Mixed convection and radiation from an isothermal bladed structure

International Journal of Heat and Mass Transfer 147 (2020) 118906

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Mixed convection and radiation from an isothermal bladed structure Juan F. Torres a,⇑, Farzin Ghanadi b, Ye Wang a, Maziar Arjomandi b, John Pye a,⇑ a b

Research School of Electrical, Energy and Materials Engineering, The Australian National University, Canberra, ACT, Australia School of Mechanical Engineering, The University of Adelaide, Adelaide, SA, Australia

a r t i c l e

i n f o

Article history: Received 13 November 2018 Received in revised form 14 October 2019 Accepted 14 October 2019

Keywords: Mixed convection Heat transfer control Convective heat losses Parallel plates CFD Solar thermal receivers

a b s t r a c t The total heat transfer from a finned or ‘bladed’ structure may be expected to rise from that of a flat geometry due to a larger area exposed to ambient air. However, this is not always the case because a trade-off can be found between convection and radiation as the bladed geometry and surface temperature are varied. Furthermore, a mixed convection regime adds complexity to the heat exchange between the surrounding air and the heated bladed structure, which is relevant to various applications such as cooling and solar thermal energy. In this study, mixed convection and radiation heat transfer were determined for a cuboid with isothermal blades protruding from one of its sides. A steady-state simulation based on a three-dimensional SST k–x turbulence model was performed in OpenFOAM to estimate the average convection heat transfer coefficient as a function of structure orientation, convection regime and bladed geometry. A Monte Carlo ray tracing method was employed to calculate view factors for determining the radiation heat transfer coefficient. Wind tunnel experiments validated the combined numerical approach. An increasing pitch angle (starting from the vertical) gradually increased convection heat transfer until a maximum value at
45 and then significantly decreased it, where both effects were due to a flow behaviour in which vortices between the blades formed (< 45 ) but then disappeared (> 45 ). The velocity and wall temperature of the mixed convection flow in between the dominantly natural and forced convection regimes were identified. In comparison to a flat geometry, the bladed structure increased the temperature value at which the heat transfer by convection and radiation equalised, and produced a lower total heat transfer coefficient than the flat configuration at high surface temperatures. Furthermore, flow bifurcations were observed between the blades as their length was increased. In contrast, a large number of blades led to a flow transition where most of the incoming air exited from the side apertures to a lid-driven-like convection with flow recirculation occurring within the blades, which created a thermal barrier that decreased convection heat transfer. These results support the design of bladed or finned structures that enhance or reduce heat transfer by mixed convection and/or thermal radiation. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction Mixed convection is an important heat transfer regime [1] relevant to heated structures in windy conditions. Furthermore, radiation generally becomes the dominant heat transfer mode as the surface temperature is increased based on Stefan–Boltzmann law where the black body emission is proportional to the temperature to the power of four, i.e. qrad / T 4 . For example, in receivers used in solar central receiver systems, which is the motivation for this study, banks of tubes (up to 30 m long) are vertically mounted atop a tower (up to 250 m) and irradiated with concentrated sunlight ⇑ Corresponding authors. E-mail addresses: [email protected] (J.F. Torres), [email protected] (J. Pye). https://doi.org/10.1016/j.ijheatmasstransfer.2019.118906 0017-9310/Ó 2019 Elsevier Ltd. All rights reserved.

from a heliostat field (with a radius of up to 1.6 km). The receiver surface may be heated up to high temperatures of
770  C [2] and can be exposed to a broad range of natural or free convection (buoyancy) and forced convection (wind) regimes during its operation. Conventional solar thermal receivers have either an external or cavity design [3]. Cavity receivers are characterised of having rather low thermal losses and high absorption efficiency [4–6] but are not well suited for a large-scale surround heliostat field. In contrast, conventional external receivers have a wider acceptance angle suitable for large heliostat fields, but radiation and convection losses are significant due to large view factors and an overexposure to the colder ambient air, respectively. Recently, new bladed receiver designs have been proposed for improving light trapping and reducing thermal emission [7–10]. A bladed design exhibits similar concave shape

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Nomenclature A B E F g G h i; j J k L N Pheat q Q Q_ RBS Ri S T TI u U W x; y; z X

surface area of isothermal boundary, m2 blade length, m emissive power, W m2 view factor, – acceleration of gravity, m s2 irradiation, W m2 heat transfer coefficient, W m2 K1 elements in the ray tracing model, – radiosity, W m2 turbulence kinetic energy, m2 s2 characteristic length, m total number of blades, cells or layers, – total power from a heating element, W heat flux, W m2 Second invariant of the velocity gradient, s2 heat transfer rate (or heat loss), W blade length to spacing ratio, – Richardson number, – blade spacing, m Temperature, K turbulence intensity, % velocity, m s1 freestream wind speed, m s1 side of the square back wall, m Cartesian coordinates, m reattachment length, m

as a cavity receiver with its benefits, but is also applicable to external receivers. Although a larger surface area could increase thermal losses (i.e. heat transfer rate), the heat transfer coefficient (i.e. heat transfer rate per unit area per unit temperature) may be reduced due to a similar blocking effect as in cavity receivers. Nonetheless, predicting mixed convection from a bladed structure is challenging due to its complex geometry. Torres et al. [11] investigated the convection from a cuboid, which is a simpler geometry than a bladed structure, with an isothermal wall for Richardson numbers Ri in the range of 5  103 < Ri < 2. Here, Ri is the ratio of buoyancy to shear flow terms. They reported that convection heat transfer strongly depends on the cuboid orientation (pitch angle and wind direction), as well as on the freestream wind speed and turbulence intensity. For the flat configuration, flow separation at the cuboid edges and reattachment on its hot side was the cause for the drop and subsequent increment of convection heat transfer rate. This behaviour was also observed in high-fidelity CFD models of flat receivers [12]. For high Richardson numbers, at which momentum exchange through buoyancy is stronger than the externally induced momentum, Torres et al. [11] reported some discrepancy between experiments and simulation at characteristic angles, but overall a good agreement was obtained. For complex geometries, such as L- or U-shaped obstacles in wind studied by Gomes et al. [13] or the bladed structure considered in this study, the flow is usually ‘disturbed’ by the irregular shape, promoting flow separation, thicker boundary layers and vortex generation. Concerning the prediction of convection around bladed structures, Nock et al. [14] used a computational fluid dynamics (CFD) simulation to study convective heat losses from non-isothermal small-scale models of flat and bladed receivers. For their chosen wall temperature profile, which was similar to a real solar thermal receiver, the blades yielded a marginal decrease of heat transfer

