the initial temperature distribution is difficult to obtain and the equation derived from constant initial temperature has to be employed. Crown Copyright Ó 2011 Published by Elsevier Inc. All rights reserved.
1. Introduction Nowadays there are mainly two types of heat transfer measurement technique: steady state measurement technique and transient measurement technique. Transient heat transfer measurements are more and more employed by researchers because of short experiment duration which is able to reduce the experiment cost and workload, and is required in short-duration test facilities. At the beginning, it was used at high temperature in shock tunnels for the measurement of surface heat flux [1]. The use of transient technique at lower temperature started from Russell et al. [2] and Clifford et al. [3]. They employed phase change paints to study heat transfer within gas turbine blade cooling passages. Ireland and Jones [4,5] were the first to present a transient heat transfer measurement technique using thermochronic liquid crystal (TLC) coating where they tracked the movement of a single band of liquid crystal during a transient experiment. In Ireland and Jones [4,5], the basic principles and the data reduction method for the transient heat transfer measurements were also described in detail. The normal assumptions are that the test plate is a semiinfinite solid and the transient temperature response is governed by the one-dimensional heat conduction into the model. When the material of the model has a low thermal diffusivity (e.g. Per⇑ Corresponding author. E-mail address: [email protected] (C.-l. Liu).
spex) a one-dimensional (l-D) assumption is often a good approximation, since the surface temperature response is limited to a thin layer near the surface and the lateral conduction is small [6]. Following the step of Ireland and Jones, Jones and Hippensteele [7] measured the heat transfer coefficient on a compound-curve surface in a transient wind tunnel with TLC. Metzeger et al. [8] employed the transient liquid crystal method for local heat transfer measurements on a rotating disk with jet impingement. Ekkad and Han [9] and Zehnder et al. [10] measured the heat transfer characteristics in a square two-pass channel through the transient technique with TLC. The experiments mentioned in [2–10] are classified as a two-temperature system because the thermal boundary conditions are set by a single gas temperature and the wall temperature. Transient heat transfer measurement techniques are also widely used in three-temperature systems where boundary conditions are set by the freestream temperature, wall temperature, and injection temperature, such as film cooling experiments. Because both of the heat transfer coefficient and the film cooling effectiveness g are unknown at every measurement location, at least two equations are needed to obtain the two parameters. Vedula and Metzger [11] proposed a two-test strategy for film cooling transient measurement in which two experimental tests were performed with different local fluid temperatures to determine both the heat transfer coefficient and the film cooling effectiveness. Chambers et al. [12] further developed this idea to a three-test strategy for film cooling transient measurement. And Drost et al.
0894-1777/$ - see front matter Crown Copyright Ó 2011 Published by Elsevier Inc. All rights reserved. doi:10.1016/j.expthermflusci.2011.01.021
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Nomenclature a d c D h U I q ReD T t X Y Z
square root of thermal diffusivity depth of the heat penetration (m) specific heat of wall (J/kg K) film hole inlet diameter (m) convective heat transfer coefficient (W/m2 K) velocity (m/s) momentum flux ratio ¼ qc U 2c =qg U 2g heat flux (W/m2) Reynolds number based on film hole inlet diameter (= qgUgD/lg) temperature (K) time (s) streamwise coordinate originating at the center of cooling hole exit (m) normal coordinate (m) spanwise coordinate originating at the center of cooling hole exit (m)
d k
range of the nonuniform initial wall temperature (m) thermal conductivity of wall (W/m K)
Subscripts aw adiabatic wall c jet g mainstream i initial t = 0 s surface 0 without injection 1 infinite ld long duration sd short duration TC1 measured by 1# TC10 thermocouple measured by 10# thermocouple
Greek symbols g film cooling effectiveness q density (kg/m3)
