Performance Evaluations of the ATST Secondary Mirror
A Draft Version of SPIE Proc. 6665, San Diego, August 2007
Performance Evaluations of the ATST Secondary Mirror Myung K. Cho1, Joe DeVries1, and Eric Hansen2 1
GSMT Program Office, National Optical Astronomy Observatory, 950 N. Cherry, Tucson AZ 85719 2 National Solar Observatory, ATST, 950 N. Cherry, Tucson AZ 85719 ABSTRACT
The Advanced Technology Solar Telescope (ATST) has a 4.24m off-axis primary mirror designed to deliver diffraction-limited images of the sun. Its baseline secondary mirror (M2) design uses a 0.65m diameter Silicon Carbide mirror mounted kinematically by a bi-pod flexure mechanism at three equally spaced locations. Unlike other common telescopes, the ATST M2 is to be exposed to a significant solar heat loading. A thermal management system (TMS) will be developed to accommodate the solar loading and minimize “mirror seeing effect” by controlling the temperature difference between the M2 optical surface and the ambient air at the site. Thermo-elastic analyses for steady state thermal behaviors of the ATST secondary mirror was performed using finite element analysis by I-DEASTM and PCRINGETM for the optical analysis. We examined extensive heat transfer simulation cases and their results were discussed. The goal of this study is to evaluate the optical performances of M2 using thermal models and mechanical models. Thermal responses from the models enable us to manipulate time dependent thermal loadings to synthesize the operational environment for the design and development of TMS. Keywords: solar telescope, off-axis mirrors, secondary mirror, PCFRINGE, thermal management system
1
INTRODUCTION
The ATST secondary mirror analyzed as a baseline model is an off-axis lightweighted concave Silicon Carbide (SiC) mirror with a physical diameter of 650 mm. The SiC M2 has been chosen based on its superb thermal characteristics and stiffness superiority through parametric trade-off studies. The M2 configuration is shown in Figure 1 and the optical descriptions are as follows: Diameter = 0.65 m (0.62m clear aperture) Depth = 50mm (center), 30mm (edge) Rib wall thickness = 5 mm Radius of curvature = 2.081 m Conic constant = -0.5392
(a) Secondary mirror configurations (b) Secondary mirror with 3 bi-pod supports Figure 1. Secondary mirror configurations and its support system Physical and material properties of the silicon carbide mirror material utilized in the analysis are as follows: Mass Density Thermal conductivity Specific heat
2530 kg/m3 170 W/mºC 700 J/kgºC
Coefficient of thermal expansion Young’s Modulus Poisson’s ratio
2 x 10-6 m/mºC 218 x 109 Pa .17
2
THERMAL ANALYSIS
The thermal environment of the secondary mirror is assumed as shown in Figure 2. The boundary conditions on this model include the absorbed heat flux on the front surface, convection of air on the front surface, convection of air in the back surface, and radiative flux from both front and back surfaces. The basic heat transfer is described with conduction, convection and radiation. Moreover, it is observed that Fourier’s equation is satisfied inside the secondary mirror.
Figure 2. Thermal environment of Secondary Mirror (mirror face down) The boundary conditions of M2 in thermal equilibrium on the front and back surface are as follows: " " q& abs (t ) − q& rad (t ) − h1 [T (t ) front − T1 (t )] = k
" q& rad (t ) + h2 [T (t ) back − T2 (t )] = −k
∂T ∂n
∂T ∂n
front
back
Where " q& abs (t ) : Heat flux absorbed on the front surface
" q& rad (t ) : Time-dependent radiative flux from the front surface and back surface h1 , h2 : Heat transfer coefficient on the front surface and back surface, respectively T1 (t ) , T2 (t ) : Air temperature on the front surface and back surface, respectively k : thermal conductivity
Using I-DEASTM finite element program, extensive thermal analyses have been performed. The thermal response of the finite element model was obtained based on the boundary conditions described above. Since the radiative terms are much smaller than the solar heat flux in the ATST M2, the radiative effect was ignored in thermal calculations. The thermal response was calculated and evaluated over a 24 hour heat cycle.
2.1
Thermal loads with assumption of boundary conditions
In order to quantify the thermal response of the mirror, thermal boundary conditions similar to the telescope site were considered. These conditions are air convection on the front and back surfaces, heat flux on the back surface,
and radiative flux from both front and back surfaces. Sample baseline cases were considered and the thermal analyses were performed. These sample models were utilized to validate the final thermal results. The analyses performed were as follows: Case A: Air convection to ambient air alone. Case B: Heat flux with ambient air convection (Case A + Heat flux) Case C: Heat flux with ambient air and controlled air convection (Case B + controlled air convection) Case D: Heat flux variation in Case C 2.1.1 Ambient air temperature variation ATST has been performing extensive CFD analyses to quantify air flows inside the enclosure. In addition, a few simulation models have been developed for the air temperature distribution around the optics. In this study, a time dependent air temperature was assumed to have a sinusoidal variation. During a 24 hour thermal cycle, we further assume that maximum air temperature difference from the primary mirror is observed to be 5ºC as shown in Figure 3. In other words, the air is warmer than the mirror by 5ºC when the mirror is exposed at 6 hours. Such air temperature was applied to the front surface and the outer edge surface of the mirror. The front and edge surfaces change heat with the ambient air by free convection with a coefficient ha = 5 W/m2 ºC
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Figure 3. Ambient air temperature variation 2.1.2 Thermally controlled air temperature variation Through extensive CFD analyses and simulation models, a thermal management system (TMS) concept was established. A time dependent air temperature controlled by TMS was assumed to have a sinusoidal variation. During a 24 hour thermal cycle, we further assume that maximum temperature difference between the ambient air and the controlled air temperature is assumed to be 2ºC as shown in Figure 4. In other words, the TMS system provides controlled air cooler by 2ºC from the ambient air when the mirror is exposed at 6 hours. This controlled air temperature was applied to the back surface of the mirror. The back surface exchanges heat with the controlled air by forced convection with coefficients, hc, ranging 25 to 45 W/m2 ºC based on impinging jet characteristics as shown in TMS. 4.00
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Figure 4. Thermally controlled t air temperature variation
2.1.2 Heat Flux The mirror will absorb heat flux on the front surface and transfer it to the entire mirror. The heat flux observed can be from two sources. The one is due to the solar heat flux and the other is due to the three actuators positioned on the back surface. The specification calls for irradiance distribution, solar flux distribution, of 134 W/m2 over the M2 aperture. We anticipated that each actuator has a maximum value 2 W/m2; therefore, this flux is ignored in the thermal calculations. The solar heat flux was assumed to have a uniform distribution with a maximum of 134 W/m2 as depicted in Figure 5.