Greek symbols b thermal expansion coefficient, K1 d near-wall layer thickness, lm D difference h pitch angle, degrees k thermal conductivity, W m1 K1 q reflectivity, –  thermal emissivity, – / wind direction, degrees x vorticity or turbulence dissipation rate, s1 Subscripts 1 ambient condition b blades c cells or critical cond conduction conv convection edge blade edge exp experiment err error film mean value for wall and ambient air L mesh layers rad radiation sim simulation sum total w wall

coefficient (i.e. heat loss per unit area per unit temperature) in comparison to the reference flat case. However, they did not validate their CFD model or investigate the effects of pitch angle, wind direction and bladed geometry on convection heat transfer. Furthermore, the more fundamental configuration of an isothermal boundary condition can be used to study the temperature dependence of the heat transfer coefficient. Heat transfer and corresponding flow behaviour are known to change in natural convection with varying pitch angle [15,16], and could change even further in a mixed convection regime. Concerning the effects of radiation on convection, Kogawa et al. [17] investigated turbulent natural convection of a participating gas in a cubical cavity at atmospheric pressure. They showed that surface radiation was dominant in the generation of the shear stress by the turbulent flow. However, the working fluid of interest in this study, i.e. air at atmospheric pressure, has a low attenuation coefficient and can be considered as non-participating medium for length scales < 30 m (i.e. the size of large receivers). Miroshnichenko and Sheremet [18] showed that, for a non-participating medium, the average convective Nusselt number increased with the Rayleigh number Ra but decreased with the surface emissivity , whereas the average radiative Nusselt number increased with both Ra and . They also showed that a presence of surface thermal radiation effect leads to an expansion of the eddy viscosity zones close to the walls [19]. These studies suggest that radiation may have effects on surface mixed convection, even in the case of non-participating media, but those effects are neglected in the present study. The objectives of this study are: (1) to understand the characteristics of mixed convection from bladed structures, associating convection heat transfer coefficients with flow behaviour, (2) to compare the heat transfer modes of convection and radiation assuming black body emissions, and (3) to identify design

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configurations that reduce or enhance heat transfer. The investigated geometry consisted of a solid cuboid (400  300  80 mm) with an isothermal wall from which horizontal blades protrude (at the same wall temperature). Wind tunnel experiments were conducted to validate the CFD model, which was used extensively for computing the convective heat rates and visualising the flow field. Six parameters in three categories were investigated. The bladed structure orientation was set by the pitch angle (h) and wind direction (/); the convection flow regime was controlled by varying the wind speed (U) and wall temperature (T w ); the bladed geometry was modified by the ratio of blade length to spacing (RBS ) and number of blades (N b ). 2. Methods A CFD model was built to simulate mixed convection from a bladed structure (Section 2.1.1), while a view factor calculation was used to study radiation heat transfer (Section 2.1.2). Wind tunnel experiments were conducted using electric heaters inside a bladed structure to obtain the combined heat transfer rates of convection and radiation (Section 2.2.1). To estimate convection and radiation in the wind tunnel experiments, a Monte Carlo ray tracing (MCRT) method was used to determine the view factor profiles on the non-isothermal bladed structure surface whose temperature profile was measured (Section 2.2.2). Although an independent validation of each model (convection and radiation) was not possible, the combined CFD/MCRT modelling approach had a good agreement with the wind tunnel experimental data (Section 2.3). 2.1. Simulations 2.1.1. Mixed convection modelling A Shear-Stress Transport (SST) k–x model is a two-equation eddy-viscosity model that combines features of the widely used k–x and k– turbulence models [20]. In this study, SST k–x was employed in OpenFOAM (version 3.0.1) to investigate the dependence of convection on the rig orientation ðh; /Þ, flow regime ðU; T w Þ and bladed geometry ðRBS ; N b Þ. Modelling details including the governing equations and discretisation methods are described in [11], which is a study on the non-bladed configuration. The SST k–x model was validated against experimental results on mixed convection from flat surfaces by Harris [21] and Torres et al. [11], as well as compared with alternative turbulence models (k– and Launder Sharma k–). The Richardson number Ri provides the ratio of contributing factors of buoyancy to forced convection [1], and is defined as

Ri ¼

gbLðT w  T 1 Þ U2

;

ð1Þ

where g is the acceleration of gravity, b is the volumetric thermal expansion (calculated as b ¼ 1=T film where T film ¼ ðT w þ T 1 Þ=2 is the film temperature in kelvin), T w and T 1 are the wall and ambient temperature, respectively, L is the characteristic length, which for a bladed structure is not defined in the literature, and U is the freestream velocity. Fig. 1 shows the geometry of the bladed structure in a perspective (a), side (b) and top (c) views. The acceleration of gravity is in the negative y direction while the freestream wind velocity is in the positive x direction. The structure orientation is defined by two angles: the pitch angle h (i.e. inclination relative to the vertical) and wind direction /, or yaw angle. Note that h ¼ 0 and 90 correspond to vertical and horizontal back walls (blades facing downwards), as shown in Fig. 1(b), while / ¼ 0 and 90 correspond to headwind and crosswind configurations, as shown in Fig. 1(c). The blade spacing S for a given bladed geometry is uni-

