1. Introduction

Solar-thermal technologies harness energy with heliostats, heliostats that heat a boiler or reactor with reflected sunlight. This approach has been successfully used to drive biomass gasification, metals production and electricity generation in a renewable manner [1–3 ]. Solar-thermal research is accelerated by high-flux solar simulators: assemblies of high power lamps whose irradiance mimics concentrated sunlight [4–7 ]. These platforms provide controlled environments for the evaluation of new solar-thermal technologies at the laboratory scale. To date six high-flux solar simulators have been reported for concentrated solar research [4].

Knowledge of the radiative power and flux available from a given solar simulator is prerequisite to its use. A calorimeter, which consists of a water cooled absorptive cavity and aperture for light inlet, can be used to acquire this information [8–11 ]. The irradiance a calorimeter intercepts is dependent on aperture diameter. With a range of aperture sizes one can characterize a solar simulator flux profile [8, 12 ], although it is recommended that the ratio of calorimeter length to aperture diameter always exceed 4:1 [12, 13 ]. At lower ratios, emission and reflective losses can significantly degrade calorimeter power measurements. Previous work has shown that different flux measurement techniques, which included a cylindrical calorimeter, yield disparate readings [9]. The cause of these measurement discrepancies, which spanned nearly 10%, was unclear [9].

The present study focuses on a well-insulated calorimeter with detachable faceplates similar to Diver et al. 1983 [8]. Figure 1(a) shows how the absorptive body and detachable faceplate are separately cooled to avert heat exchange between these components. Radiation that enters the calorimeter through the faceplate aperture, Q _in, heats circulating coolant within the body, causing a coolant temperature rise that reveals the intercepted irradiance [10–12 ]:

 

Fig. 1 A simulation framework for calorimeter evaluation. a) piping and instrumentation diagram of calorimeter cooling flows showing confounding radiation effects. In this work the operational temperature target was ΔT = 25°C and the faceplate coolant flowrate was fixed at 6 L/min. b) Radiative and thermal phenomena within and outside the calorimeter.

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Herein the effect of calorimeter design attributes on thermal losses was explored with Monte Carlo ray tracing and computer simulation. CUtrace, a parallel, grey body, Monte Carlo ray tracer was written to represent the solar simulator at Australian National University (ANU) and École Polytechnique Fédérale de Lausanne (EPFL) [4]. CUtrace is freely available online at Matlab Central (Mathworks Incorporated) and features a graphical user interface. Cylindrical calorimeter designs were traced in the solar simulator and the results were processed by finite volume simulation. Simulations were performed over a range of optical properties to evaluate calorimeter robustness against radiative losses, emissive losses, convective losses and the observer effect.

2. Methods

2.1 Monte Carlo ray tracing

A software package, CUtrace, was written using Matlab’s Parallel Programming Toolbox [14]. The grey body ray tracer implements 14 geometric primitives and three filaments (ray sources). The code is freely available for detailed inspection online at Matlab Central (Mathworks Incorporated). In brief, the object-oriented program traces rays through a given user defined scene by affine reflections [15, 16 ]. To determine where a ray terminates, a random number is drawn [0-1] upon each ray-shape intersection. If the random number is greater than the incident surface reflectivity, the ray is absorbed. Otherwise the ray is reflected in analogy with Phong Illumination [17]. Specifically, a second random number [0-1] is drawn. Reflection is Lambertian if the number is less than a user inputted fraction for the incident surface. Otherwise, reflection is specular with error bounded by a user inputted angular standard deviation [18]. All random numbers originate from the Mersenne Twister algorithm for a uniform distribution. Each geometric object maintains a private record of absorbed rays. Each new ray intersection is discovered by scanning objects in the scene for the nearest parametric ray-shape collision.