[13] performed 6–8 tests by varying the injection flow temperatures to reduce the measurement error of the results. In [11–13] a single narrow-band TLC was used. Ireland and Jones [14] further developed a double-TLC transient measurement technique which used a mixture of two carefully chosen liquid crystals to obtain the heat transfer coefficient and the film cooling effectiveness by only one test. Wide band TLC and infrared thermography method can also be used in transient heat transfer measurements to get full surface distributions of the heat transfer coefficient and the film cooling effectiveness through a single test [15,16]. Some basic methods used in the transient heat transfer measurements and the applications of the transient heat transfer measurements were reviewed by Baughn [17], Ireland and Jones [18], Ekkad and Han [19]. And the recent developments in the technique of the transient test can be found in Ireland et al. [20], Buttsworth et al. [21], Jenkins et al. [22] and Jonsson et al. [23]. The mathematical model of the transient heat transfer measurements mentioned above can be described as:
qffiffiffiffi 8 2 @Tðy;tÞ k 2 @ Tðy;tÞ > ¼ a ; a ¼ 2 > qc; y P 0; t P 0; @t @y < ; Tð0; tÞ ¼ T s ðtÞ; y ¼ 0; t ! 1 : qðtÞ ¼ h½T s ðtÞ T g ðtÞ k @Tðy¼0;tÞ > @y > : t ¼ 0; y ! 1 : T i ðyÞ ¼ T i : ð1Þ For three temperature systems, Tg(t) is the adiabatic wall temperature Taw(t). And for transient experiments without heat foil: q(t) = 0. There are four implicit assumptions in that mathematical model: h and other unknown parameters, such as film cooling effectiveness, are invariant with time; the test plate is a semi-infinite solid; the conduction process in the wall is one-dimensional; the initial wall temperature is uniform. Although h can depend on the level and the distribution of the surface temperature, these effects are always insignificant for force convection, so the first assumption is acceptable. In many cases, the second assumption is easy to achieve by making the duration of one test be sufficiently short that the thermal pulse is not affected by the thermal boundary conditions at remote surface of the test plate. Recently researchers have paid much attention to the third assumption and made great effort to consider the influence of the lateral conduction. Lin and Wang
[24] and Ling et al. [25] solved the 3D unsteady heat conduction equation to avoid the lateral conduction error. To reduce the large computation cost introduced by solving 3D unsteady heat conduction equation, Kingsley-Rowe et al. [26] introduced an approximate analysis and derived a correction parameter to consider the lateral conduction error. Von Wolfersdorf [27] further presented an analytical model to address the effect of lateral conduction in the transient film cooling measurements. For the fourth assumption, researchers tried many methods [17] to achieve a uniform initial wall temperature, such as preheating the model to a uniform temperature and exposing the model to the flow field quickly or introducing the flow by switching fast valves. Those methods made the test facilities very complicated and expensive. However, very few published works try to consider and solve the problem in the presence of nonuniform initial wall temperature except Jones and Hippensteele [7] and Liu et al. [28]. Jones and Hippensteele [7] regarded the initial wall temperature as an unknown parameter and used the double-TLC method to get the initial wall temperature and h simultaneously. However, they just considered the nonuniformity normal to the conduction direction and assumed that the wall temperature at every location was still a constant in the conduction direction. Liu et al. [28] further developed a new transient technique to consider the temperature nonuniformity in the conduction direction. The core of this new technique is an equation obtained by Laplace transformation from 1-D unsteady conduction equation with nonuniform initial wall temperature. However, Liu et al. [28] did not present how much influence the nonuniform initial temperature has on the transient measurement results. In the present paper, the authors investigate the influence of the nonuniform initial temperature in the conduction direction on the results of the transient heat transfer measurement through both the numerical and experimental methods. 2. Transient measurement theory in the presence of nonuniform initial wall temperature for film cooling measurements To give a general equation, the case of film cooling is considered. When considering the case of no secondary injection, it just needs to make the cooling effectiveness g be 0. Although some
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descriptions about the transient measurement theory in the presence of nonuniform initial wall temperature can be found in Liu et al. [28], a relatively detailed description is introduced in this section. The mathematical model of the transient film cooling measurement in the presence of nonuniform initial wall temperature is:
qffiffiffiffi 8 @Tðy;tÞ 2 Tðy;tÞ > ¼ a2 @ @y ; a ¼ qkc; y P 0; t P 0; 2 > < @t y ¼ 0; t ! 1 : h½T s ðtÞ T aw ðtÞ ¼ > > : t ¼ 0; y ! 1 : T i ðyÞ ¼ f ðyÞ:
k @Tðy¼0;tÞ ; @y
Tð0; tÞ ¼ T s ðtÞ;
where the Taw(t) stands for the adiabatic wall temperature. This value can be replaced by the film cooling effectiveness, which is defined by:
T aw ðtÞ T g ðtÞ : T c ðtÞ T g ðtÞ
ð3Þ
Usually, it is difficult to create a step change in the air temperature. To be more general, the function of Tg(t) and Tc(t) are in the form of power series in the present study,
T g ðtÞ ¼
M X m¼0
Gm
tm ; Cðm þ 1Þ
T c ðtÞ ¼
N X n¼0
Cn
tn : Cðn þ 1Þ
ð4Þ
When step change in the temperature is needed, it just needs to make Gm(m P 1) and Cn(n P 1) be 0. The key point for our analysis is the initial condition. In the former work including [11–13,15,16], the initial wall temperature was always considered as constant. The temperature profile is similar with the curve shown in Fig. 1a. But in the real experiment, the initial temperature in the region near the test surface is usually nonuniform due to the temperature difference between the test plate and the air in the test duct. The temperature profile is similar to the curve shown in Fig. 1b. In Fig. 1b, d is the extent of the nonuniform initial temperature in the wall. The inner boundary of d is defined as the location at which |(T T1)/(Ts T1)| = 0.01 . In the present paper, the nonuniform initial wall temperature is approximated by the polynomial of third order:
e þ By e 2 þ Dy e þ Cy e 3: T i ðyÞ ¼ f ðyÞ ¼ A
ð5Þ
For transient heat transfer measurements, semi-infinite test plate is assumed. But the polynomial (5) cannot describe the temperature distribution properly in the whole semi-infinite region due to its mathematical character. In most cases, including the experiment introduced in the following section, the nonuniform initial temperature usually exists in a limited extent in the test plate, and its distribution is similar with the temperature profile as shown in Fig. 1b. The polynomial (5) is sufficient to express this
0
Ti ( y )
0
Ti ( y )
δ
" # pffiffik 2n X b t E0 T s ðtÞ ¼ C n gb C 2k þ 1 n¼0 k¼0 " # pffiffik M 2m X X b t k E0 Gm ð1 gÞb2m þ C 2þ1 m¼0 k¼0 ( pffiffi 2e e e 0 þ a B ½1 E0 þ 2a C 2pffiffiffitffi 1 þ E0 þ AE 2 b p b b ) pffiffi e 6a3 D 2 t 1 E0 ; t pffiffiffiffi þ þ b b p b2 N X
ð2Þ
g¼
kind of initial wall temperature distribution in the region [0, d]. And it is also sufficient to get accurate results according to the analysis in the following sections. Eqs. (2)–(5) can be solved analytically using the Laplace transform technique. The solution is the function of wall surface temperature Ts with time t. It is also the equation to calculate h and g: 2n
ð6Þ
pffiffiffiffiffiffiffiffi pffiffi pffiffi 2 qck; E0 ¼ eb t erfc b t and erfc b t is the compffiffi e and D e C e are equal with plementary error function of b t . When B; 0, Eq. (6) regresses to expression derived from constant initial wall e temperature with the value A. where b ¼ h=
3. Numerical investigation on the influence of nonuniform initial temperature 3.1. Method For the case of film cooling, step changes in Tg(t) and Tc(t) are assumed. If the values of the following parameters are specified: the property of the wall material: q; c; k; the initial wall temperature distribution; the air temperatures: Tg and Tc; the convective heat transfer coefficient h; the film cooling effectiveness g and the time t, the corresponding wall surface temperature history Ts(t) can be calculated accurately from (2) by numerical method, such as finite difference method. So the first step of investigation is to specify the values of q; c; k, Ti(y), Tg, Tc, h, g and t to calculate Ts(t) using the numerical method. Then fit the Ti(y) with the e B; e and D. e C e At last substitute the polynomial (5) to obtain A; e B; e ; D, e C e Tg, Tc, t and Ts(t) into the Eq. (6) to values of q; c; k; A; calculate h and g. Because two equations, at least, are needed to calculate h and g, t and Tg and Tc are assigned two different values respectively to obtain two Ts(t) s with the same h and g. To show the influence of nonuniform initial wall temperature, calculations using the Eq. (7) which is derived from (2) with constant initial temperature Ti:
T s ðtÞ ¼ ð1 gÞT g ½1 E0 þ gT c ½1 E0 þ T i E0 ;
ð7Þ
are also carried out with the same q; c; k, Tg, Tc, t and Ts(t). The differences between the calculated h and g and the imposed values can show the influence of the nonuniform initial temperature. The sketch of the computational model is shown in Fig. 2. For all the calculations, wall material property q; c; k were set as 1200 Kg/ m3, 1430 J/kg K, 0.17 W/m K which are equal with perspex’s q; c; k. The extent of computation domain L is 0.04 m. And the penetration time computed from L2/16a2 is about 1000 s.
Δy Δy
y
(a)
y
Fig. 1. Two kinds of initial wall temperature.
(b)
m-1 B1
m
m+1
y B2
Fig. 2. Sketch of the computational model.