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(a) Solar flux distribution on M2 ( q& abs (t ) max=134 W/m2)
(b) Heat flux in a uniform distribution
Figure 5. Secondary mirror configurations and its support system 2.2
Finite element thermal model
In order to calculate the temperature distribution of the ATST secondary mirror, a finite element was generated using I-DEASTM FE program. The mirror was modeled with solid thermal elements. In addition, 2-D thin shell elements were created for all of the free surfaces of the mirror. In I-DEAS, these shell elements were required to facilitate the application of thermal loads. Current study does not include FE models for features and support hardware behind the mirror. These will impact on thermal behavior of M2 locally and globally, and the impact will be investigated with extended higher fidelity FE models. The finite element model used in this analysis is shown in Figure 6. Thermal analysis with I-DEAS TMGTM includes thermal responses as follows: the thermal time constant, convection, head flux, and the temperature distribution for the different cases described previously.
(a) Bottom view (optical surface in bottom) (b) Top view (optical surface on top) Figure 6. Finite element model of M2 for thermal analysis (1/6 mirror model) 2.3
Thermal Responses
2.3.1 Thermal time constant For the thermal time constant calculation, we assume that the front surface and edge surface exchange heat with the air by free convection, and the rest of the mirror surfaces with the controlled air. We performed an air convection analysis with an initial mirror temperature of 10ºC and a constant air temperature of 0ºC. For this calculation, air convection was applied on the front and edge surfaces having a heat transfer coefficient of 5 W/m2 ºC (ha=5), and the rest are 25 W/m2 ºC (hc=25). These convection coefficients can be further justified from CFD calculations. The thermal time constant (τ) was calculated by obtaining the required time for the mirror to reach 3.7ºC as 1/e of the initial temperature. The thermal time constant was calculated to be 300 seconds for the top surfaces at center (in red) and the edge (in green), respectively, as shown in Figure 7. This indicates that SiC is favorable for the ATST M2.
Top center
Top edge
Figure 7. Thermal time constant for the top surface at the center of M2 (in red) and the edge (in green) 2.3.2 Air convection (Case A) In this case, air convection to ambient air was applied on both the front and edge surfaces with heat transfer coefficients of 5 W/m2 ºC as seen in time constant calculations. In addition, a sinusoidal air temperature variation as described in (2.1.1) was employed. From a heat convection analysis, the temperature distribution of the mirror was obtained. The thermal responses were calculated and plotted over time for a node located at the center on the front surface of the mirror. From the FE result the peak temperature reached 4.8ºC at 7 hours as shown in Figure 8(a). Figure 8(b) shows the peak response of the mirror along with the ambient air temperature. The peak response is delayed by 1 hour from the peak air temperature. Temperature Profile Ambient Air
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(a) temperature responses at the mirror top center node (b) temperature responses with ambient air Figure 8. Temperature time responses at the center on the optical surface due to free heat convection to ambient air
2.3.3 Heat Flux (Case B) A thermal response analysis with heat flux loading was performed. The temperature distribution due to a maximum heat flux of 134 W/m2 over the mirror front surface was calculated. The uniform heat flux distribution over the 12 hours described in section 2.1.2 was used along with the free air convection (Case A). The maximum temperature from a heat flux analysis was obtained to be 22.3ºC at 7 hours as shown in Figure 9. Thermal responses of the mirror at the maximum (7 hours) were calculated. Figure 10 shows the temperature distribution due to the time dependent heat input (heat flux and air convection) over the mirror. A lowest temperature of 21.7ºC was observed around the edge, which gives a maximum temperature difference of 0.6ºC (compared to maximum of 22.3ºC).
Figure 9. Temperature time responses at the center on the optical surface due to heat flux and air convection
(a) temperature distribution (top view) (b) temperature distribution (bottom view) Figure 10. Thermal responses at 7 hours due to heat flux and air convection 2.3.4 Combined thermal loads (Case C) Some design parameters and thermal boundary conditions are currently under development. Study plans and experiment procedures are being assessed to validate the concepts and refine parameters for a useful thermal performance prediction. In order to simulate an operational environment, a synthesized thermal loading case was considered. The thermal load applied herein is a combined case of the previous cases (air convection, heat flux) with a thermally controlled air convection. The input thermal loadings are shown in Figures 3, 4 and 5(b). The thermal management system (TMS) controls thermal flow at the back of the mirror to maintain the optical surface temperature for the best seeing. We assume the back surfaces exchange heat with the thermally controlled air by forced convection with coefficients hc ranging 25 to 45 W/m2 ºC based on impinging jet characteristics. A finite element model was developed to accommodate the heat convection coefficient variations. Three different coefficients were assumed in the thermal calculations as shown in Figure 11.