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form and defined as the shortest distance between two opposing blade faces, as indicated in Fig. 1(b). The blade length B to spacing ratio RBS was varied together with the number of blades N b (Fig. 1 shows a specific bladed geometry of RBS ¼ 3 and N b ¼ 5). The dimensions indicated in Fig. 1 were chosen to: first, achieve a mixed convection regime dictated by the value of Ri (as explained below); second, compare with previously published results for the non-bladed configuration [11]; and third, compare simulation results with wind tunnel experimental data (which also include the radiation component). The experimental rig does not include the top and bottom blades shown in Fig. 1 (see Fig. 4). Fig. 2 shows the three-dimensional computational domain (a), mesh detail surrounding the bladed structure (b) and mesh refinement around the blade edge (c). An automatic generation procedure of a structured mesh was devised using OpenFOAM’s blockMesh and snappyHexMesh together with Gmsh [22], which was used to generate the blade surface geometry. The blade front and side edges were smooth to avoid sharp edges that may not resolve the breakdown of vortices and wake development. The inlet was set to a constant speed U with a fixed turbulence intensity of TI ¼ 1% for boundary conditions with turbulence kinetic energy k (the effects of varying TI in the range of 1 6 TI 6 5% is discussed by Torres et al. [11]). When investigating the effect of other parameters, the wind speed was set to U ¼ 6 m/ s, which yielded a Richardson number of Ri  0:04, producing a rather forced convection regime [1] if the characteristic length is taken as the side of the square back wall L ¼ 0:3 m. The outlet was fixed to the atmospheric pressure and other boundaries set at zero pressure gradient. The inlet flow was set at ambient temperature of T 1 ¼ 25  C. The isothermal surfaces of the bladed structure, i.e. the blades and back wall, were set to T w ¼ 100; 200, and 300  C when investigating the effects of other parameters; the remaining sidewalls were set adiabatic, as indicated in Fig. 2(b). Only in the validation, the blades were set non-isothermal to match the measured temperature profiles in the experiment (plotted on the model surface in Fig. 4(d)). Concerning the parametric study for the six variables of interest, the angles were increased in steps of Dh; D/ ¼ 5 in the range 0 6 h; / 6 90 . In an application, the orientation (h and /) of a bladed receiver would generally be decided based on the heliostat field, with blades pointing in the direction of the optical axis of the field to optimise light trapping [23], e.g. a pitch angle of h ¼ 0 would be suitable for a horizontal optical axis such as the megawatt solar furnace in Ref. [24], while the wind direction / depends on the surrounding topography and atmospheric conditions. The aspect ratio was increased by DRBS ¼ 0:25 from RBS ¼ 0:25 to 4. The number of blades was increased in odd numbers to have a blade protruding from the mid-back wall, i.e. Blade 0 indicated in Fig. 2(b), from N b ¼ 3 to 37. The temperature was increased in steps of DT w ¼ 50  C in the range 50 6 T w 6 800  C; the freestream wind speed was varied by DU ¼ 0:5 m/s in the range 0:5 6 U 6 9:5 m/s. If the characteristic length is assumed to be L ¼ 0:3 m, the convection regime varies from forced at Ri  2:27  103 (for T w ¼ 50  C and U ¼ 0:5 m=s) to natural convection at Ri  8:5.

2.1.2. Radiation modelling Air (wind) was assumed to be a non-participating medium and hence the radiation is simulated independently to the CFD simulations. There were two radiation models devised in this study. The first was a diffuse, grey and non-isothermal surface model applied to the experimental test case, with the purpose of validating the combined convection/radiation model. The second was a black body isothermal surface model applied to the parametric study, with the purpose of comparing the heat transfer modes of convection and radiation.

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Fig. 1. Geometry of the bladed structure when the aspect ratio and number of blades are RBS ¼ 3 and N b ¼ 5, respectively. The freestream wind is in the positive x direction, while gravity is in the negative y direction. (a) Perspective view indicating the orientation of the structure in terms of the pitch angle h (measured from vertical axis y) and the wind direction or yaw angle / (rotation about vertical axis y). The blades and back walls (i.e. surface between adjacent blades) are heated; the sidewalls are adiabatic. (b) Side view when the wind direction is / ¼ 0 , i.e. headwind. (b) Top view when the pitch angle is h ¼ 0 , i.e. vertical. The experimental rig does not have the top and bottom blades indicated in (a).

Fig. 2. Computational domain with mesh for a bladed structure with aspect ratio of RBS ¼ 3, number of blades of N b ¼ 5 and headwind orientation / ¼ 0 . (a) Large computational domain; half of its surrounding boundary are shown due to symmetry and to indicate the location of the bladed structure inside the domain. (b) Detailed view of the mesh around the bladed structure showing the finer mesh as the flow (not shown) approaches the walls. (c) Detail of layered mesh around the blade curved edges for the grid dependence study. The validation model does not have blades þ2 and 2 indicated in (b).

In a non-isothermal bladed structure, as in the experiments reported in this paper, there is thermal radiation exchange between neighbouring blades and the back wall. This internal radiative exchange must be considered to obtain the rate of total radiative heat losses from the bladed structure to the ambient. In the experimental test case, thermal radiation was calculated by the view factor matrix and radiosity network method demonstrated in Bergman et al. [25] with the assumption of diffuse and grey surfaces. As shown in Fig. 3, the bladed structure was treated in three separate parts: (a) the blade front and side edges, where the view factor to the ambient is equal to 1; (b) a section constituted by the back wall and the top of the first (or the bottom of the last) blade, which only applies to the experimental test case;

(c) a section constituted by the back wall in between two adjacent blades (i.e. the bottom of the upper blade and the top of the lower blade). The rate of radiative losses from the blade edges (a) are calculated by edges X

 T 4i  T 41 Ai :

Q_ rad;edges ¼ r

ð2Þ

i

Non-isothermal temperature along the side edges can be accounted for in Eq. (2) (e.g. as in the experiment shown in Fig. 4 (c)). The open cavity sections were discretised in surface elements, as shown in Fig. 3. The net radiative heat flux qrad;i for an element i

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Fig. 3. Radiation model for the non-isothermal bladed structure in the experiments (shown in Fig. 4). (a) The edge section comprising two side edges and one front edge per blade, with view factors of F edge ¼ 1. (b) A section corresponding to the upper and lower sections of the rig with one blade surface and one back wall. (c) The section corresponding to the open cavity between two adjacent blades with a back wall. The surface element i is highlighted while indicating the incoming and outgoing radiation.