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Table 1 Combinations of h and g for the present study. (h, g) (60, 0.1) (120, 0.1) (200, 0.1)
(60, 0.4) (120, 0.4) (200, 0.4)
(60, 0.8) (120, 0.8) (200, 0.8)
Table 2 Combinations of t and Tg and Tc for the present study. (t, Tg, Tc)
The nonuniform initial wall temperature was produced by computation with the method mention above from 0 s to a certain time at a constant initial wall temperature Ti which was also the initial temperature used in Eq. (7) and a specified Taw which was carefully chosen to obtain representative initial wall temperature distributions. In the calculations, Ti was set as 295 K. And two initial wall temperature distributions which are shown in Fig. 3 were produced and employed in the calculations to show the influence of different nonuniformity degrees. Fig. 3 only shows the fitting curves in the region [0, d] where d = 0.013 and the fitting curves agree well with the initial wall temperature distributions in this region.
Short duration
(12, 315, 335) (10, 340, 320)
3.2. Results and discussion
Long duration
(220, 315, 335) (200, 340, 320)
Figs. 4–6 show the relative deviations of h and g: Dg = v|gcalculated greal|/greal and Dh = |hcalculated hreal|/hreal, which were calculated from different initial conditions and different test durations. ‘Ti-nonuni1’ or ‘Ti-nonuni2’ in the legend means the h and g were calculated using Eq. (6) with initial wall temperature distribution ‘Ti-nonuni1’ or ‘Ti-nonuni2’. ‘Ti’ in the legend means constant initial wall temperature Ti of value 295 K is employed to calculate h and g using Eq. (7) in the presence of real initial temperature distribution ‘Ti-nonuni1’ or ‘Ti-nonuni2’. ‘tld’ or ‘tsd’ in the legend means long test duration or short test duration in Table 2 is employed to calculate h and g using Eq. (7) or Eq. (6). All the three figures show that the relative deviations of h and g calculated by Eq. (6) are very small, no matter under which initial temperature distribution or whether long or short duration is employed. The maximal deviations of h and g calculated by Eq. (6) are about 4% and 2% respectively. This indicates that although polynomial (5) can only well fit the initial temperature distribution in the region [0, d], it is already sufficient to get accurate results. The reason can be obtained from the perspective of thermal resistance analysis. Change of wall surface temperature Ts with t is influenced by two heat transfer processes: convection at the wall surface and conduction in the wall. Biot number Bi ¼ hy=k is the ratio of the thermal resistance of conduction per unit area y=k to the thermal resistance of convection per unit area 1/h. The value of Bi determines which heat transfer process has more influence on the change of unsteady temperature field in the solid. When Bi < 1, conduction has more influence on the change of Ts with t. This
In the present work, Eq. (2) was discretized using fully implicit scheme: nþ1 n T nþ1 2T nþ1 T nþ1 m þ T mþ1 m Tm ; ¼ a2 m1 2 Dt Dy
ð8Þ
where Dt = 0.01 s and Dy = 0.05 mm. Discretization scheme (8) is of first-order accuracy in time and second-order accuracy in space and stable with time step. Gauss–Seidel iterative method was used to calculate the algebraic equations formed from (8) at each time step. The convergence of iteration was determined based on 7 n max T nþ1 6 10 T . m m The third kind of boundary condition, including the heat transfer coefficient h and the adiabatic wall temperature Taw which is computed from Taw = (1 g)Tg + gTc, was imposed at B1 position shown in Fig. 2. And adiabatic condition was imposed at B2 position. In the present work, h and g were assigned three typical values respectively according to the measurement results of h and g in Liu et al. [28]. So there are nine combinations of h and g (see Table 1). Two combinations of Tg and Tc were used for all the calculations. Moreover, to study the influence of the nonuniform initial temperature on the measurements with different durations, two kinds of duration: tld and tsd were studied here. The combinations of t and Tg and Tc are shown in Table 2.
300
300 calculated values and fitting curve
299
299
298
298
Ti
Ti
calculated values and fitting curve
297
297
296
296
295
295
294
0
0.01
0.02
y
Ti − nonuni1
0.03
0.04
294
0
0.01
0.02
y
Ti − nonuni 2
Fig. 3. Initial wall temperature distributions employed in the calculations.
