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TRANSCRIPT
Analytical Methods for Determination of Heat
Transfer Fields from TSP Measurements
in Hypersonic Tunnels
Tianshu Liu, Z. Cai & J. Lai
Western Michigan University, Kalamazoo, MI 49008
J. Rubal & J. P. Sullivan
Purdue University, West Lafayette, IN 47907
Current Objectives
To develop analytical methods and algorithms
for the determination of heat flux fields from
transient TSP measurements in hypersonic
tunnels
• Overview Outline
• General Exact Analytical Solution
• Simulations: Validation of Analytical Method
• Preliminary Experimental Results
• Conclusion & Further Investigation
• Effect of Lateral Heat Conduction
Overview of Current Methods
Approximate Fourier Law
L/]T)t(T[k)t(q 21is
Assumption: T2 is approximately the initial temperature for
a high-conductive base (e.g. Al) in a short run time.
Liu et al. (1994, 1995), Matsumura et al. (2003, 2005)
Hubner et al. (2002)
Cook-Felderman Method
The solution on a semi-infinite base
for TSP in hypersonic tunnel
n
1i 1inin
1iis
tttt
)t(T)t(Tck2)t(q
Cook & Felderman for thin-film sensor (1966)
Merski et al. for thermographic phosphor (1998, 1999)
Overview of Current Methods
Nagai et al. for TSP (2007)
Unsteady 1D Heat Conduction Equation
and Laplace Transform Inverse Solution
The heat flux at the polymer surface
in the transform plane:
where
)L,s()s( pps is the transformed surface temperature
)1/()1( bbbppp ck/ck
The relevant parameters:
)s(K)s(s)a/k()s(Q pspps
p
p
a/sL2exp1
a/sL2exp1
s
1)s(K
Inverse Laplace Transform
i
i
ds)s(K)tsexp(i2
1)t(k
Evaluation of the integral: Contour
An exact solution:
t
0
ps
p
2p
s dd
)(d
t
),t(W
a
)1(k)t(q
)ta/L2cos(21
d)exp(2),t(W
p2
2
0
The function that includes the effects of the polymer layer:
for a thin polymer layer on any base
The Discrete Form of the Exact Analytical Solution
Note:
The general solution is recommended in applications since it is
not only accurate, but also simple in numerical implementation
)ta/L2cos(21
d)exp(2),t(W
p2
2
0
where
For a semi-infinite base
0
The Cook-Felderman Method
1)0,t(W
)tt(W)tt(Wtttt
)t()t(
a
)1(k)t(q 1inin
n
1i 1inin
1ipsips
p
2p
ns
Effect of Lateral Heat Conduction
'dz'dx'z,'x,tt,'zz,'xxg ps1gps1
'dz'dx'z,'x,tt,'zz,'xxg ps2gps2
ta4
zxexp
ta4
1t,z,xg
p
22
p
1
ta4
zxexp
ta4
zx1
ta4
1t,z,xg
p
22
p
22
2p
2
Spatially-Filtered
Temperatures:
Filters:
t
0gps
t
0
gps
p
2p
s d)(t
),t(Wd
d
)(d
t
),t(W
a
)1(k)t(q
2
1
An exact solution of the unsteady 3D heat conduction equation:
TSP Surface Temperature & TSP-Measured Temperature
TSP-Measured Temperature
TSP Surface Temperature
An iterative method is required for temperature correction
for high heat flux
L
0
1TSP dy)y,t(TL)t(T
psTSPps k/L)t(q5.0)t(T)t(T
Simulations: Validation of Analytical Method
0.01 mm Thick PVC Layer on Al Base
Step Changes followed by a Sinusoidal Change in Heat Flux
Simulated heat flux in
a hypersonic tunnel
Temperatures obtained by numerically
solving the unsteady 1D heat conduction
equation
Recovered Heat Flux using the Analytical Method for
0.01 mm Thick PVC Layer on Al Base
Simulations: Validation of Analytical Method
0.01 mm Thick PVC Layer on Nylon Base
Temperatures
Step Changes followed by a Sinusoidal Change in Heat Flux
Recovered heat flux
larger time constant
Simulations: Validation of Analytical Method
Effect of Starting Process for 0.01 mm Thick PVC
Layer on Al Base
Linear starting process Random starting process
Sensitivity Analysis for the Analytical Method
The total uncertainty in heat flux: ss q/q
The elemental errors: L/L pp k/k / pp a/a
thickness thermal
conductivity
ratio between
thermal
properties
thermal
diffusivity
thickness thermal conductivity
Sensitivity Analysis for the Analytical Method
The total uncertainty in heat flux: ss q/q
Ratio between the thermal
properties Thermal diffusivity
25o/45o indented Cone at Mach 11
in 48-inch Shock Tunnel
at Calspan-University of Buffalo Research Center
Hubner, J. P., Carroll, B. F., and Schanze, K. S., “Heat-Transfer Measurements
in Hypersonic Flow Using Luminescent Coating Techniques,”
Journal of Thermophysics and Heat Transfer, Vol. 16, No. 4, 2002, pp. 516–522
t = 2 ms t = 6 ms
Surface Temperature
Correction
Surface Heat Flux Distribution
along a Ray
at the max heating point
25o/45o indented Cone at Mach 11
Gauge Data (Hubner et al. 2002)
25o/45o indented Cone at Mach 11
Time-Averaged Surface Heat Flux Distribution
Typical TSP image
Assumptions:
(1) Thermal properties of Mylar = those of TSP
(2) TSP thickness: 40 microns
14o Nylon Cone at Mach 6
in the AFOSR/Boeing Mach-6 Quiet Tunnel at Purdue
Typical temperature history
And heat flux at a laminar BL
Averaged heat flux image
that is downsampled by
averaging over windows
of 3x3 pixels
Averaged heat flux
distribution along the
centerline of the
turbulent wedge
streamwise direction
14o Nylon Cone at Mach 6
Infrared Laser Heating: Bench Test
Surface temperature history
at the center of the heated spot
in step heating
Insulator: 600 microns
TSP: 40 microns
Al substrate: 0.2 in
Step Heating
Infrared Laser Heating: Bench Test
Surface heat flux at the center
of the heated spot in step heating
Time-averaged surface
heat flux
Step Heating
Infrared Laser Heating: Bench Test
Surface heat flux at the center
of the heated spot
Surface temperature history
at the center of the heated spot
Oscillating Heating
Conclusion
The exact analytical method for determination of heat flux
from TSP measurements is developed and validated
in simulations and experiments