Fig. 4. Wind tunnel experiment with infrared (IR) camera measurement. (a) Schematic of the rig installed in the Adelaide large-scale wind tunnel, with IR camera pointing from the side of the rig. (b) CAD model showing the rig dimensions when RBS ¼ 0:8. (c) IR-image for the middle blade for which the temperature distribution can be extracted; freestream flow from left to right. (d) CFD model showing the surface temperature (boundary conditions) imported from IR-images, used for a validation case (Section 2.3); the blue surface was set adiabatic. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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is equal to the difference between the outgoing radiosity J i (i.e. emitted and reflected radiation) and the incoming irradiation Gi , or the sum of the emissive power Ei and reflected portion of the incoming irradiation:

qrad;i ¼ J i  Gi ¼ Ei þ qGi ;

ð3Þ

where q is the reflectivity of the surface. The emissive power can be obtained by the Stefan–Boltzmann law, surface emissivity and the corresponding temperature profile. The incoming irradiation was P calculated as Gi ¼ nj¼1 F ij Jj , where F ij is the view factor between elements i and j, while n is the total number of elements. A set of linear equations was solved in terms of radiosity, thereby allowing for obtaining the net radiative heat flux from each element. The rate of total radiative losses to the ambient is the summation of the radiative heat flux times the surface area of each element, as in Eq. (2) for the side edge. The view factor F ij was obtained by a Monte Carlo ray tracing (MCRT) method [26] using Tracer, an open-source Python-based MCRT library [27,28]. The number of rays was increased progressively until the view factors satisfied the summation rule and reciprocity rule. The view factor obtained by the MCRT program was verified with the view factors of geometries that have analytical solutions presented in [29]. The number of elements was eventually chosen as the same resolution of the temperature distribution that was obtained from the experimental test, which is 1  20 on a blade and 1  1 on the back wall. This seemingly coarse mesh was chosen after a much more refined mesh was tested with 30  20 on the blade and 30  20 on the back wall sections. Note that the number of mesh elements is 30 and 600 times that of the coarse resolution. However, the computational time for the fine mesh was 18,000 times that of the coarse mesh due to an interactive heat exchange between the elements including multiple internal reflections. Importantly, the change in the calculated total radiative losses from coarse to fine mesh was only 0.33%, which indicates that the influence of the mesh size is insignificant on an isothermal surface. In the parametric study (second radiation model), a constant surface temperature over a bladed structure and black body emission were assumed. The isothermal condition allows the surface to be treated as a single (large) element k, instead of being discretised into a number of smaller elements i as for the validation case. This simplification was verified using the more refined mesh of the first model, as well as a comparison with an analytical solution [29]. Hence, the view factor of each surface (either blade surface or back wall) to the open aperture F i1 was numerically obtained. According to the black body emission model, the radiative heat flux from   each surface i is simply qrad;k ¼ rF k1 T 4k  T 41 , so that the net radiative heat transfer rate Q_ rad can be obtained by adding the radiative losses from all surfaces. 2.2. Experiments 2.2.1. Wind tunnel Fig. 4 shows the experimental apparatus and an infrared (IR) camera measurement. The bladed structure was placed in the open test section of the Adelaide Wind Tunnel at the University of Adelaide, as shown in the sketch of Fig. 4(a) (top and front views). The streamwise length of the test section is 5:5 m with an inlet crosssection of 2:75  2 m (width  heigh), which allows the rig and its stand to be placed in the wind tunnel without blockage limitations. The IR camera, located as shown in Fig. 4(a), captured the temperature distribution along the blades of the bladed structure shown in Fig. 4(b). Only the headwind configuration (/ ¼ 0 ) is reported in the experimental validation. Preliminary

two-dimensional measurements of the blade surface temperature showed that the short blade (B ¼ 58 mm) produced a quasi-onedimensional thermal condition when / ¼ 0 . Hence, the temperature along the side edges, e.g. shown in Fig. 4(c), could be used to set the temperature boundary conditions on the blades’ surface in the CFD model for the validation, as shown in Fig. 4(d). Aluminium was chosen for the blades and back wall due to its low thermal emissivity ( ¼ 0:12) and good thermal conductivity (k ¼ 205 W=m  K). The temperature of the back wall surface (300  300 mm mm) was measured with thermocouples embedded in the aluminium plates and PID controlled with heating elements. The insulated surfaces consist of a 8-mm thick high-strength reinforced silica matrix composite of low thermal conductivity (k  0:6 W=m K at 300  C) (RSLE-57, ZIRCAR Refractory Composites). Details of the internal structure of the rig can be found in [11]. For an aspect ratio of RBS ¼ 0:8, three horizontal blades of B ¼ 58 mm (thickness of 3 mm) were fixed to the back wall. The blade front edge was rounded, as in the CFD model shown in Fig. 2(c). Heat was conducted from the isothermal back walls through the blades, so that the temperature gradually dropped as shown in Fig. 4(c) due to heat losses to the surroundings (there was no heat generation in the blades). The IR-camera measurement was first calibrated by tuning the emissivity setting in the camera so that its surface temperature measurement of an aluminium plate (same material used in the bladed structure) matched the temperature reading of a thermocouple welded on the same aluminium plate, i.e. the thermal contact resistance between the plate and the thermocouple was neglected in the calibration. The surface temperature was set to T w ¼ 200  C so that the reflected emission from the surroundings at room temperature were also neglected. The measurements of the temperature along the blades, e.g. shown in Fig. 4(c) for the middle blade, indicated that the thermal contact resistance between the base and blades was negligible relative to the temperature drop along the blades. In a quasi-steady state, an energy balance of the heated plates yields the total losses from the surface exposed to the ambient air: convective Q_ conv;exp plus radiative Q_  heat losses, which rad;exp

can be written as

Q_ conv;exp þ Q_ rad ¼ Pheat  Q_ cond;exp ;

ð4Þ

where P heat is the total power dissipated by the four heating elements during a wind tunnel experiment and Q_ cond;exp is the conductive heat loss through the rig. Q_ rad is the radiative heat loss to the ambient (at T 1 ¼ 25  C) and the superscript () indicates that this value can be estimated by inputting experimentally measured temperature profiles into a radiation model (see Sections 2.1.2 and 2.2.2). P heat was adjusted by a PID control system that sets the back wall temperature to a desired T w . The PID control had a 2:5% fluctuation in power during steady-state operation and the power had a 2:75% uncertainty at fixed power values. Q_ cond;exp was experimentally determined by removing the blades and covering the front with a high-insulation material (k ¼ 0:035 W=m K, thickness of 8 mm; Fibertex 450 Rockwool, CSR Bradford), as described in [11]. It is estimated that the experimental error of Pheat  Q_ cond;exp is within
8%, accounting for the error due to fluctuations in the PID control and uncertainties in the values of power and thermophysical properties (thermal conductivity for determining Q_ cond;exp and thermal emissivity for Q_  ). rad

