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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 Pye1 1

Research School of Engineering, The Australian National University, Canberra, ACT 0200, Australia *[email protected]

Abstract: A stochastic optimisation method adapted to illumination and radiative heat transfer problems involving Monte-Carlo ray-tracing is presented. A solar receiver shape optimisation case study illustrates the advantages of the method and its potential: efficient receivers are identified using a moderate computational cost. ©2015 Optical Society of America OCIS codes: (350.6050) Solar energy; (000.5490) Probability theory, stochastic processes, and statistics.

References and links 1. 2. 3.

M. F. Modest, Radiative Heat Transfer (University of California, 2003). J. Holman, Heat transfer, Mechanical engineering series (McGraw-Hill, 1989). Y. Shuai, X.-L. Xia, and H.-P. Tan, “Radiation performance of dish solar concentrator/cavity receiver systems,” Sol. Energy 82(1), 13–21 (2008). 4. C.-A. Asselineau, E. Abbassi, and J. Pye, “Open cavity receiver geometry influence on radiative losses,” in Proceedings of Solar2014, 52nd Annual Conference of the Australian Solar Energy Society, Solar2014, ed. (Melbourne, 2014). 5. C.-A. Asselineau, J. Zapata, and J. Pye, “Geometrical shape optimization of a cavity receiver using coupled radiative and hydrodynamic modeling,” in SolarPACES 2014, (Beijing, 2014). 6. F.-Q. Wang, R.-Y. Lin, B. Liu, H.-P. Tan, and Y. Shuai, “Optical efficiency analysis of cylindrical cavity receiver with bottom surface convex,” Sol. Energy 90, 195–204 (2013). 7. Q.-J. Mao, Y. Shuai, and Y. Yuan, “Study on radiation flux of the receiver with a parabolic solar concentrator system,” Energy Convers. Manage. 84, 1–6 (2014). 8. K. Lovegrove, G. Burgess, and J. Pye, “A new 500m2 paraboloidal dish solar concentrator,” Sol. Energy 85(4), 620–626 (2011). 9. D. Buie, A. Monger, and C. Dey, “Sunshape distributions for terrestrial solar simulations,” Sol. Energy 74(2), 113–122 (2003). 10. C. K. Ho, A. R. Mahoney, A. Ambrosini, M. Bencomo, A. Hall, and T. N. Lambert, “Characterization of Pyromark 2500 paint for high-temperature solar receivers,” J. Sol. Energy Eng. 136(1), 014502(2014). 11. Y. Meller, Tracer package: an open source, object oriented, ray-tracing library in python language, https://github.com/yosefm/tracer, (2013).

1. Introduction Monte-Carlo Ray-Tracing (MCRT) is a widely used approach for the problem of simulation of surface-to-surface radiative heat transfer in complex scenes [1]. This approach is computationally intensive when simulations include specular and diffuse surfaces, adiabatic and diathermal elements, geometrical and material limitations, and a high degree of spatial resolution. With design optimisation in mind, the study of a large number of system configurations becomes time consuming, and significant simplifications such as parametric studies and simple geometries are usually adopted. In this study we present an optimisation method to integrate MCRT of a large number of different geometrical configurations, or “scenes”, with a stochastic algorithm. The objective of the method is to identify the best scenes in a population. The number of rays used increases progressively and under-performing scenes are discarded as soon as they are identified. Using such a heuristic in the optimisation process reduces the computational effort required. This

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Received 2 Feb 2015; revised 16 Mar 2015; accepted 25 Mar 2015; published 9 Apr 2015 1 Jun 2015 | Vol. 23, No. 11 | DOI:10.1364/OE.23.00A437 | OPTICS EXPRESS A437

study includes the optimisation of the geometrical configuration of a concentrated solar receiver for maximized efficiency as a case study, to show the advantages of the method. 2. Stochastic optimisation method The objective of the optimisation method presented in the following section is to progressively screen the best-performing scenes in a population, by discarding underperforming candidates as soon as they can be identified. By doing so, MCRT simulations are only performed on potentially interesting candidate scenes, and computational time is saved. The optimisation method is formulated using a random scene generator suitable for solving the optics problem under consideration, such as radiative heat transfer, light trapping or illumination. To initialize the method, a population p j = 0 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 p j = 0 is large enough to cover this parameter space, to perform meaningful statistics and avoids statistical biases. MCRT runs are performed step-by-step with small rays count (~10,000). A ray-tracing index j records the number of ray-traces performed and increases for each new ray-trace. A metric M j ,i is used to assess the performance of each scene i in the calculated population, according to the objectives of the optimisation, at each step j. Typical metrics of relevance in optical problems include irradiance homogeneity, peak or threshold radiative flux values in designated areas and efficiency of the radiative heat transfer from a source to a receiver. After each MCRT step j and for every scene i of the population p j , the stochastic algorithm evaluates the optimisation metric average M j , i , its sample standard deviation S j , i and confidence interval IC j , i . The central limit theorem applied to large number of independent events, such as in MCRT, states that the distribution of the results follows a normal distribution and consequently the three sigma rule can be used to estimate the confidence interval IC j , i associated with the each estimations of the metric value M j , i . M j,i =