CUtrace was validated by simulating radiation exchange in three test systems. Additionally, the tracer was used to generate view factors with known analytical solutions. Figure 2 shows the validations. Radiation exchange in the 18 lamp solar simulators housed at ANU and EPFL served as the first test case [4]. CUtrace matched an existing code for modeling this installation [Fig. 2(a)] [4]. Next the VEGAS ray tracer was used to model radiation exchange within the 18 lamp solar simulator [19]. VEGAS can only approximate radiation exchange within this facility because the program lacks an exact model of the xenon filaments used at ANU and EPFL. CUtrace matched VEGAS when cylindrical radiation sources were substituted for rigorous filament modeling [Fig. 2(b)]. Of the objects implemented in the new tracer, the compound parabola is most complicated. This solid of revolution harbors four nontrivial roots [20]. A compound parabola was drawn in both Soltrace, a tool for the design of commercial solar facilities [21], and the new tracer. CUtrace reproduced the Soltrace output [Fig. 2(c)]. In a final test, the ray tracer was used to generate view factors. The new tracer matched known analytical solutions representing radiative exchange within cones, cylinders and from spheres to cylinders and paraboloids [Fig. 2(d)] [22]. Although it is impossible to exhaustively test the new tracer, it successfully recapitulates results from proven computer codes and documented view factors.

 

Fig. 2 Validation of the new, parallel, grey body Monte Carlo ray tracer (CUtrace). a) predicted power intercepted by a disc at a solar simulator focus. Trace was the ANU and EPFL solar simulator with rigorous xenon filament modeling as described by Bader 2014 [4]. b) predicted power intercepted by a disc at a solar simulator focus. Trace was the ANU and EPFL solar simulator with simplified volumetric cylindrical sources in each lamp [4, 19 ]. Source cylinder length was 0.0045 meters and radius was 0.00075 meters. c) predicted power intercepted by a disc at a compound parabola outlet [21]. The compound parabola acceptance angle was 35° and the outlet radius was 1.3 meters. d) analytical view factors from Appendix C of Howell 2002 [22] are reproduced by CUtrace, wherein any surface can act as a diffuse source of radiation. The view factors represent radiative exchange within cones, cylinders and from spheres to cylinders and paraboloids,

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Figure 3 shows the calorimeter in simulation at the focus of ANU’s/EPFL’s 18 lamp solar simulator [4]. One million rays were used to simulate 1.05 kW from each lamp. Unless noted, all calorimeter surfaces were diffusely reflective with wavelength and temperature independent reflectivities of 0.05 for body cavity, 0.95 for faceplate interior and 0.05 for face.

 

Fig. 3 CUtrace depiction of the calorimeter design at the focus of the ANU/EPFL solar simulator [4].

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2.2 Calorimeter finite volume simulation

Figure 4 shows how 3D Monte Carlo ray tracing results were fed to a 2D finite volume simulation by applying annularly the peak axial and radial fluxes. This results in a higher radiative power input, higher surface temperatures, and consequently higher losses consistent with worst-case calorimeter performance.

 

Fig. 4 Peak Monte Carlo fluxes were mapped annularly around exposed calorimeter surfaces. a) actual fluxes and b) mapped fluxes.

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Figure 5 details the calorimeter design. Two detachable faceplates were modeled, each with a different aperture diameter and independent coolant system. Coolant flow was modeled by convective upwinding assuming a water inlet temperature of 20°C. For the range of flowrates examined, the Dean number was consistently turbulent (Dean > 500). Thus, the simulation implemented turbulent heat transfer relationships for cylindrical and spiral ducts. Details of the simulation equations are listed in Appendix A. Temperature dependent physical properties were implemented for the copper calorimeter body, water coolant, and ambient air [23]. Body outer surfaces, which are typically insulated, were assumed adiabatic. Convection to ambient air employed a heat transfer coefficient for cylindrical calorimeters (Appendix A) [24]. Air was assumed to be optically transparent. Thermal emission between and within cylindrical and radial annuli was modeled using analytical view factors [22]. In all cases, the view factors of a radial or cylindrical annulus summed to unity. Simulation physicality was supported by energy closure. The mesh was refined until power lost through the aperture was statistically invariant (95% confidence), yielding a 540 element grid.