Figure 11. FE model (1/6 mirror) with controlled air convection coefficients, hc = 25 W/m2°C surfaces in magenta; hc = 45 W/m2°C surfaces in blue; and hc = 45 W/m2°C surfaces in red (note: ambient air convection coefficient, ha = 5 W/m2°C surfaces in yellow)The temperature distribution of the mirror was calculated based on the combined load. Shown in Figure 12 is the thermal response of M2 plotted over time at the center of the front surface and edge of the front surface. As shown in the figure, the peak response occurred at 6 hours on both center and edge locations. The maximum temperatures are 4.34ºC and 4.23ºC, respectively.
Top center
Top edge
Figure 12. Temperature time responses at the center and edge of the optical surface to the combined case (heat flux and air convection, and controlled air convection) The thermal responses of the mirror were plotted along with the ambient air temperature as shown in Figure 13(a). The figure shows no evidence in time delay between the mirror response and the air temperature. Figure 13(b) shows the temperature differences between the mirror surface and the ambient air. The maximum deviation from the ambient air temperature over the M2 is 1.2ºC, which is well within the design goal of 2.0ºC. Thus we can predict that the design of the TMS is valid.
Thermal reponses combined (qmax=134W/m2) 6.0
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(a) temperature responses at the mirror top center and top edge nodes (b) temperature responses with ambient air Figure 13. Temperature time responses at the center on the optical surface due to free heat convection to ambient air The temperature distribution over the mirror is shown in Figure 14. The lowest temperature of 4.21ºC was observed around the edge of the back surface. In addition, we obtained a maximum temperature difference of 0.06ºC between the front surface and the bottom.
(a) temperature distribution (top view) (b) temperature distribution (bottom view) Figure 14. Temperature distribution over the mirror for the combined case (peak response at 6 hours) 2.4 Thermo-elastic analysis The thermal deformation and the optical surface deformation of the secondary mirror were calculated. As a conservative approach, we utilized a thermal loading with the peak temperature during a 24 hour thermal cycle. From the thermal analysis performed in Case C (combined loads), the temperature distribution at 6 hours, as shown in Figure 14, was applied to the FE model in order to perform a thermo-elastic analysis. For this thermo-elastic analysis, the model was kinematically constrained by three points on the back surface. The kinematic supports were located at 60 percent of the distance along the radius and separated 120 degrees, and were mounted at the CG of the mirror. From the thermo-elastic analysis, the mirror deformation was obtained and the deformed shape was depicted in Figure 15. Maximum mechanical deformation of 11.4nm and a minimum of –67.4 nm were obtained.
(a) thermal deformations (top view) (b) thermal deformations (bottom view) Figure 15. T thermal deformations due to temperature distribution of the combined case (peak response at 6 hours) For the optical surface evaluation, PCFRINGE program was employed. A Peak-valley surface error of 75nm and RMS of 22nm were calculated on the optical surface. After piston, tilts, and focus were removed, the Peak-valley surface error and RMS reduced to 13nm and 3nm, respectively. The optical surface maps are shown in Figure 16.
(a) The optical surface map (raw data) (b) The optical surface map (PTF removed) Figure 16. The optical surface map for the combined case (peak response at 6 hours) 2.5. Heat Flux variation effect In previous heat flux calculation in 2.3.1, we assumed a uniform heat flux solar flux distribution of 134 W/m2 over the M2 aperture. The detailed irradiance calculation indicated that the solar flux of 134 W/m2 has a uniform distribution along the X-axis and a non-uniform distribution along the Y-axis over the M2 aperture. The nonuniform distribution has approximately 10% variation in Y as shown in Figure 17(a).
(a) Solar heat flux distributions (b) The optical surface map with a tilt of 2nm Figure 17. Solar flux non-uniformity effect over the secondary mirror aperture We estimated the flux variation effect on the thermal distribution based on the combined case (Case C). This nonuniform flux load would produce a maximum temperature of 4.49ºC and a minimum of 4.09ºC over the M2. From this flux variation, the net optical surface variation would be a P-V of 7nm and RMS of 2nm of tilt as depicted in Figure 17(b).
3
GRAVITY DEFORMATIONS
In the previous chapter, several FE thermal models were employed to calculate temperature distributions and thermal deformations. In order for gravity induced deformations of the ATST secondary mirror, a few more finite element models were established by I-DEAS. Typical finite element model of the mirror (one half model) is shown in Figure 18. This is a 3-D plate model and consists of 5124 thin shell elements with 5060 nodes. This FE mirror model was properly modeled to simulate the bi-pod flexure support system. The line of action of the flexure support forces due to the mirror lateral gravity passes the center of gravity of the mirror. In this current model, three constraint boundary conditions were employed for the flexures. These constrain conditions will be further enhanced with structural elements in a high fidelity FE model which we plan to develop for a detail mechanical analysis.