2.2.2. Estimation of heat loss components: radiation and convection Q_ rad was estimated by the method described in Section 2.1.2 with the temperature profile that was experimentally measured

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using the IR camera. Fig. 5 shows the calculation results of the selected view factors F and radiation heat fluxes qrad for the experiment shown in Fig. 4(c). Q_ conv;exp can be determined from Eq. (4) by substituting the simulation-aided result of Q_ rad . Therefore, in a strict sense, Q_ conv;exp is a pseudo-experimental result which is subject to errors in the radiation modelling. Separate measurements of radiation and convection would have required a cryogenic wind tunnel [30]. Despite of this apparent limitation, MCRT-aided estimation of Q_ conv;exp is thought to be a good approximation of experimental convection because (1) prediction of radiative heat transfer generally carries a much lower degree of uncertainty than prediction of turbulent mixed convection due to the non-linearities associated with the Navier–Stokes equations, and (2) the magnitude of convection heat transfer is much larger than radiation in the experiments reported in this study (Section 2.3). Hence, the difference that stems from the comparison of the combined CFD/MCRT approach and the wind tunnel experiments, i.e. total heat transfer rate (radiation and convection), is thought to be mainly due to inaccuracies in the convection modelling. 2.3. Verification and validation For accurately simulating the thermal boundary layer, a mesh refinement varying the number of layers around the isothermal blades was conducted, as indicated in Fig. 2(c). Table 1 summarises the grid convergence study where the number of mesh layers N L , number of cells N c , total heat transfer rate Q_ sum and corresponding difference Dsum relative to the finest mesh (defined as jQ_ sum;NL ¼16  Q_ sum;NL j=Q_ sum;NL ¼16  100%) are listed. The near-wall layer thickness d around the blades became smaller as the number of layers was increased, but at a higher computational cost. In this paper, a refined mesh with eight layers (N L ¼ 8) was chosen as a good compromise between uncertainty and computational expense; only the wind speed (Section 3.2.1) and number of blades (Section 3.3.2) calculations used a four-layered mesh (N L ¼ 4). The refinement shown in Table 1 for the thermal boundary layer was preceded by a grid independence study (without the boundary mesh) for the mesh farther from the rig. Preliminary results for the coarsest mesh (N L ¼ 0) are reported in [31,32].

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A comparison between experimental Q_ conv;exp and simulation _ Q conv;sim (CFD) results for convection is shown in Table 2. Note that Q_ conv;exp was calculated from Eq. (4), i.e. using the experimentally measured Pheat  Q_ cond;exp and the simulation-aided estimation of experimental radiation loss Q_ rad . The error Derr , defined as jQ_ conv;exp  Q_ conv;sim j=Q_ conv;exp  100%, was within 4%, which is less than the estimated experimental error of
8%, validating the combined CFD/MCRT model approach. Although independent validation of convection (CFD model) and radiation (MCRT model) was not achieved, Q_ conv;exp was considered accurate enough because heat transfer by convection was more than 10 times that of radiation (see Table 2), e.g. a 20% error in the radiative heat loss estimates would only cause a
2% error in the convective heat loss calculation. The good agreement between experiment and simulation shown in Table 2 suggests that the combined model is a good approach. Although the precision of the CFD simulation may decrease for different flow regimes, e.g. at higher wind speeds and wall temperatures, the small error Derr reported in Table 2 suggests a good accuracy of the CFD simulations for the parameter range investigated in this study. 3. Results and discussion The heat transfer coefficient was determined by numerical simulations for varying rig orientation (Section 3.1), convection flow regime (Section 3.2) and bladed geometry (Section 3.3). Two parameters in each category were investigated. Previously reported convection heat transfer rates Q_ conv;sim for the nonbladed case [11] and simulation results obtained for the bladed model (Section 2.1.1) were further post-processed to obtain the convection heat transfer coefficient hconv defined as

hconv ¼

Q_ conv;sim ; AðT w  T 1 Þ

ð5Þ

where A is the surface area of the isothermal surface, i.e. blades and back walls. Likewise, the radiative heat transfer coefficient hrad was determined as in Eq. (5) but using the Q_ rad obtained with the view factor calculation explained in Section 2.1.2 (note that the superscript ⁄ is now removed because the Q_ rad reported hereinafter is for an ideal isothermal structure). 3.1. Rig orientation

Fig. 5. Radiative heat transfer for the validation experimental case, corresponding to the measured non-isothermal temperature profile shown in Fig. 4(d), when U ¼ 6 m=s, T w ¼ 200  C (back wall temperature) and RBS ¼ 0:8. (a) View factor for each internal surface in relation to the front and side apertures, and (b) radiation R heat flux qrad . The radiative heat loss is calculated by Q_ rad ¼ qrad dA, where dA is a surface area element.

3.1.1. Pitch angle With regards to the non-bladed (flat) cuboid, Fig. 6(a) shows hconv as a function of h for T w ¼ 100; 200 and 300  C with a headwind of U ¼ 6 m=s. The radiation heat transfer coefficient for these temperatures was hrad ¼ 8:7; 13:7 and 20:6 W=m2 K, respectively, indicating that convection was dominant. The heat transfer coefficients are reported up to three significant figures, which is consistent with the
3:6% error in Table 2. For h < 55 , hconv at T w ¼ 300  C (solid line) was slightly larger than at T w ¼ 100 and 200  C in Fig. 6(a), suggesting that buoyancy enhanced the upward flow which in turn increased heat transfer at larger temperatures, a result consistent with observations for natural convection [33,34]. The decrease of hconv in the range 30 < h < 55 was due to both opposing buoyancy against the incoming flow and an upward shift of the stagnation point. For h > 55 , there was an enhancement of convection due to a reattachment of the separated flow from the leading edge, as described in [11]. Interestingly, in the convection-enhanced range, T w ¼ 100  C yielded a higher heat transfer coefficient than other temperatures.

