Analytical Methods for Determination of Heat Transfer Fields from TSP Measurements in Hypersonic Tunnels

Transcription

1 Analytical Methods for Determination of Heat Transfer Fields from TSP Measurements in Hyersonic Tunnels Tianshu Liu, Z. Cai & J. Lai Western Michigan University, Kalamazoo, MI J. Rubal & J. P. Sullivan Purdue University, West Lafayette, IN 47907

2 Current Objectives To develo analytical methods and algorithms for the determination of heat flux fields from transient TSP measurements in hyersonic tunnels Outline Overview General Exact Analytical Solution Effect of Lateral Heat Conduction Simulations: Validation of Analytical Method Preliminary Exerimental Results Conclusion & Further Investigation

3 Overview of Current Methods Aroximate Fourier Law qs(t ) ki [T1(t ) T ] / L Assumtion: T is aroximately the initial temerature for a high-conductive base (e.g. Al) in a short run time. Liu et al. (1994, 1995), Matsumura et al. (003, 005) Hubner et al. (00)

4 Overview of Current Methods Cook-Felderman Method The solution on a semi-infinite base for TSP in hyersonic tunnel q s ( t ) k c n T( ti ) T( ti1 ) t i 1 n ti tn ti1 Cook & Felderman for thin-film sensor (1966) Merski et al. for thermograhic hoshor (1998, 1999) Nagai et al. for TSP (007)

5 Unsteady 1D Heat Conduction Equation and Lalace Transform Inverse Solution The heat flux at the olymer surface in the transform lane: Q s ( s ) ( k / a )s s ( s )K( s ) where s ) ( s,l ) is the transformed surface temerature s( K( s ) 1 s 1 ex 1 ex L L s s / / a a The relevant arameters: ( 1 ) /( 1 ) k c / k b c b b

6 Evaluation of the integral: Inverse Lalace Transform Contour k( t ) 1 i i i ex( st )K( s )ds An exact solution: q s (t ) k (1 a ) t 0 W ( t, ) t d s( ) d d The function that includes the effects of the olymer layer: W (t, ) ex( ) d 0 1 cos( L / a t )

7 The Discrete Form of the Exact Analytical Solution for a thin olymer layer on any base n k (1 ) s(ti ) s( ti1 ) qs( tn ) W ( tn ti ) W ( tn ti1 a t t t t i1 n i n i1 ) where W (t, ) ex( ) d 0 1 cos( L / a t ) For a semi-infinite base 0 W (t,0 ) 1 The Cook-Felderman Method Note: The general solution is recommended in alications since it is not only accurate, but also simle in numerical imlementation

8 Effect of Lateral Heat Conduction An exact solution of the unsteady 3D heat conduction equation: q s ( t ) k (1 a ) t 0 W ( t, ) t d s ( ) t g W ( t, ) 1 d s ( ) d d 0 t g Satially-Filtered Temeratures: Filters: g 1 x,z,t 1 4 a t s s ex g g 1 x 4a z t g 1 g g x x',z z',t t,x',z' s dx' x x',z z',t t,x',z' x,z,t 1 4 a t 1 s x z 4a t dz' dx' ex dz' x z 4a t

9 TSP Surface Temerature & TSP-Measured Temerature TSP-Measured Temerature T TSP ( t ) L 1 0 L T( t, y ) dy TSP Surface Temerature T s (t ) T TSP (t ) 0.5q s (t )L / k An iterative method is required for temerature correction for high heat flux

10 Simulations: Validation of Analytical Method 0.01 mm Thick PVC Layer on Al Base Ste Changes followed by a Sinusoidal Change in Heat Flux Simulated heat flux in a hyersonic tunnel Temeratures obtained by numerically solving the unsteady 1D heat conduction equation

11 Recovered Heat Flux using the Analytical Method for 0.01 mm Thick PVC Layer on Al Base

12 Simulations: Validation of Analytical Method 0.01 mm Thick PVC Layer on Nylon Base Ste Changes followed by a Sinusoidal Change in Heat Flux Temeratures Recovered heat flux larger time constant

13 Simulations: Validation of Analytical Method Effect of Starting Process for 0.01 mm Thick PVC Layer on Al Base Linear starting rocess Random starting rocess

14 Sensitivity Analysis for the Analytical Method The total uncertainty in heat flux: The elemental errors: thickness L / L k thermal conductivity / q k s / q s ratio between thermal roerties / a / a thermal diffusivity thickness thermal conductivity

15 Sensitivity Analysis for the Analytical Method The total uncertainty in heat flux: q s / q s Ratio between the thermal roerties Thermal diffusivity

16 5 o /45 o indented Cone at Mach 11 in 48-inch Shock Tunnel at Calsan-University of Buffalo Research Center Hubner, J. P., Carroll, B. F., and Schanze, K. S., Heat-Transfer Measurements in Hyersonic Flow Using Luminescent Coating Techniques, Journal of Thermohysics and Heat Transfer, Vol. 16, No. 4, 00, t = ms t = 6 ms

17 5 o /45 o indented Cone at Mach 11 Surface Temerature Correction Surface Heat Flux Distribution along a Ray at the max heating oint Gauge Data (Hubner et al. 00)

18 5 o /45 o indented Cone at Mach 11 Time-Averaged Surface Heat Flux Distribution

19 Tyical TSP image 14 o Nylon Cone at Mach 6 in the AFOSR/Boeing Mach-6 Quiet Tunnel at Purdue Tyical temerature history And heat flux at a laminar BL Assumtions: (1) Thermal roerties of Mylar = those of TSP () TSP thickness: 40 microns

20 14 o Nylon Cone at Mach 6 Averaged heat flux image that is downsamled by averaging over windows of 3x3 ixels Averaged heat flux distribution along the centerline of the turbulent wedge streamwise direction

21 Infrared Laser Heating: Bench Test Surface temerature history at the center of the heated sot in ste heating Insulator: 600 microns TSP: 40 microns Al substrate: 0. in Ste Heating

22 Infrared Laser Heating: Bench Test Time-averaged surface heat flux Ste Heating Surface heat flux at the center of the heated sot in ste heating

23 Infrared Laser Heating: Bench Test Oscillating Heating Surface temerature history at the center of the heated sot Surface heat flux at the center of the heated sot

24 Conclusion The exact analytical method for determination of heat flux from TSP measurements is develoed and validated in simulations and exeriments