Figure 18. FE mirror model (1/2 mirror) used to calculate gravity induced deformations. Left shows top view of the optical surface and Right shows the back surface. Gravity induced mirror deformation was evaluated at zenith and horizon pointing positions. These two extreme positions are outside the maximum operating range of 75 degrees maximum zenith angle. Axial supports were optimized at the zenith position to obtain minimum RMS surface deformation. The optimization predicts a maximum surface deflection of 503nm. An optical surface RMS of 55 nm was obtained after removing piston, tilts, and focus aberrations. Lateral support optimization done at horizon pointing predicted a P-V of 90nm and a surface RMS of 21 nm after removing piston, tilts, and focus aberrations. The optical surface maps for both positions are shown in Figure 19.
Figure 19. The optical surface deformations from axial gravity and lateral gravity (right) after piston, tilts, and focus removed. For axial gravity (left), P-V and RMS are 227nm and 55nm surface, respectively. For lateral gravity (right), P-V and RMS are 90nm and 21nm surface, respectively. To investigate the gravity print-through effects during the telescope operation, we calculated zenith angle dependent gravity deformations over the optical surface. We assume the secondary mirror was being polished and tested at zenith with the optical surface down. Therefore, no gravity print-through exists at zenith. Figure 20 shows the gravity print-through effect snapshots for zenith positions with a zenith angle increment of 15 degrees.
Figure 20. Gravity print-through effects (M2 polished and tested at zenith) at various zenith angles. Far left shows the optical surface map with a RMS of 6nm at zenith angle of 15 degrees (ZA=15) after piston, tilts, and focus removed. Those that follow are the optical surface maps with RMS of 13nm at ZA=30; 22nm at ZA=45; 33nm at ZA=60; 46nm at ZA=75; 59nm at ZA=90 (horizon), respectively. These gravity print-through effects are typical low order natural or Zernike modes which can be corrected by the active optics system (aO)4. We applied aO to the print-through at horizon case. The optical surface improved significantly to 20nm from 59nm by a maxim active optics force of 1.4N. A factor of three (3) improvement in the optical surface quality was obtained with low aO forces. The results are shown in Figure 21.
Figure 21. Active optics correction for the gravity print-through at horizon case. Raw uncorrected print-through case (left) shows a P-V of 287nm and RMS of 59nm. AO Corrected surface (right) shows a P-V of 99nm and RMS of 20nm.
4
MIRROR FREQUENCY ANALYSIS
Fundamental frequencies of the mirror were calculated by using a full FE mirror model with a free-free boundary condition. Several mirror bending mode shapes were obtained after removing rigid body motions. The lowest mode
was found at 629 Hz as an astigmatic shape. Several selected fundamental frequencies and their mode shapes are shown in Figure 22.
629Hz
1204Hz
2368Hz
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Figure 22. Natural frequencies and mode shapes with a free-free boundary condition. Far left shows the lowest astigmatic mode at 629 Hz.
5
SUMMARY AND CONCLUSIONS
Thermal management system (TMS) is being developed for the stringent thermal requirement on the secondary mirror. The TMS is designed to maintain temperature distribution for the best seeing. Finite element models were established by I-DEAS in order to predict the temperature distributions of the ATST secondary mirror. Several synthesized thermal loadings and heat transfer boundary conditions were employed to simulate the thermal effects of the secondary mirror. Thermal time constant calculation confirmed that the SiC M2 has an excellent thermal performance and it reaches a thermal equilibrium in 300 seconds. Under the combined thermal loading (air convection, heat flux, and thermally controlled air convection), maximum temperature was found at 4.34ºC (at center) and the lowest temperature of 4.21ºC was observed around the edge. In addition, we obtained a maximum temperature difference of 0.06ºC between the front surface and the bottom. The maximum temperature deviation from the ambient air was calculated to be 1.2ºC at the peak response (after 6 hours exposure). The results indicated the current TMS of secondary mirror adequately met the seeing requirements of 2ºC. From a thermo-elastic analysis with the peak thermal loading, maximum mechanical deformation of 11.4nm and a minimum of –67.4 nm were obtained. Peak-valley surface error of 13nm and RMS of 3nm after piston, tilts, and focus removed. The non-uniform flux load would produce a maximum temperature of 4.49ºC and a minimum of 4.09ºC over the M2. From this flux variation, the net optical surface variation would be a P-V of 7nm and RMS of 2nm of tilt. The FE results described herein were based on synthesized parameters and thermal boundary conditions. We calculated gravity induced deformations for a bi-pod flexure support system. The support optimization predicted the optical surface RMS of 55nm for the axial support (at zenith) and a RMS surface error of 15nm from the lateral support (at horizon). We also demonstrated zenith angle dependent gravity deformations. The gravity print-through of 59nm RMS surface was predicted at horizon when M2 is polished and tested at zenith with the optical surface down. This surface error was further improved to a RMS of 20nm by the active optics system. Additionally, the lowest natural frequency of 629Hz was found as an astigmatic mode. The results can be further refined and become useful when the input is fully defined. A high fidelity finite element model will be established to evaluate more extensive and detail performance evaluations.
