Integration of Monte-Carlo ray tracing with a stochastic optimisation method: application to the design of solar receiver geometry Charles-Alexis Asselineau, Jose Zapata, and John Pye Research School of Engineering, The Australian National University, Canberra, ACT 0200, Australia.
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Abstract: A stochastic optimisation combined with Monte-Carlo ray-tracing is applied to solar receiver shape design. Efficient receivers are determined using a moderate computational cost. A case study is presented to show the results of the method. OCIS codes: (350.6050) Solar energy; (000.5490) Probability theory, stochastic processes, and statistics.
1. Introduction Monte-Carlo Ray-Tracing (MCRT) is the general approach to the problem of surface-to-surface radiative heat transfer simulation of complex scenes [9]. This is computationally intensive when simulations include specular and diffuse surfaces, adiabatic and diathermal elements, physical limitations as well as a high level of surface discretisation. With design optimisation in mind, the study of a large number of system configurations becomes time consuming and significant simplifications such as parametric studies of simple geometries are usually adopted. In this study we present a method to integrate MCRT of a large number of different scenes with a stochastic algorithm, to progressively screen the potential optimal configurations during the simulation, thus mitigating the computational effort spent. A case study presents the optimisation of the geometrical configuration of a concentrated solar receiver for maximized efficiency to show the advantages of the method. 2. Stochastic optimisation method The optimisation method is formulated using a random scene generator suitable for the radiative problem ounder consideration. To initialize the method, a population of random scenes i is declared. The stochastic nature of the scene declaration enables a comprehensive exploration of the parameter space, provided that the initial population is large enough and avoiding statistical biases. A metric to assess the performance of each scene i in the population is set, according to the objectives of the optimisation. In the specific case of concentrated solar energy, a metric of interest is the thermal efficiency of the system , ratio of the rate of thermal energy harvested ̇ to the incoming solar radiation input ̇ : ̇ ̇ The efficiency value is obtained by MCRT radiative heat transfer simulations, eventually coupled with additional physical phenomena (convection, conduction or thermochemical reactions for example) depending on the complexity of the problem. After each MCRT simulation n and for every scene i of the population , the stochastic algorithm evaluates the average ̅ and sample standard deviation of the chosen metric.
̅
̅
√( ̅
)
Once all scenes in the population have been simulated with an identical number of rays, they are compared based on their average efficiency and sample standard deviation. MCRT relying on large number of independent events, the central limit theorem states that the distribution of the results follows a normal distribution. Using the three sigma rule, underperforming candidates are discarded from the population. As more ray bundles n are cast, the
confidence interval of the results from MCRT method decreases with a ⁄√ ratio and the precision of the calculation increases for potential optimal candidates still present in the population .
Figure 1: Typical convergence rate and overall result distribution obtained by MCRT on 5 different scene configurations of a radiative transfer problem.
As a consequence, only the potentially optimal scenes are simulated at greatest precision and significant computational time is saved. The algorithm continues screening the receiver population until reaching a standard deviation termination threshold .
Figure 2: Stochastic algorithm flowchart.
3. Case study: Concentrated solar receiver geometry optimisation In concentrated solar power systems, receivers convert concentrated solar radiation into thermal energy and, consequently, have a major impact on overall system efficiency and economic viability. Increasing the operation temperature of a receiver offers downstream thermodynamic efficiency gains in accordance with Carnot efficiency limits. However, higher receiver temperatures also translate into higher thermal losses from the hotter receiver external surfaces. At high temperatures, receiver geometries able to promote a cavity effect facilitate reduction of thermal emission losses. Cavity receiver shapes are discussed in several studies that examine cavity optimisation approaches: to make the flux on the internal walls of the receiver uniform [10], minimize overall radiative losses [1], optimise the geometry taking into account a coupled radiative and hydrodynamic heat transfer model [2], improve optical efficiency using a cylindrical geometry with a convex element at the bottom [11] or to study optimal geometrical aspect ratio for cylindrical cavity receivers at the focus of multi-dish concentrators [7]. The case study focuses on axi-symmetric water/steam tubular receivers located at the focal plane of the ANU SG4 dish [6] as shown in
Figure 3.