0.03
0.04
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8
8 Ti-nonuni1 -tsd Ti-nonuni1 -tld Ti-tsd Ti-tld
6
6
5
5
4
4
3
3
2
2
1
1
0
60
80
100
120
h
140
160
180
Ti-nonuni2 -tsd Ti-nonuni2 -tld Ti-tsd Ti-tld
7
Δη×100%
Δη×100%
7
0
200
60
80
Δη = ηcalculated − 0.1 / 0.1
100
120
h
140
160
40 Ti-nonuni1 -tsd Ti-nonuni1 -tld Ti-tsd Ti-tld
Ti-nonuni2 -tsd Ti-nonuni2 -tld Ti-tsd Ti-tld 30
Δh×100%
30
Δh×100%
200
Δη = ηcalculated − 0.1 / 0.1
40
20
10
0 60
180
20
10
80
100
120
h
140
160
180
200
Δh = hcalculated − hreal / hreal
0 60
80
100
120
h
140
160
180
200
Δh = hcalculated − hreal / hreal
Fig. 4. Relative deviations of g and h calculated by different initial conditions at g = 0.1.
means only the initial wall temperature in the region [0, y] where Bi < 1 has great influence on the change of Ts with t. In the region (y, 1) where Bi > 1, the influence of initial temperature on Ts(t) is very small. So accurate results can be calculated by Eq. (6) when the polynomial (5) fits the initial temperature distribution well in the region [0, y] where Bi < 1. In the present calculations, the maximum of 1/h is 1/60, and the fitting region is [0, 0.013], and k = 0.17 W/m K. The region [0, y] where Bi < 1 is in the fitting region. So the results calculated by Eq. (6) are very accurate. Figs. 4–6 also show that the relative deviations of h and g are very large when Eq. (7) is used to calculate h and g in the condition of short test duration tsd. The maximal relative deviation of h is about 12% at the presence of Ti-nonuni1 and 36% at the presence of Ti-nonuni2. However, the relative deviations of h and g calculated by Eq. (7) in the condition of long test duration tld are very small. Especially in the case of Ti-nonuni1, they are as small as the deviations of h and g calculated by Eq. (6). That is because long test duration produced a larger heat penetration depth dld than d, and in the region (d, dld) the initial wall temperature is just Ti. Moreover, when the nonuniformity of the initial temperature is not very severe, the absolute difference between Ti and the real initial tem-
perature in the region [0, d] is not very large. So the relative deviations of h and g calculated by Eq. (7) in the condition of long duration are very small, especially at the presence of Ti-nonuni1. In the condition of short test duration, dsd is smaller than d. In the region [0, dsd], the temperature difference between Ti and the real initial temperature is large, especially when the nonuniformity degree of the initial temperature is large like Ti-nonuni2. So the h and g calculated by Eq. (7) in the condition of short test duration are inaccurate. From the above discussion we know that if the initial wall temperature distribution could be measured, accurate results can be calculated by equations which contain the information of the nonuniform initial wall temperature, such as Eq. (6). When the initial wall temperature distribution is difficult to obtain and constant initial wall temperature has to be employed to calculated h and g, some methods have to be employed to reduce the influence. Two approaches are suggested here. One is to reduce the region [0, y] where Bi < 1 by using low heat conductivity material as the test plate or/and performing the transient measurements in a high h condition. All the figures in Figs. 4–6 show that the relative deviations of h and g calculated with the same initial condition and the
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8
8
Ti-nonuni1-tsd Ti-nonuni1-tld Ti-tsd Ti-tld
6
6
5
5
4
4
3
3
2
2
1
1
0 60
80
100
120
h
140
160
180
Ti-nonuni2-tsd Ti-nonuni2-tld Ti-tsd Ti-tld
7
Δη×100%
Δη×100%
7
0
200
60
80
100
120
h
140
160
40
40 Ti-nonuni1-tsd Ti-nonuni1-tld Ti-tsd Ti-tld
Ti-nonuni2-tsd Ti-nonuni2-tld Ti-tsd Ti-tld 30
Δh×100%
30
Δh×100%
200
Δη = ηcalculated − 0.4 | / 0.4
Δη = ηcalculated − 0.4 | / 0.4
20
10
0 60
180
20
10
80
100
120
h
140
160
180
200
Δh = hcalculated − hreal / hreal
0 60
80
100
120
h
140
160
180
200
Δh = hcalculated − hreal / hreal
Fig. 5. Relative deviations of g and h calculated by different initial conditions at g = 0.4.