REFERENCES [1] Nathan E. Dalrymple, “ATST Primary Mirror Thermal Analysis: Zero-Dimensional, Time-Dependent Model”, April 17, 2002 [2] Myung K. Cho and Andrew C. Corredor, “ATST Primary Mirror Thermal Analysis”, October, 2006 [3] Andrew C. Corredor and Myung K. Cho, “TMT M1 Thermal Analysis (Work in Progress)”, TMT.SEN.OPT.06.053.DRF01, November 30, 2006 [4] Myung K. Cho, Ronald S. Price, and Il K. Moon, “Optimization of the ATST Primary Mirror Support System”, SPIE Proc. 6263, 2006; Orlando, May 2006
Performance Evaluations of the ATST Secondary Mirror Myung K. Cho1, Joe DeVries1, and Eric Hansen2 1
GSMT Program Office, National Optical Astronomy Observatory, 950 N. Cherry, Tucson AZ 85719 2 National Solar Observatory, ATST, 950 N. Cherry, Tucson AZ 85719 ABSTRACT
The Advanced Technology Solar Telescope (ATST) has a 4.24m off-axis primary mirror designed to deliver diffraction-limited images of the sun. Its baseline secondary mirror (M2) design uses a 0.65m diameter Silicon Carbide mirror mounted kinematically by a bi-pod flexure mechanism at three equally spaced locations. Unlike other common telescopes, the ATST M2 is to be exposed to a significant solar heat loading. A thermal management system (TMS) will be developed to accommodate the solar loading and minimize “mirror seeing effect” by controlling the temperature difference between the M2 optical surface and the ambient air at the site. Thermo-elastic analyses for steady state thermal behaviors of the ATST secondary mirror was performed using finite element analysis by I-DEASTM and PCRINGETM for the optical analysis. We examined extensive heat transfer simulation cases and their results were discussed. The goal of this study is to evaluate the optical performances of M2 using thermal models and mechanical models. Thermal responses from the models enable us to manipulate time dependent thermal loadings to synthesize the operational environment for the design and development of TMS. Keywords: solar telescope, off-axis mirrors, secondary mirror, PCFRINGE, thermal management system
1
INTRODUCTION
The ATST secondary mirror analyzed as a baseline model is an off-axis lightweighted concave Silicon Carbide (SiC) mirror with a physical diameter of 650 mm. The SiC M2 has been chosen based on its superb thermal characteristics and stiffness superiority through parametric trade-off studies. The M2 configuration is shown in Figure 1 and the optical descriptions are as follows: Diameter = 0.65 m (0.62m clear aperture) Depth = 50mm (center), 30mm (edge) Rib wall thickness = 5 mm Radius of curvature = 2.081 m Conic constant = -0.5392
(a) Secondary mirror configurations (b) Secondary mirror with 3 bi-pod supports Figure 1. Secondary mirror configurations and its support system Physical and material properties of the silicon carbide mirror material utilized in the analysis are as follows: Mass Density Thermal conductivity Specific heat
2530 kg/m3 170 W/mºC 700 J/kgºC
Coefficient of thermal expansion Young’s Modulus Poisson’s ratio
2 x 10-6 m/mºC 218 x 109 Pa .17
2
THERMAL ANALYSIS
The thermal environment of the secondary mirror is assumed as shown in Figure 2. The boundary conditions on this model include the absorbed heat flux on the front surface, convection of air on the front surface, convection of air in the back surface, and radiative flux from both front and back surfaces. The basic heat transfer is described with conduction, convection and radiation. Moreover, it is observed that Fourier’s equation is satisfied inside the secondary mirror.
Figure 2. Thermal environment of Secondary Mirror (mirror face down) The boundary conditions of M2 in thermal equilibrium on the front and back surface are as follows: " " q& abs (t ) − q& rad (t ) − h1 [T (t ) front − T1 (t )] = k
" q& rad (t ) + h2 [T (t ) back − T2 (t )] = −k
∂T ∂n
∂T ∂n
front
back
Where " q& abs (t ) : Heat flux absorbed on the front surface
" q& rad (t ) : Time-dependent radiative flux from the front surface and back surface h1 , h2 : Heat transfer coefficient on the front surface and back surface, respectively T1 (t ) , T2 (t ) : Air temperature on the front surface and back surface, respectively k : thermal conductivity
Using I-DEASTM finite element program, extensive thermal analyses have been performed. The thermal response of the finite element model was obtained based on the boundary conditions described above. Since the radiative terms are much smaller than the solar heat flux in the ATST M2, the radiative effect was ignored in thermal calculations. The thermal response was calculated and evaluated over a 24 hour heat cycle.
2.1
Thermal loads with assumption of boundary conditions
In order to quantify the thermal response of the mirror, thermal boundary conditions similar to the telescope site were considered. These conditions are air convection on the front and back surfaces, heat flux on the back surface,
and radiative flux from both front and back surfaces. Sample baseline cases were considered and the thermal analyses were performed. These sample models were utilized to validate the final thermal results. The analyses performed were as follows: Case A: Air convection to ambient air alone. Case B: Heat flux with ambient air convection (Case A + Heat flux) Case C: Heat flux with ambient air and controlled air convection (Case B + controlled air convection) Case D: Heat flux variation in Case C 2.1.1 Ambient air temperature variation ATST has been performing extensive CFD analyses to quantify air flows inside the enclosure. In addition, a few simulation models have been developed for the air temperature distribution around the optics. In this study, a time dependent air temperature was assumed to have a sinusoidal variation. During a 24 hour thermal cycle, we further assume that maximum air temperature difference from the primary mirror is observed to be 5ºC as shown in Figure 3. In other words, the air is warmer than the mirror by 5ºC when the mirror is exposed at 6 hours. Such air temperature was applied to the front surface and the outer edge surface of the mirror. The front and edge surfaces change heat with the ambient air by free convection with a coefficient ha = 5 W/m2 ºC
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Figure 3. Ambient air temperature variation 2.1.2 Thermally controlled air temperature variation Through extensive CFD analyses and simulation models, a thermal management system (TMS) concept was established. A time dependent air temperature controlled by TMS was assumed to have a sinusoidal variation. During a 24 hour thermal cycle, we further assume that maximum temperature difference between the ambient air and the controlled air temperature is assumed to be 2ºC as shown in Figure 4. In other words, the TMS system provides controlled air cooler by 2ºC from the ambient air when the mirror is exposed at 6 hours. This controlled air temperature was applied to the back surface of the mirror. The back surface exchanges heat with the controlled air by forced convection with coefficients, hc, ranging 25 to 45 W/m2 ºC based on impinging jet characteristics as shown in TMS. 4.00
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Figure 4. Thermally controlled t air temperature variation
2.1.2 Heat Flux The mirror will absorb heat flux on the front surface and transfer it to the entire mirror. The heat flux observed can be from two sources. The one is due to the solar heat flux and the other is due to the three actuators positioned on the back surface. The specification calls for irradiance distribution, solar flux distribution, of 134 W/m2 over the M2 aperture. We anticipated that each actuator has a maximum value 2 W/m2; therefore, this flux is ignored in the thermal calculations. The solar heat flux was assumed to have a uniform distribution with a maximum of 134 W/m2 as depicted in Figure 5.