Figure 3: The SG4 dish at the ANU STG facilities (left) and the parametric receiver model (right).
The SG4 parabolic dish is modeled using experimental results of focal plane flux distributions and focal distance resolved concentration ratio measurements. The incoming solar radiation is modeled using a Buie sunshape source model ratio [3]. Receiver internal surfaces are considered semi-gray to differentiate optical phenomena occurring before the absorption of light by receiver surfaces and thermal emissions occurring after. Steady state operations are considered. MCRT is used to incident solar radiation taking into account the concentration and absorption processes, for which long-wave component is considered negligible [5]. Thermal emissions are calculated using the radiosity method, given the diffuse nature of receiver surfaces considered. View factor matrices, required to solve the radiosity balance, are calculated using a distinct MCRT method able to adapt to any geometry in the parameter space. Receiver surfaces are covered by a single coiled tube in which a water/steam mixture circulates, from the aperture into the back of the receiver. Tube surfaces are considered diffuse at all wavelengths and coated with a Pyromark2500® selective coating [4]. Effective emissivity and absorptivity of tube covered surfaces are considered to take into account the self-viewing grooved absorbing/emitting surface [5]. A small adiabatic region is placed at the bottom of the cavity where the tube curvature would have exceeded manufacturable limits. The temperature profile of the external walls of the tube depends on the coupling of radiative heat transfer and internal convective heat transfer, and is determined using an iterative solution for the overall energy balance that incorporates a onedimensional finite-difference model for the internal water/steam flow. The MCRT code used in this study is “Tracer” by Y. Meller, an open-source ray-tracing code written in Python language [8]. The ray-tracing method used by Tracer is sometimes referred to as “path-tracing” where each ray bears a fraction of the energy of the source and absorption/reflection events are directly computed, gradually depleting the ray energy as it proceeds through the scene. 4. References [1]
Charles-Alexis Asselineau, Ehsan Abbassi, and John Pye. Open cavity receiver geometry influence on radiative losses. In Solar2014, editor, Proceedings of Solar2014, 52 nd Annual Conference of the Australian Solar Energy Society. Melbourne, May 2014. [2] Charles-Alexis Asselineau, Jose Zapata, and John Pye. Geometrical shape optimization of a cavity receiver using coupled radiative and hydrodynamic modeling. In SolarPACES2014, editor, International Conference on Concentrating Solar Power and Chemical Energy Systems. Beijing, September 2014. [3] Daniel Buie, A.G. Monger, and C.J. Dey. Sunshape distributions for terrestrial solar simulations. Solar Energy, 74(2):113 – 122, 2003. [4] Clifford K. Ho, A. Roderick Mahoney, Andrea Ambrosini, Marlene Bencomo, Aaron Hall, and Timothy N. Lambert. Characterization of Pyromark 2500 paint for high-temperature solar receivers. Journal of Solar Energy Engineering, 136(1):014502–014502, July 2013. [5] J.P. Holman. Heat transfer. Mechanical engineering series. McGraw-Hill, 1989. [6] Keith Lovegrove, Gregory Burgess, and John Pye. A new 500m2 paraboloidal dish solar concentrator. Solar Energy, 85(4):620 – 626, 2011. SolarPACES 2009. [7] Qian-Jun Mao, Yong Shuai, and Yuan Yuan. Study on radiation flux of the receiver with a parabolic solar concentrator system. Energy Conversion and Management, 84(0):1 – 6, 2014. [8] Yosef Meller. Tracer package: an open source, object oriented, ray-tracing library in python language. https://github.com/yosefm/tracer. [9] M. F. Modest. Radiative Heat Transfer. Academic Press, 2003. [10] Yong Shuai, Xin-Lin Xia, and He-Ping Tan. Radiation performance of dish solar concentrator/cavity receiver systems. Solar Energy, 82(1):13 – 21, 2008. [11] Fu-Qiang Wang, Ri-Yi Lin, Bin Liu, He-Ping Tan, and Yong Shuai. Optical efficiency analysis of cylindrical cavity receiver with bottom surface convex. Solar Energy, 90(0):195 – 204, 2013.