( j − 1) M j −1, i + M j , i

 (M j

k,i

− M k,i )

2

; S j,i = ; IC j ,i = 3S j , i (1) j j −1 Using these confidence interval estimates, underperforming candidates are discarded from the population at each step j. As more ray bundles are cast, the confidence interval of the results from MCRT method decreases with a 1 / j ratio and the precision of the calculation increases for potential optimal candidates still present in the population p j [1]. k =1

Fig. 1. Flowchart of the stochastic optimisation algorithm.

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In the present implementation of the method, the number of rays cast at each step j is constant and the index j is increased by one. As a consequence, only the potentially optimal scenes are simulated at greatest precision and significant computational time is saved (see 3.1). The algorithm continues its “screening” of the successive receiver populations until reaching a standard deviation termination threshold for all scenes σ T at which the precision is considered sufficient for the study. This algorithm shown in Fig. 1 only depends on the estimations of the metric and the standard deviations at each step, consequently MCRT can be coupled with other physical and chemical models to take into account more phenomena and still converge to optimal solutions, as shown the following case study. 3. Case study: Concentrated solar receiver geometry optimisation

3.1 System model In concentrated solar power systems, irradiated receivers convert concentrated solar radiation into thermal energy. 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 [2]. Cavity receiver shapes are discussed in several studies with various optimisation approaches: to make the flux on the internal walls of the receiver uniform [3], minimize overall radiative losses [4], optimise the geometry taking into account a coupled radiative and hydrodynamic heat transfer model [5], improve optical efficiency using a cylindrical geometry with a convex element at the bottom [6] or to study optimal geometrical aspect ratio for cylindrical cavity receivers at the focus of multi-dish concentrators [7]. In the specific case of solar receivers, an optimisation metric of interest is the thermal efficiency of the system η th,i , ratio of the rate of thermal energy harvested Q th, i to the incoming solar radiation input Q sun, i : Q

η th,i =  th, i Q

(2)

sun, i

The thermal efficiency value η th,i is obtained by MCRT radiative heat transfer simulations, here coupled with additional physical phenomena: convection, conduction and thermal emissions, suitable to solve a coupled energy balance and obtain detailed results. While additional considerations such as cost could be incorporated, a simple optimisation metric was preferred to preserve clarity of the case study. This case study focuses on axi-symmetric water/steam tubular receivers located at the focal plane of the ANU SG4 dish [8]. The geometry of the receiver is composed of stacked frusta (truncated cones) and a cone to close the geometry at the back, as shown in Fig. 2(b).

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Fig. 2. (a) The SG4 dish at the ANU STG facilities, and (b) cross-section of a parametric open cavity receiver model.

By assigning random values to each of these parameters, random shapes can be generated. The parametric open cavity receiver model is set to have one frustum element in front of the focal plane. The SG4 parabolic dish is modeled under a steady state operating regime using experimental results of focal plane flux distributions and focal distance resolved concentration ratio measurements [8]. The incoming solar radiation is modeled using a Buie sunshape [9]. Receiver internal surfaces are considered semi-gray to differentiate optical phenomena occurring before the absorption of solar radiation by receiver surfaces and thermal emissions occurring after. MCRT is used to simulate incident solar radiation taking into account the concentration and absorption processes, for which long-wave component is considered negligible [2]. 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 at the beginning of the algorithm using a distinct MCRT routine able to adapt to any geometry in the parameter space. Discretised receiver surfaces are covered by a single coiled tube in which a water/steam mixture transitions from liquid to saturated and superheated steam conditions in a single pass, from the aperture into the back of the receiver, as shown in Fig. 3(a). Tube surfaces are considered diffuse at all wavelengths and coated with a Pyromark2500® selective coating for an absorptivity of 0.95 and an emissivity of 0.85 [10]. Effective emissivity and absorptivity of tube covered surfaces are considered to take into account the self-viewing grooved absorbing/emitting surface arising from the curved surface formed by adjacent tubes [2]. A small adiabatic region is placed at the bottom of the cavity where the tube curvature would have exceeded manufacturable limits.