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(a) Solar flux distribution on M2 ( q& abs (t ) max=134 W/m2)
(b) Heat flux in a uniform distribution
Figure 5. Secondary mirror configurations and its support system 2.2
Finite element thermal model
In order to calculate the temperature distribution of the ATST secondary mirror, a finite element was generated using I-DEASTM FE program. The mirror was modeled with solid thermal elements. In addition, 2-D thin shell elements were created for all of the free surfaces of the mirror. In I-DEAS, these shell elements were required to facilitate the application of thermal loads. Current study does not include FE models for features and support hardware behind the mirror. These will impact on thermal behavior of M2 locally and globally, and the impact will be investigated with extended higher fidelity FE models. The finite element model used in this analysis is shown in Figure 6. Thermal analysis with I-DEAS TMGTM includes thermal responses as follows: the thermal time constant, convection, head flux, and the temperature distribution for the different cases described previously.
(a) Bottom view (optical surface in bottom) (b) Top view (optical surface on top) Figure 6. Finite element model of M2 for thermal analysis (1/6 mirror model) 2.3
Thermal Responses
2.3.1 Thermal time constant For the thermal time constant calculation, we assume that the front surface and edge surface exchange heat with the air by free convection, and the rest of the mirror surfaces with the controlled air. We performed an air convection analysis with an initial mirror temperature of 10ºC and a constant air temperature of 0ºC. For this calculation, air convection was applied on the front and edge surfaces having a heat transfer coefficient of 5 W/m2 ºC (ha=5), and the rest are 25 W/m2 ºC (hc=25). These convection coefficients can be further justified from CFD calculations. The thermal time constant (τ) was calculated by obtaining the required time for the mirror to reach 3.7ºC as 1/e of the initial temperature. The thermal time constant was calculated to be 300 seconds for the top surfaces at center (in red) and the edge (in green), respectively, as shown in Figure 7. This indicates that SiC is favorable for the ATST M2.
Top center
Top edge
Figure 7. Thermal time constant for the top surface at the center of M2 (in red) and the edge (in green) 2.3.2 Air convection (Case A) In this case, air convection to ambient air was applied on both the front and edge surfaces with heat transfer coefficients of 5 W/m2 ºC as seen in time constant calculations. In addition, a sinusoidal air temperature variation as described in (2.1.1) was employed. From a heat convection analysis, the temperature distribution of the mirror was obtained. The thermal responses were calculated and plotted over time for a node located at the center on the front surface of the mirror. From the FE result the peak temperature reached 4.8ºC at 7 hours as shown in Figure 8(a). Figure 8(b) shows the peak response of the mirror along with the ambient air temperature. The peak response is delayed by 1 hour from the peak air temperature. Temperature Profile Ambient Air
Temperature ('C)
6.00
M2 response
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2.00
0.00 0
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Time (hour)
(a) temperature responses at the mirror top center node (b) temperature responses with ambient air Figure 8. Temperature time responses at the center on the optical surface due to free heat convection to ambient air
2.3.3 Heat Flux (Case B) A thermal response analysis with heat flux loading was performed. The temperature distribution due to a maximum heat flux of 134 W/m2 over the mirror front surface was calculated. The uniform heat flux distribution over the 12 hours described in section 2.1.2 was used along with the free air convection (Case A). The maximum temperature from a heat flux analysis was obtained to be 22.3ºC at 7 hours as shown in Figure 9. Thermal responses of the mirror at the maximum (7 hours) were calculated. Figure 10 shows the temperature distribution due to the time dependent heat input (heat flux and air convection) over the mirror. A lowest temperature of 21.7ºC was observed around the edge, which gives a maximum temperature difference of 0.6ºC (compared to maximum of 22.3ºC).
Figure 9. Temperature time responses at the center on the optical surface due to heat flux and air convection
(a) temperature distribution (top view) (b) temperature distribution (bottom view) Figure 10. Thermal responses at 7 hours due to heat flux and air convection 2.3.4 Combined thermal loads (Case C) Some design parameters and thermal boundary conditions are currently under development. Study plans and experiment procedures are being assessed to validate the concepts and refine parameters for a useful thermal performance prediction. In order to simulate an operational environment, a synthesized thermal loading case was considered. The thermal load applied herein is a combined case of the previous cases (air convection, heat flux) with a thermally controlled air convection. The input thermal loadings are shown in Figures 3, 4 and 5(b). The thermal management system (TMS) controls thermal flow at the back of the mirror to maintain the optical surface temperature for the best seeing. We assume the back surfaces exchange heat with the thermally controlled air by forced convection with coefficients hc ranging 25 to 45 W/m2 ºC based on impinging jet characteristics. A finite element model was developed to accommodate the heat convection coefficient variations. Three different coefficients were assumed in the thermal calculations as shown in Figure 11.