same duration decrease as h increases, especially when constant initial temperature is used. Also, properly increasing the transient test duration is an effective and economical method to reduce the influence of the nonuniform initial temperature when the nonuniform initial temperature is unavoidable and difficult to measure. 4. Experimental investigation on the influence of nonuniform initial temperature Numerical investigation on the influence of nonuniform initial temperature has been introduced in the above section. However, to be more convictive, experimental validation on the influence of nonuniform initial temperature is also required. This section will present and analyze the measured h and g results of cylindrical hole film cooling with different initial wall temperatures which were measured in the real experiment tests. 4.1. Experimental apparatus and procedure The experimental apparatus and procedure in the present paper are the same with those in Liu et al. [28]. Only a brief description is
present. Fig. 7 shows a schematic of the overall test setup. Mainstream passed through valves, settling chamber, contraction section, mesh heater, second contraction and then entered the test section. The heated secondary flow discharged into environment through solenoid valve before starting a test. The test plate is a perspex plate with thickness of 40 mm in Y direction shown in Fig. 8. A single layer of narrow-band liquid crystal which was used to measure the surface temperature was sprayed on the test plate surface. Configuration of the cylindrical hole row in the present study is shown in Fig. 8. The hole diameter D is 10 mm. The lateral spacing and the inclined angle of the holes is 3D and 35° respectively. The mainstream velocity was maintained at 17 m/s, the corresponding Reynolds number based on film hole inlet diameter, ReD, was 10,000. The turbulence intensity of mainstream was of the order of 2%. Two momentum flux ratios I were tested: 1 and 4. Because the temperature difference between mainstream and jet is not very large, the density ratio qc/qg is nominally equal to 1. A transient test was initiated by switching the solenoid valve and the butterfly valve simultaneously to introduce the mainstream and the heated secondary flow injection into the test section. Different tests were distinguished by varying the heating
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8
8 Ti-nonuni1-tsd Ti-nonuni1-tld Ti-tsd Ti-tld
6
6
5
5
4
4
3
3
2
2
1
1
0
60
80
100
120
h
140
160
180
Ti-nonuni2-tsd Ti-nonuni2-tld Ti-tsd Ti-tld
7
Δη×100%
Δη×100%
7
0
200
60
80
Δη = ηcalculated − 0.8 | / 0.8
100
120
h
140
160
40
Ti-nonuni1-tsd Ti-nonuni1-tld Ti-tsd Ti-tld
Ti-nonuni2-tsd Ti-nonuni2-tld Ti-tsd Ti-tld
30
30
Δh×100%
Δh×100%
200
Δη = ηcalculated − 0.8 | / 0.8
40
20
20
10
10
0 60
180
80
100
120
140
160
180
200
h
Δh = hcalculated − hreal / hreal
0 60
80
100
120
140
160
180
200
h
Δh = hcalculated − hreal / hreal
Fig. 6. Relative deviations of g and h calculated by different initial conditions at g = 0.8.
power of the mainstream and the secondary flow to produce different Tg(t) and Tc(t) which were measured by K-type thermocouples. And the time between two tests was 40 min at least to guarantee the recovery of the test plate temperature. Eq. (6) was used as the calculation formula in the present measurements. And 6–8 tests were carried out at identical momentum flux ratios. This multiple-test method is according to Drost et al. [13]. The initial wall temperature distribution Ti(y) was measured by K-type thermocouples shown in Fig. 9a. The location of this crosssection which is perpendicular to the mainstream is 180 mm from the start of the test plate. The test plate had been in the environment for 40 min at least to ensure a uniform temperature in the plate before installation. But the temperature difference between the test plate and the air in the test duct produced nonuniform initial temperature distribution in the Y direction region [0, 10 mm] during the short installation process according to the measurement. Because the test region of interest is at least 10 mm from the edge of test plate in the X, Z directions and the air temperature in the test duct was uniform, the temperature in the X–Z plane under the test region surface were considered as uniform. So the nonuniform initial temperature distribution only existed in the Y
direction of the test region, and the present initial temperature measurement method is reasonable. In every test case, the measured initial wall temperature distribution was similar with the curve in Fig. 9b, and the biggest temperature difference in the distribution was between 1 °C and 2 °C.
4.2. Results and discussion Error analysis has been performed with the method proposed by Kline and McClintock [29]. In the present experiment, the measurement uncertainties include temperature uncertainties: DTg, DTc, DTi, DTs; and time uncertainty pffiffiffiffiffiffiffiffi Dt in Tg(t), Tc(t), Ts(t); and material property uncertainty D qck. The estimated uncertainty intervals for the present experiment are: DTg = DTc = DTi = DTs = ±0.2 °C; pffiffiffiffiffiffiffiffi Dt = ±0.1 s; D qck ¼ 20. The temperature and time uncertainties in Tg(t) and Tc(t) have been assessed to be the uncertainties of the fitting coefficients in Eq. (6). According to the uncertainty intervals given above, the relative uncertainties in heat transfer coefficient is about 6%, and in local film cooling effectiveness is about 15% at g = 0.1 and 3% at g = 0.7.