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Table 1 Grid convergence study using the experimental validation case with the geometry and boundary conditions shown in Fig. 4(d) where T w ¼ 200  CC, U ¼ 6 m=s, h ¼ 70 and / ¼ 0 . The mesh was refined adding N L layers around the blades, as illustrated in Fig. 2(c). The total cell number N c , near-wall layer thickness d, total heat rate Q_ sum , and the corresponding difference with the finest mesh Dsum are listed for each mesh. NL

0

1

2

4

8

16

3.99

4.43

4.86

5.74

7.48

10.78

d Q_ sum

[106 ] [lm] [W]

440.0 739.7

100.0 719.7

87.0 713.5

65.8 722.4

37.6 720.4

12.3 718.6

Dsum

[%]

2.94

0.15

0.70

0.53

0.26

–

Nc

Table 2 Comparison between experiments and simulations. The experimental results are reported for the simulation-aided radiation loss Q_ rad (from measured temperature distributions) and convective heat loss Q_ conv;exp when T w ¼ 200  C, h ¼ 70 and / ¼ 0 . Convection is determined by Q_ conv;exp ¼ P heat  Q_ cond;exp  Q_ rad , from Eq. (4). The CFD simulation results Q_ conv;sim correspond to the layered mesh of N L ¼ 8. Experiments Wind speed

raw input

MCRT-aided

Simulation

Difference

CFD

Derr

U [m/s]

P heat

Q_ cond;exp

Q_ rad

Q_ conv;exp

Q_ conv;sim

3:0 6:0

796.4 1125.3

347.0 347.0

39.6 41.2

409.8 737.1

424.6 720.4

3.6% 2.3%

units: W

Fig. 6. Convection heat transfer coefficient hconv obtained by simulations as a function of the pitch angle h for three wall temperatures (see legend) and headwind of U ¼ 6 m/s. (a) For the flat (non-bladed) cuboid previously reported in [11]. (b) For a bladed structure with a geometry of RBS ¼ 3 and N b ¼ 5. The hollow points, included for comparison, indicate hconv for the flat case at T w ¼ 300  C in (a).

Once the blades were introduced in the freestream flow, local flow behaviour and heat transfer significantly differed from the flat configuration. Fig. 6(b) shows hconv as a function of h for the bladed geometry of RBS ¼ 3 and N b ¼ 5, with the same wind conditions and wall temperatures as in Fig. 6(a). For pitch angles of h < 65 , the bladed structure had a larger hconv than the flat case. This was expected because the flow was perturbed by numerous blade edges, resulting in a large portion of the isothermal surface having thin boundary layers that increased convection heat transfer. However, for steeper angles of h > 65 , the flat case had a larger heat transfer coefficient than the bladed case, e.g. almost double at h ¼ 80 . The reason for this behaviour is discussed below in relation to Fig. 7, where the velocity and temperature contours are plotted on the mid-plane (symmetrical for headwind). Note that the isothermal surface area A of the bladed structure was larger than the flat case (0:736 m2 versus 0:090 m2 ), hence the heat transfer rate (in watts) was larger for the bladed case despite having lower values of hconv for h > 65 . For h < 45 ; hconv gradually increased with h, as shown in Fig. 6(b). This increment can be attributed to thinner boundary layers on the upper blade surfaces due to the impingement of separated flow (from blade front edges), as well as to the mixing caused by vortices between the blades, both phenomena indicated in Fig. 7(a). At h ¼ 0 , the flow is almost symmetrical respect to its middle blade, suggesting that buoyancy effects were weak at this wind speed, as a strongly buoyant flow would break that symmetry. For 0 < h < 30 , the originally horizontal plane symmetry was clearly broken with the generation of vortices between the blades, commencing from the lower blade spacing and finalising with the upper spacing, until each compartment had fully structured vortices, as indicated in Fig. 7(a). In this study, vortices were identified by using the Q-criterion [35]. Fig. 7(b) shows contours of Q, i.e. connected fluid regions with a positive second invariant of ru, along with the velocity and vorticity on the mid-plane when h ¼ 45 . Two large vortices were identified in each blade spacing for this aspect ratio, a strong vortex closer to the back wall and a weaker vortex formed by the separating flow from the leading edge. Note also that there are several smaller vortices at the corners near the back wall and between the larger vortices. For h > 55 , there was a significant decrease of hconv , as shown in Fig. 6(b), dropping by more than 50% at h ¼ 90 from the maximum heat transfer coefficient observed at h ¼ 45 . This was due

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9

Fig. 7. Flow patterns for increasing pitch angle h, with simulation U ¼ 6 m=s, / ¼ 0 ; T w ¼ 300  C, RBS ¼ 3 and N b ¼ 5. Contours are plotted on the vertical mid-plane z ¼ 0 of the symmetrical flow for selected profiles. (a) From h ¼ 0 to 40 , (b) at h ¼ 45 and (c) from h ¼ 55 to 85 . In (a) and (c) the velocity u (left) and temperature T (right) are plotted; U is from left to right. In (b) u, vorticity x and Q, i.e. second invariant of ru [35] at h ¼ 45 , are shown.

to the disappearance of vortices between the blade spacings, as the pitch angle was increased beyond 55 , which is evidenced by the flow contours in Fig. 7(c). The blade spacing S and the reattachment length X from the leading edge, which increased with h, played a crucial role in convection heat transfer from the bladed structure. If X < S, then all blade spacings would contain vortices, as for h ¼ 45 in Fig. 7(b). At h ¼ 60 , the reattachment length was S < X < 2S; hence, a pocket of hot air was observed in the first spacing only. The vortices then gradually disappeared at characteristic pitch angles as the pitch angle was increased. Finally, for h > 85 , the reattachment length became X > 4S and, hence, the vortices within the blades disappeared for the bladed geometry shown in Fig. 7. The relationship between the velocity contours (left plots) and the temperature profiles (right plots) can be clearly

seen in Fig. 7(c). That is, pockets of hot air formed between the blades as the pitch angle was increased due to the disappearance of vortices. Concerning the effect of temperature in Fig. 6(b), the heat transfer coefficient for the highest temperature (T w ¼ 300  C) had a slightly lower value than at lower temperatures (T w ¼ 100; 200  C). This may be due to a blocking effect at the inlet of the blades caused by the buoyancy force normal to the incoming flow. The weak dependence on temperature of convection heat transfer, as shown in Fig. 6, suggests that the reported values of hconv can be applied to non-isothermal surfaces with mean wall temperatures in the range 100 6 T w 6 300  C if the temperature difference within the structure is small, although the exact limits for this temperature difference cannot be determined with the present study.