Figure 11. FE model (1/6 mirror) with controlled air convection coefficients, hc = 25 W/m2°C surfaces in magenta; hc = 45 W/m2°C surfaces in blue; and hc = 45 W/m2°C surfaces in red (note: ambient air convection coefficient, ha = 5 W/m2°C surfaces in yellow)The temperature distribution of the mirror was calculated based on the combined load. Shown in Figure 12 is the thermal response of M2 plotted over time at the center of the front surface and edge of the front surface. As shown in the figure, the peak response occurred at 6 hours on both center and edge locations. The maximum temperatures are 4.34ºC and 4.23ºC, respectively.
Top center
Top edge
Figure 12. Temperature time responses at the center and edge of the optical surface to the combined case (heat flux and air convection, and controlled air convection) The thermal responses of the mirror were plotted along with the ambient air temperature as shown in Figure 13(a). The figure shows no evidence in time delay between the mirror response and the air temperature. Figure 13(b) shows the temperature differences between the mirror surface and the ambient air. The maximum deviation from the ambient air temperature over the M2 is 1.2ºC, which is well within the design goal of 2.0ºC. Thus we can predict that the design of the TMS is valid.
Thermal reponses combined (qmax=134W/m2) 6.0
Delta T from ambient air 1.00 Top center
Ambient Air
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2.0
Temperature ('C)
Temperature ('C)
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(a) temperature responses at the mirror top center and top edge nodes (b) temperature responses with ambient air Figure 13. Temperature time responses at the center on the optical surface due to free heat convection to ambient air The temperature distribution over the mirror is shown in Figure 14. The lowest temperature of 4.21ºC was observed around the edge of the back surface. In addition, we obtained a maximum temperature difference of 0.06ºC between the front surface and the bottom.
(a) temperature distribution (top view) (b) temperature distribution (bottom view) Figure 14. Temperature distribution over the mirror for the combined case (peak response at 6 hours) 2.4 Thermo-elastic analysis The thermal deformation and the optical surface deformation of the secondary mirror were calculated. As a conservative approach, we utilized a thermal loading with the peak temperature during a 24 hour thermal cycle. From the thermal analysis performed in Case C (combined loads), the temperature distribution at 6 hours, as shown in Figure 14, was applied to the FE model in order to perform a thermo-elastic analysis. For this thermo-elastic analysis, the model was kinematically constrained by three points on the back surface. The kinematic supports were located at 60 percent of the distance along the radius and separated 120 degrees, and were mounted at the CG of the mirror. From the thermo-elastic analysis, the mirror deformation was obtained and the deformed shape was depicted in Figure 15. Maximum mechanical deformation of 11.4nm and a minimum of –67.4 nm were obtained.
(a) thermal deformations (top view) (b) thermal deformations (bottom view) Figure 15. T thermal deformations due to temperature distribution of the combined case (peak response at 6 hours) For the optical surface evaluation, PCFRINGE program was employed. A Peak-valley surface error of 75nm and RMS of 22nm were calculated on the optical surface. After piston, tilts, and focus were removed, the Peak-valley surface error and RMS reduced to 13nm and 3nm, respectively. The optical surface maps are shown in Figure 16.
(a) The optical surface map (raw data) (b) The optical surface map (PTF removed) Figure 16. The optical surface map for the combined case (peak response at 6 hours) 2.5. Heat Flux variation effect In previous heat flux calculation in 2.3.1, we assumed a uniform heat flux solar flux distribution of 134 W/m2 over the M2 aperture. The detailed irradiance calculation indicated that the solar flux of 134 W/m2 has a uniform distribution along the X-axis and a non-uniform distribution along the Y-axis over the M2 aperture. The nonuniform distribution has approximately 10% variation in Y as shown in Figure 17(a).
(a) Solar heat flux distributions (b) The optical surface map with a tilt of 2nm Figure 17. Solar flux non-uniformity effect over the secondary mirror aperture We estimated the flux variation effect on the thermal distribution based on the combined case (Case C). This nonuniform flux load would produce a maximum temperature of 4.49ºC and a minimum of 4.09ºC over the M2. From this flux variation, the net optical surface variation would be a P-V of 7nm and RMS of 2nm of tilt as depicted in Figure 17(b).
3
GRAVITY DEFORMATIONS
In the previous chapter, several FE thermal models were employed to calculate temperature distributions and thermal deformations. In order for gravity induced deformations of the ATST secondary mirror, a few more finite element models were established by I-DEAS. Typical finite element model of the mirror (one half model) is shown in Figure 18. This is a 3-D plate model and consists of 5124 thin shell elements with 5060 nodes. This FE mirror model was properly modeled to simulate the bi-pod flexure support system. The line of action of the flexure support forces due to the mirror lateral gravity passes the center of gravity of the mirror. In this current model, three constraint boundary conditions were employed for the flexures. These constrain conditions will be further enhanced with structural elements in a high fidelity FE model which we plan to develop for a detail mechanical analysis.