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Pressure measurement station
Temperature acquisition
Computer Blower
Air tank
valve
CCD camera
Mesh heater
-
Tungsten halogen lamp
TC Test section Butterfly valve
Test plate TC
Settling Contraction + Secondary chamber section contraction
controller
Solenoid valve
Blower Air tank Flow meter
-
valve
Solenoid valve
+ Air heater
Fig. 7. Sketch of the experiment system.
Z
Y X
3D D
30mm
0
D
40mm
X 35°
Fig. 8. Sketch of the test plate and the film-cooling hole row.
27
(a)
(b) 100mm
26.5
20mm T/ oC
100mm
40mm
Y
Ti Ti0
11 1 2 3 4 5 6 7 8 9 10
26
25.5
25
0
0.002
0.004
Y/m
0.006
Fig. 9. Sketch of thermocouples’ distribution in the test plate and the measured initial temperature distribution.
0.008
0.01
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I =1
I =4
I =1
I =4
Fig. 10. Laterally averaged g and h calculated with different initial wall temperatures.
Three kinds of initial temperature distributions were used to calculate the film cooling effectiveness and the heat transfer coefficient: one is the real nonuniform initial wall temperature marked as ‘‘Ti-real’’, one is the temperature measured by the 1# thermocouple marked as ‘‘Ti-TC1’’, and one is the temperature measured by the 10# thermocouple marked as ‘‘Ti-TC10’’. Fig. 10 shows the laterally averaged results of film cooling effectiveness g and heat transfer coefficient h. Relative to the results of ‘‘Ti-real’’, the cooling effectiveness results of ‘‘Ti-TC1’’ are lower by about 10% under I = 1 and about 20% under I = 4, and the heat transfer coefficient results of ‘‘Ti-TC1’’ are lower by about 10% under both momentum ratios. However, the cooling effectiveness results of ‘‘Ti-TC10’’ are higher by about 7% under I = 1 and about 10% under I = 4, and the heat transfer coefficient results of ‘‘Ti-TC10’’ are higher by about 10% under both momentum ratios. The above deviations are explained as the following. The temperature of ‘‘TiTC1’’ is higher than the real initial wall temperature. The liquid crystal color-change time should be shorter under ‘‘Ti-TC1’’. Using the real but relatively long color-change time in the calculation makes the calculated convective heat flux smaller than the real.
And thus the results of cooling effectiveness and heat transfer coefficient are lower accordingly. Similar explanation can be drawn to the results of ‘‘Ti-TC10’’ which are relatively higher. Actually, similar with the experimental results of ‘‘Ti-TC10’’, the numerical results of Dg = (gcalculated greal)/greal and Dh = (hcalculated hreal)/hreal calculated with constant initial temperature Ti in Section 3.2 are all positive because the value of Ti is lower than the real initial wall temperature. In Fig. 10, film cooling effectiveness results were compared with the data of Goldstein et al. [30]. In Goldstein et al. [30], the film cooling effectiveness of cylindrical hole row, which has the same or similar geometry parameters with the current hole row, has been measured using mass analogy method under similar operating conditions. Although the deviation between the ‘‘Ti-real’’ results and the Goldstein et al. [30] data is on the same order as the deviation between the ‘‘Ti-real’’ results and the ‘‘Ti-TC1’’ results, the ‘‘Ti-real’’ results have relatively better agreement in authors’ opinion. These results show that obtaining accurate initial temperature distribution is very important for transient heat transfer measurement of film cooling. Small overestimate or underesti-
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Fig. 11. Local g and h calculated with different initial wall temperatures under I = 1 at X/D = 20.
mate of the initial temperature can make the experiment results deviate from the true values badly. Moreover, from Fig. 10 we also find that the deviation caused by the inaccurate initial wall temperature is large in the downstream region and small in the upstream region under I = 4 case. Under large momentum ratio case, the large-flux jet lifts off the wall immediately after ejecting and reattaches on the wall in the downstream region. Higher jet temperature in the experiment makes the change of liquid crystal color on the downstream wall more quickly than that on the upstream wall. So the test duration time in the downstream region is shorter than the upstream region. According to the analysis in Section 3.2, the influence of inaccurate initial temperature could be larger on the experiment results in the downstream region. This also provides experimental validation for the numerical investigation in Section 3. For the I = 1 case, because the jet flux is relatively small, the liquid crystal color-change time in most of downstream region is relatively long. So the deviations keep almost the same order in the downstream region. Figs. 11 and 12 also show the local spanwise distributions of g and h at X/D = 20 position for both momentum ratios, which were calculated by the three different initial wall temperature distributions respectively. In the streamwise midspan region where the liquid crystal colorchange time is relatively long and close, the deviations keep almost the same order in the whole spanwise region.