 

Fig. 5 details of the dimensions and discretization used in Monte Carlo ray tracing and finite volume simulation. Unless otherwise specified body cavity reflectivity was 0.05 diffuse, faceplate interior reflectivity was 0.95 diffuse, and face reflectivity was 0.05 diffuse. The faceplate coolant flow was a constant 6 L/min. Both the body and faceplate coolant flows [Fig. 1(a)] were drawn from a 20°C water supply.

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3. Results

3.1 Observer effects on the solar simulator flux profile

Monte Carlo simulations performed with a perfectly absorbing calorimeter face resulted in Q _observer = 0 kW for both the large (⌀10 cm) and small (⌀3 cm) aperture faceplates. Any face reflectivity greater than zero caused radiation exchange between the calorimeter and lamp reflectors [Fig. 6 ], exchange that distorted the native solar simulator flux profile. When the calorimeter face had a reflectivity of 0.95 more radiation was intercepted irrespective of whether that surface was diffuse (Q _obersever = 0.20 kW) or specular (Q _observer = 1.6 kW). Thus, a significant fraction of radiation that reflected off the face ultimately returned to the calorimeter aperture [Fig. 6]. Calorimeter face reflectivity presents a tradeoff. A highly reflective face will absorb little heat, but can perturb the solar simulator flux profile. Conversely, an absorptive face preserves the flux profile, but can overheat. Based on these results a diffusely reflective calorimeter face with a reflectivity of 0.05 was adopted in subsequent simulations.

 

Fig. 6 The calorimeter induces an observer effect. a) percent distortion (◆) and power intercepted (■) by the large (⌀10 cm) aperture faceplate as calorimeter face reflectivity increased. b) percent distortion (◆) and power intercepted (■) by the small (⌀3 cm) aperture faceplate as calorimeter face reflectivity increased. c) flux distortion at the small (⌀3 cm) calorimeter aperture induced by different face reflectivities.

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3.2 Calorimeter performance and robustness

Figure 7 shows coolant temperature rise and calorimeter losses for a range of coolant flowrates and body reflectivities. Effects of both the large (⌀10 cm) and small (⌀3 cm) apertures are shown. Predictably, Q _reflective increased with higher body reflectivities because light was not immediately absorbed by the cavity surfaces. This effect was more pronounced for the large aperture than small aperture, commensurate with more light escape. For the small aperture, the aperture lip absorbed and scattered more radiation than the large aperture, which inflated absolute reflective losses (Q _reflective) [Fig. 7]. Aperture lip effects are further explored in section 3.3. Q _convective was dominated by coolant flow and significant at low flowrates, flowrates which resulted in high calorimeter temperatures [Fig. 7]. Convective losses were always higher for the large aperture calorimeter, consistent with natural convection through a larger outlet. For all cases Q _emissive was comparatively small and increased with lower coolant flows and lower reflectivities (higher emissivities).

 

Fig. 7 Simulation results for variations in body reflectivity. a) results for the large (⌀10 cm) aperture faceplate. b) results for the small (⌀3 cm) aperture faceplate. In all cases face and faceplate interior reflectivities assumed their nominal values (0.05 and 0.95 diffuse, respectively). Faceplate coolant flow was a constant 6 L/min.

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The influence of faceplate interior reflectivity on calorimeter losses are shown in Fig. 8 . Throughout these simulations the body assumed its nominal reflectivity; 0.05 diffuse. Decreasing the faceplate interior reflectivity increased Q _reflective because more incident radiation was absorbed by the aperture lip prior to detection. Furthermore, reflection from the cavity to the faceplate was absorbed, not returned. No significant influence on Q _convective was observed, while Q _emissive decreased with decreasing faceplate interior reflectivity (increased interior absorptivity).

 

Fig. 8 Simulation results for variations in faceplate interior reflectivity. a) results for the large (⌀10 cm) aperture faceplate. b) results for the small (⌀3 cm) aperture faceplate. In all cases face and body reflectivities assumed their nominal values (0.05 and 0.05 diffuse, respectively). Faceplate coolant flow was a constant 6 L/min.