Fig. 3. The one-dimensional finite difference model used to model the helical heat extraction coil comprising the receiver walls: (a) full cross-section and (b) flow-segment energy balance.

The temperature profile of the external walls of the tube depends on the coupling of radiative heat absorption, thermal emissions, conductive heat transfer through the tube walls and internal convective heat transfer, and is determined using an iterative solution for the

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overall energy balance that incorporates a one-dimensional finite-difference model for the internal water/steam flow. The MCRT code used in this study is “Tracer” by Yosef Meller, an open-source raytracing code written in Python language with efficient Numpy numerical routines [11]. 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.

3.2 Stochastic optimisation results An optimisation run with a starting population p j = 0 of 1000 scenes was performed to obtain the results presented in this section. For this study the receiver was set to heat water from an inlet temperature of 50 °C to an outlet temperature of 500 °C. Ambient temperature was set to 27°C. The solar irradiation was set to 1000 W/m2 and the circumsolar ratio set to 1%. The maximum radius of the receiver was set at 0.75 m including a 0.1 m region dedicated to house thermal insulation. Similarly the maximum depth of the receiver was set to 1.5 m allowing 0.1 m for thermal insulation purposes. The computational effort spent by the optimisation, shown in Fig. 4(b), highlights the efficiency of the algorithm when compared with a brute force random search evaluation. The presented optimisation obtains its results in 7.8% of the time it takes to obtain them using a brute force approach. The fluctuations in the population count shown in Fig. 4(a) are caused by re-evaluation of previously discarded scenes: the routine evaluates every simulated scene at each step and is consequently able to “recoup” previously discarded scenes if their efficiency has become acceptable. This occurs when a new best candidate appears and its sample standard deviation evaluation is larger than the previous best candidate, thus increasing the confidence interval used to select potential optima.

Fig. 4. (a) Evolution of population count during optimisation and (b) computational effort spent on the optimisation case study as a function of the number of rays cast for each scene. The brute force simulation time was estimated by multiplying the number of MCRT passes by the average time spent per MCRT pass in the actual optimisation.

Figure 5 illustrates the convergence evolution of the algorithm. The maximum potential thermal efficiency for each candidate scene is shown at each MCRT step.

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Fig. 5. Convergence of the presented stochastic optimisation algorithm.

The gray area highlights the efficiency cutoff: red-marked scenes in the gray area are the ones that get discarded for the next iteration of the routine while black ones are kept as potential optima. The convergence observed in Fig. 5 shows that the optimisation is successfully eliminating under-performing scenes and finding an adequate optimum. Analysing the information stored during the optimisation offers useful insights for the design problem of interest. To illustrate this, thermal efficiencies are shown in Fig. 6(a) as function of the general aperture and the focal plane aperture for each candidate scene. The correlation between the aperture of the receiver and thermal efficiency appears on the left figure: smaller apertures limit radiative energy rate input in the receiver. Figure 6(b) illustrates the trade-off between concentrated solar flux input and thermal emission losses: if the radius on the focal plane is too large, the hot regions in the cavity tend to have higher view factor to the surroundings and lose more energy; however, if this radius is too small, a higher portion of the incoming solar flux is reflected outside and does not enter the cavity.

Fig. 6. Sensitivity of simulated thermal efficiencies to (a) the aperture radius and (b) the focal plane aperture radius.

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4. Conclusion and further improvements Design of optical sub-systems is often challenging when considering the interactions with other elements of the system. The algorithm presented in this paper offers a solution to tackle this complexity with a mitigated computational effort for the MCRT component of the problem, usually a clear bottleneck. A 92.2% saving in computation time arising from the use of the present stochastic algorithm compared to a brute force approach was shown in Fig. 4. However, most of the underperforming scenes are discarded at the very start of the routine. This paves the way towards the use of swarm or genetic meta-heuristics for scenes generation, which could additionally provide interesting opportunities to capitalise on the information gathered through the algorithm and generate better candidates at every step. In future work, new scenes could be introduced to the simulation as it progresses, as “mutations” of the surviving elements in the population at each step j of the simulation. The stochastic algorithm presented in this study is based on statistical sampling of an arbitrary optimisation metric and looks promising for the introduction of multi-constrained and multi-objective optical system design optimisation. Evaluating several metrics at each step for each candidate seems straightforward but work is ongoing to adapt the behaviour of the algorithm to take into account the specific correlation and interaction between parameters necessary to achieve combined multi-objective evaluation.

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