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3.1.2. Wind direction The heat transfer coefficient hconv as a function of the yaw angle / is shown in Fig. 8(a) for a vertical back wall (h ¼ 0 ) when U ¼ 6 m/s. For / < 15 ; hconv was almost constant with a weak dependence on temperature. In contrast, a large drop was observed in the range 15 < / < 50 , which was related to the flow behaviour, as discussed below. The contours in Fig. 8(b) depict the magnitude of velocity on the isothermal surfaces of 50  C for wind directions of interest. The flow field revealed that the boundary layer first developed from the blade front edges at / ¼ 0 but then shifted to the leading vertices for a change in wind direction of / : 15 ! 60 , which gradually decreased hconv due to the thickening of the boundary layer. At wind directions close to a crosswind, e.g. / ¼ 80 , the airflow passed somewhat unimpeded through the blade spacing, producing thinner boundary layers near the side edges (shorter than the front edges) that increased hconv . However, a flow separation from the supporting structure reduced hconv at / ¼ 90 , a phenomenon also observed for the flat configuration [11], which is due to the boundary layer separation from the supporting structure (i.e. sidewalls in Fig. 2(b)). This flow behaviour was also observed by Cagnoli et al. [12] with high-fidelity CFD of a billboard (flat) receiver in a crosswind configuration.

3.2. Convection flow regime 3.2.1. Wind speed The convection heat transfer coefficient hconv as a function of the freestream wind speed U is shown in Fig. 9 for three wall temperatures, a bladed geometry of RBS ¼ 3 and N b ¼ 5, and pitch angles of either h ¼ 30 or 60 in a headwind configuration (/ ¼ 0 ). For h ¼ 30 in Fig. 9(a), the wall temperature dependence of the hconv was marginal in the range 100 6 T w 6 300  C. The higher temperature value (T w ¼ 300  C) yielded a slightly larger hconv when U < 1 m/s; thus, a buoyancy-driven flow was present. In contrast, the lower end of the temperature range (T w ¼ 100  C)

produced a slightly larger hconv when U > 2 m=s, which can be attributed to a weakening of the blocking effect of the hot air within the blades against a colder incoming airflow, or to a thickening of the boundary layer caused by a more viscous hot air. Thus, a wind-dominant heat transfer was identified for this range of temperature and wind speeds. If the Richardson number at which buoyancy and forced convection balance each other is taken as Ric ¼ 1, then the characteristic velocity at which this balance occurs becomes

Uc ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi gbDTL:

ð6Þ

If the characteristic length L is arbitrarily taken as the total length of the blade edge, i.e. L ¼ 2  B þ W where W is the width, then the characteristic velocity becomes U c ¼ 1:2; 1:6 and 1:9 m/s at T w ¼ 100; 200 and 300  C, respectively, which is consistent with the results shown in Fig. 9(a) for the mixed convection band. The choice of characteristic length, however, is arbitrary because no consensus exists for such a choice (considering other parameters, e.g. N b ; RBS ; h or /, may be more suitable). In contrast, as shown in Fig. 9(b), wall-temperature-driven buoyancy was significant at h ¼ 60 because a larger wall temperature significantly increased hconv . This could have been expected because forced convection has largely disappeared in the first blade spacing, as evidenced by the pocket of hot air in Fig. 7 when h ¼ 60 . Furthermore, the assisting flow configuration of buoyancy between the spacings at this angle is thought to be a contributing factors for the increase of hconv . Thinner boundary layers caused by the impinging flow at steeper angles may have also increased hconv . 3.2.2. Wall temperature Fig. 10 shows hconv and hrad as a function of T w for both flat and bladed structures (constant parameters are indicated in the caption). It can be seen that the bladed structure increased convection but decreased radiation, per unit of area per unit temperature. For the flat case at
340  C, both convection and radiation coefficients were equal. In contrast, for the bladed case, both modes of heat

Fig. 8. Convection behaviour as the wind direction / was varied when U ¼ 6 m=s and h ¼ 0 (vertical). The bladed geometry was fixed to RBS ¼ 3 and N b ¼ 5. (a) hconv at wall temperatures indicated in the legend; at / ¼ 15 and T w ¼ 300  C, the steady-state calculation did not converge. (b) Velocity contours plotted on the isothermal surface of 50  CC indicating locations where the thermal boundary layer is thin.

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11

Fig. 9. Convection heat transfer coefficient hconv as a function of the freestream wind speed for / ¼ 0 and three wall temperatures T w indicated in the legends when (a) h ¼ 30 and (b) h ¼ 60 . The radiation heat transfer coefficients are hrad ¼ 8:7; 13:7 and 20:6 W=m2 K for T w ¼ 100; 200 and 300  C, respectively.

Fig. 10. Convection (solid lines) and radiation (dashed lines) heat transfer coefficients for a flat (j) and bladed (H) structure as a function of wall temperature T w for U ¼ 6 m=s, T 1 ¼ 25  C, h ¼ 30 ; / ¼ 0 ; RBS ¼ 3 and N b ¼ 5. The double-line arrow shows the change of ðT w ; hÞ when convection and radiation have the same magnitude for each geometry. The total heat transfer coefficient, i.e. hconv þ hrad , is plotted in the inset.

transfer were equal at
680  C. The change of ðT w ; hÞ when hconv ¼ hrad , from flat to bladed is indicated by the arrow in Fig. 10, while the total heat transfer coefficient hconv þ hrad is plotted in the inset. Interestingly, at temperatures of T w > 200  C, the flat structure (i.e. non-bladed cuboid) was a better heat exchanger with the surroundings than the bladed structure, per unit of area per unit

temperature, as indicated by hconv þ hrad in the inset of Fig. 10. At high temperatures, the radiation greatly dominated the heat transfer, and its coefficient reduction was more pronounced for the bladed case as T w was increased. Furthermore, if non-isothermal blades (colder at the front edges) and grey body emissions had been considered, a further reduction of heat losses (radiation and convection) would have been expected. Note, however, that the

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introduction of blades increased the surface area by
717:9%, from 0:090 m2 to 0:736 m2 , so that the heat transfer rate (in watts) increased for this specific geometry. Regarding the change of convection as T w was increased, Fig. 10 shows an interesting trend (but minor compared to the radiation effect). In the flat case, hconv increased linearly from 23:0 W=m2 K at 50  C to 24:6 W=m2 K at 800  C, which is consistent with a natural convection behaviour prominent at shallow angles (see Fig. 6(a)). In contrast, for the bladed case, hconv decreased monotonically (almost linearly) from 32:4 W=m2 K at 50  C to 30:0 W=m2 K at 800  C, which supports the conjecture that there is a blocking effect on the incoming flow by the hot air within the blades for this particular configuration and wind conditions.