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3.3 Calorimeter performance with a beveled aperture

To test the importance of the aperture lip, beveled (conical) apertures were explored as a means of making the calorimeter more accurate and robust. The beveled (conical) aperture design featured an acceptance angle that matched that of the solar simulator installed at ANU and EPFL (45°) [4, 20 ]. Thus, light entered the calorimeter directly without being scattered, absorbed, or reflected by the aperture lip. Figure 9 compares losses from the different calorimeter designs for a coolant temperature rise of ΔT = 25°C. In all cases this modification rendered reflective losses, which were consistently dominant, nearly insubstantial [Fig. 9]. Notably, in all cases the beveled aperture dropped the ratio of overall losses to incident flux (Q _in) below 1.5%, whereas the initial design showed relative losses of up to 15%. This modification reveals that calorimeter losses are a strong function of the calorimeter faceplate geometry, not necessarily calorimeter body design, coolant flow or reflectivity.

 

Fig. 9 Losses as a percentage of Q _in for a calorimeter body coolant temperature rise of ΔT = 25°C and the nominal diffuse reflectivities (body = 0.05, face = 0.05, faceplate = 0.95). Symbols across all graphs: Q = Q _reflective (◆), Q = Q _convective (■), Q = Q _emissive (▲) and Q = Q _observer (X). a) Large (⌀10 cm) square aperture faceplate. b) Small (⌀3 cm) square aperture faceplate. c) Large (⌀10 cm) square aperture faceplate. d) Small (⌀3 cm) square aperture faceplate. e) Large (⌀10 cm) beveled aperture faceplate. f) Small (⌀3 cm) beveled aperture faceplate.

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4. Conclusion

A grey body Monte Carlo ray tracer was developed and used to analyze a cylindrical calorimeter for solar simulator calibration. Worst-case emissive and convective losses from the calorimeter were predicted by finite volume simulation. The calorimeter was most accurate when mounted with a beveled (conical) aperture faceplate. For both a large aperture and small aperture this geometry brought calorimeter losses within 1.5% of the incident power. Conversely, use of a square aperture could induce relative losses of 15%. Generally, calorimeter performance was robust to mild changes in cavity reflectivity. It is noteworthy that the calorimeter, like any radiation target, can alter the solar simulator flux profile significantly. This effect should be considered when studying any device within a high-flux solar simulator.

5 Appendix A

In this appendix we list variables and equations of the finite volume simulation. Table 1 lists variables in the solver. Heat transfer modeling followed the principles outlined in Bird Stewart and Lightfoot (2002) [25].

Table 1. Variable definitions in the finite volume simulation.

View Table

The governing conservation equation for a given finite volume was:

(3)$$\frac{\partial [rN(r,x)]}{\partial r}+r\frac{\partial [N(r,x)]}{\partial x}=0$$

For radial conductive heat transfer flux was taken as the temperature gradient (Fourier’s law):

(4)$$N_r=-k\frac{\partial T}{\partial r}$$

Axial conductive heat transfer was also taken as the gradient of temperature (Fourier’s law):

(5)$$N_x=-k\frac{\partial T}{\partial x}$$

The convective flux at exposed surfaces was taken as (Newton’s law of cooling):

(6)$$N_s=h(T_s-T_f)$$

Equation (58) of Mori and Nakayama (1967) served as the heat transfer coefficient, h, in spiral coolant ducts [26]. Equation (45) of Kaya and Teke (2005) served as the heat transfer coefficient, h, in cylindrical coolant ducts [27]. Equation (7) of Prakash et Al. (2008) served as the heat transfer coefficient, h, for natural convection to the ambient air [24]. Grey body radiative exchange between exposed annuli was dictated by the Stefan-Boltzman law with view factors A-20, B-94 and A-16 of Howell (2002) as appropriate [22]:

(7)$$N_{rad}=\sigma \epsilon F{T}^4$$

Acknowledgments

Work on this project was funded by Department of Energy Advanced Research Program Agency – Energy (ARPA-E) Award AR0000404. The authors are grateful to the Department of Energy for their support and Dr. Roman Bader for generously sharing his code. Additionally, the authors thank the University of Texas Elements of Computing program for their commitment to accessible programming education, especially Michael Scott whose instruction enabled the present work.

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