Fig. 12. Local g and h calculated with different initial wall temperatures under I = 4 at X/D = 20.
5. Conclusion A new equation used to calculate h and g in the transient heat transfer measurement was derived by Laplace transformation from 1-D unsteady conduction model with nonuniform initial wall temperature in the conduction direction. Both numerical and experimental methods were performed to investigate the influence of the nonuniform initial temperature on the transient measurement results of film cooling. The results show that accurate initial temperature distribution is very important for transient heat transfer measurement. Small overestimate or underestimate of the initial temperature by about 1–2 °C makes the experiment results deviate from the true values more than 10%. Using initial wall temperature higher than the real values causes the results of h and g lower than the true values. And using initial wall temperature lower than the real values causes the results higher than the true values. When the nonuniform initial temperature distribution is well fitted in the region where the conduction thermal resistance is smaller than the convection thermal resistance, accurate h and g can be calculated by the newly derived equation. If the initial temperature distribution is difficult to obtain and constant initial temperature has to be employed, the following methods are suggested to reduce the influence of nonuniform initial temperature: using material with
C.-l. Liu et al. / Experimental Thermal and Fluid Science 35 (2011) 1151–1161
low heat conductivity as the test plate, performing the transient measurements in a high h condition as well as properly increasing the transient test duration. Acknowledgments This work is sponsored by the National Basic Research Program of China (China 973 Program) under number of 2007CB707701. References [1] D.L. Schultz, T.V. Jones, Heat Transfer Measurements in Short Duration Hypersonic Facilities, AGARD NO. 165, 1985. [2] C.M.B. Russell, T.V. Jones, G.H. Lee, Heat transfer enhancement using vortex generators, in: Proceeding of International Heat Transfer Conference, Munich, 1982. [3] R.J. Clifford, T.V. Jones, S.T. Dunne, Techniques for Obtaining Detailed Heat Transfer Coefficient Measurements within Gas Turbine Blade Vane Cooling Passages, ASME Paper No. 83-GT-58, 1983. [4] P.T. Ireland, T.V. Jones, The measurement of local heat transfer coefficients in blade cooling geometries, in: AGARD Conf. on Heat Transfer and Cooling in Gas Turbines CP 390, paper 28, 1985. [5] P.T. Ireland, T.V. Jones, Detailed measurements of heat transfer on and around a pedestal in fully-developed channel flow, in: International Heat Transfer Conference, San Francisco, 1986. [6] R.P. Vedula, D.E. Metzger, W.B. Bickford, Effects of lateral and anisotropic conduction on determination of local convection heat transfer characteristics with transient tests and surface coatings, in: Winter Annual Meeting of ASME, Chicago, 1988, HTD-I, pp. 21–27. [7] T.V. Jones, S.A. Hippensteele, High-resolution heat-transfer-coefficient maps applicable to compound-curve surface using liquid crystal in a transient wind tunnel, NASA Technical Memorandum 89855, 1988. [8] D.E. Metzeger, R.S. Bunker, G. Bosch, Transient liquid crystal measurement of local heat transfer on a rotating disk with jet impingement, ASME Journal of Turbomachinery 113 (1991) 52–59. [9] S.V. Ekkad, J.-C. Han, Detailed heat transfer distributions in two-pass square channels with Rib turbulators, International Journal of Heat and Mass Transfer 40 (1997) 2525–2537. [10] F. Zehnder, M. Schüler, B. Weigand, J. von Wolfersdorf, S.O. Neumann, The Effect of Turning Vanes on Pressure Loss and Heat Transfer of a Ribbed Rectangular Two-Pass Internal Cooling Channel, ASME Paper No. GT200959482, 2009. [11] R.J. Vedula, D.E. Metzger, A Method for the Simultaneous Determination of Local Effectiveness and Heat Transfer Distributions in Three Temperature Convection Situations, ASME paper No. 91-GT-345, 1991. [12] A.C. Chambers, D.R.H. Gillespie, P.T. Ireland, A novel transient liquid crystal technique to determine heat transfer coefficient distributions and adiabatic wall temperature in a three temperature problem, ASME Journal of Turbomachinery 125 (2003) 538–546.
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