Figure 18. FE mirror model (1/2 mirror) used to calculate gravity induced deformations. Left shows top view of the optical surface and Right shows the back surface. Gravity induced mirror deformation was evaluated at zenith and horizon pointing positions. These two extreme positions are outside the maximum operating range of 75 degrees maximum zenith angle. Axial supports were optimized at the zenith position to obtain minimum RMS surface deformation. The optimization predicts a maximum surface deflection of 503nm. An optical surface RMS of 55 nm was obtained after removing piston, tilts, and focus aberrations. Lateral support optimization done at horizon pointing predicted a P-V of 90nm and a surface RMS of 21 nm after removing piston, tilts, and focus aberrations. The optical surface maps for both positions are shown in Figure 19.
Figure 19. The optical surface deformations from axial gravity and lateral gravity (right) after piston, tilts, and focus removed. For axial gravity (left), P-V and RMS are 227nm and 55nm surface, respectively. For lateral gravity (right), P-V and RMS are 90nm and 21nm surface, respectively. To investigate the gravity print-through effects during the telescope operation, we calculated zenith angle dependent gravity deformations over the optical surface. We assume the secondary mirror was being polished and tested at zenith with the optical surface down. Therefore, no gravity print-through exists at zenith. Figure 20 shows the gravity print-through effect snapshots for zenith positions with a zenith angle increment of 15 degrees.
Figure 20. Gravity print-through effects (M2 polished and tested at zenith) at various zenith angles. Far left shows the optical surface map with a RMS of 6nm at zenith angle of 15 degrees (ZA=15) after piston, tilts, and focus removed. Those that follow are the optical surface maps with RMS of 13nm at ZA=30; 22nm at ZA=45; 33nm at ZA=60; 46nm at ZA=75; 59nm at ZA=90 (horizon), respectively. These gravity print-through effects are typical low order natural or Zernike modes which can be corrected by the active optics system (aO)4. We applied aO to the print-through at horizon case. The optical surface improved significantly to 20nm from 59nm by a maxim active optics force of 1.4N. A factor of three (3) improvement in the optical surface quality was obtained with low aO forces. The results are shown in Figure 21.
Figure 21. Active optics correction for the gravity print-through at horizon case. Raw uncorrected print-through case (left) shows a P-V of 287nm and RMS of 59nm. AO Corrected surface (right) shows a P-V of 99nm and RMS of 20nm.
4
MIRROR FREQUENCY ANALYSIS
Fundamental frequencies of the mirror were calculated by using a full FE mirror model with a free-free boundary condition. Several mirror bending mode shapes were obtained after removing rigid body motions. The lowest mode
was found at 629 Hz as an astigmatic shape. Several selected fundamental frequencies and their mode shapes are shown in Figure 22.
629Hz
1204Hz
2368Hz
3172Hz
1387Hz
3594Hz
2213Hz
3831Hz
Figure 22. Natural frequencies and mode shapes with a free-free boundary condition. Far left shows the lowest astigmatic mode at 629 Hz.
5
SUMMARY AND CONCLUSIONS
Thermal management system (TMS) is being developed for the stringent thermal requirement on the secondary mirror. The TMS is designed to maintain temperature distribution for the best seeing. Finite element models were established by I-DEAS in order to predict the temperature distributions of the ATST secondary mirror. Several synthesized thermal loadings and heat transfer boundary conditions were employed to simulate the thermal effects of the secondary mirror. Thermal time constant calculation confirmed that the SiC M2 has an excellent thermal performance and it reaches a thermal equilibrium in 300 seconds. Under the combined thermal loading (air convection, heat flux, and thermally controlled air convection), maximum temperature was found at 4.34ºC (at center) and the lowest temperature of 4.21ºC was observed around the edge. In addition, we obtained a maximum temperature difference of 0.06ºC between the front surface and the bottom. The maximum temperature deviation from the ambient air was calculated to be 1.2ºC at the peak response (after 6 hours exposure). The results indicated the current TMS of secondary mirror adequately met the seeing requirements of 2ºC. From a thermo-elastic analysis with the peak thermal loading, maximum mechanical deformation of 11.4nm and a minimum of –67.4 nm were obtained. Peak-valley surface error of 13nm and RMS of 3nm after piston, tilts, and focus removed. The non-uniform flux load would produce a maximum temperature of 4.49ºC and a minimum of 4.09ºC over the M2. From this flux variation, the net optical surface variation would be a P-V of 7nm and RMS of 2nm of tilt. The FE results described herein were based on synthesized parameters and thermal boundary conditions. We calculated gravity induced deformations for a bi-pod flexure support system. The support optimization predicted the optical surface RMS of 55nm for the axial support (at zenith) and a RMS surface error of 15nm from the lateral support (at horizon). We also demonstrated zenith angle dependent gravity deformations. The gravity print-through of 59nm RMS surface was predicted at horizon when M2 is polished and tested at zenith with the optical surface down. This surface error was further improved to a RMS of 20nm by the active optics system. Additionally, the lowest natural frequency of 629Hz was found as an astigmatic mode. The results can be further refined and become useful when the input is fully defined. A high fidelity finite element model will be established to evaluate more extensive and detail performance evaluations.
REFERENCES [1] Nathan E. Dalrymple, “ATST Primary Mirror Thermal Analysis: Zero-Dimensional, Time-Dependent Model”, April 17, 2002 [2] Myung K. Cho and Andrew C. Corredor, “ATST Primary Mirror Thermal Analysis”, October, 2006 [3] Andrew C. Corredor and Myung K. Cho, “TMT M1 Thermal Analysis (Work in Progress)”, TMT.SEN.OPT.06.053.DRF01, November 30, 2006 [4] Myung K. Cho, Ronald S. Price, and Il K. Moon, “Optimization of the ATST Primary Mirror Support System”, SPIE Proc. 6263, 2006; Orlando, May 2006