3.3. Bladed geometry 3.3.1. Aspect ratio The heat transfer coefficients hconv and hrad of the bladed structure are shown in Fig. 11 as a function of the aspect ratio RBS at the pitch angles of h ¼ 30 and 60 when T w ¼ 300  C (other constant parameters are indicated in the caption). The values of hrad were smaller than hconv at this wall temperature, and decreased exponentially due to a reduction in view factors close to the back wall as the blade length was increased. The isothermal surface area was linearly proportional to RBS because N b was constant. The results at h ¼ 30 showed a relatively weak dependence of hconv on RBS , varying between 27:9 and 32:2 W=m2 K. In contrast at h ¼ 60 ; hconv varied between 27:4 and 36:7 W=m2 K (at RBS  0:75 and 4, respectively). Heat transfer coefficients were associated with the flow behaviour shown in Fig. 12. For RBS ¼ 0:5, there was only one main vortex within each blade spacing. The vortices follow an elliptical pattern that adjusted itself to the rectangular cross-section blade spacing. However, there was one exception for h ¼ 30 at which the first spacing adjacent to the leading blade had a very weak flow, almost stagnant, as indicated in Fig. 12(a). In contrast, at h ¼ 60 shown in Fig. 12(b), a vortex was observed in the first spacing, which explains the larger hconv at h ¼ 60 than

30 for small RBS (see Fig. 11). The dynamics of the leading vortices (i.e. in the top blade spacing) for h ¼ 30 and 60 shifted as RBS was increased, appearing for the former but disappearing for the latter. Another interesting result was the flow bifurcation observed at RBS  1:5, where the flow changed from a single vortex to two vortices per spacing. The Q-criterion for vortex identification, as plotted in Fig. 12(c) for varying aspect ratio, confirmed this observation. Likewise, flow bifurcation has been reported in [36,15] for natural convection inside a tilted rectangular enclosure when the aspect ratio was changed from 1 to 2. In this study, flow bifurcation was confirmed for both h ¼ 30 and 60 , and was thought to be responsible for the trend change of hconv at RBS  1:5. Multiple smaller vortices were also identified, as shown in Fig. 12(c). The increase of surface area, however, is the driving factor for the total heat transfer rate (in watts), as shown in the inset of Fig. 11.

3.3.2. Blade number The heat transfer coefficients for convection and radiation, as well as the surface area A of the isothermal bladed structure, are plotted in Fig. 13 as a function of the number of blades N b when RBS ¼ 1 (other constant parameters are specified in the caption). The first observation was that hconv and A decreased as N b increased for N b > 7, so that Q_ conv further decreased at large N b without a sig-

nificant change in Q_ rad . For a small number of blades, e.g. N b 6 13, convection heat transfer at h ¼ 60 was larger than at 30 (consistent with results in Fig. 11 for RBS ¼ 1). When N b P 15, however, both pitch angles had roughly the same value of hconv , which decreased linearly with N b . Interestingly, for N b P 25, the heat transfer coefficient of the bladed structure became less than that of the flat plate, which is indicated by the horizontal dashed lines in Fig. 13. The heat transfer rate Q_ conv in watts was still greater for the bladed structure due to its larger surface area (for the flat case A ¼ 0:090 m2). The flow behaviour for one pitch angle (h ¼ 30 ) is then discussed below. When the number of blades is quite reduced, e.g. N b ¼ 3 in Fig. 14(a), a large portion of the airflow exited through the side

Fig. 11. Heat transfer coefficients h (left ordinate) as a function of RBS when N b ¼ 5 and T w ¼ 300  C. For convection, pitch angles of h ¼ 30 ( ) and 60 (M) are plotted when U ¼ 6 m=s and / ¼ 0 (headwind). Radiation is included (). The surface area of the isothermal walls (right ordinate) is plotted by a dotted line. The inset shows the total heat transfer rate Q_ ¼ ADTðhconv þ hrad Þ for both inclinations.

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13

Fig. 12. Flow patterns with increasing aspect ratio, RBS : 0:25 ! 4, for constant values of N b ¼ 5; U ¼ 6 m=s, T w ¼ 300  CC and / ¼ 0 (headwind; freestream flow from left to right). Contours are plotted on the vertical mid-plane z ¼ 0. (a) u and T at h ¼ 30 . (b) Contours at h ¼ 60 with the same variables and scales as in (a); the quasi-stagnant flow in the front blade spacing is a distinctive feature. (c) Q-criterion for vortex identification in the second spacing from the top, showing the transition from one-roll to multipleroll flows for h ¼ 30 as RBS was increased.

apertures, normal to the contour plane (not shown in the figure). In contrast, for a large number of blades, the vortices within the blades stretched, due to a larger aspect ratio of W=S, while reducing their vorticity. This resulted in the external flow ‘sliding’ across the front aperture, as indicated in the inset of Fig. 14(a) for N b ¼ 31, which is a similar behaviour to a lid-driven cavity flow [37]. As a consequence, the air temperature between the blades rose, as shown in Fig. 14(b). Since the mass transfer through the front aperture was reduced, and air is a poor thermal conductor, then the heat exchange between the bladed structure and the incoming airflow dropped significantly, resulting in the observed decrease of hconv . Furthermore, the radiative heat transfer coefficient hrad had a minimum value of 8:6 W=m2 K at N b ¼ 17, which is a reduction of 32:4% from its maximum value of 12:8 W=m2 K at N b ¼ 3. Depending on the application and temperature ranges, the bladed structure design can be tailored to enhance or reduce the heat transfer between the structure and its surroundings. For hightemperature applications where a reduction of heat transfer is desirable, such as in solar thermal receivers, a number of blades of N b  17 would favour the reduction of thermal losses if the temperature is high enough, i.e. radiation is dominant (see Fig. 10). On the other hand, if convection is dominant, as in large-scale

receivers at mid-temperatures, then a greater number of blades keeping a moderate RBS would favour the reduction of losses. In contrast, for applications where heat transfer enhancement is desired, such as in cooling, then N b ¼ 7 would favour total heat transfer in a convection-dominant configuration, whereas a flat surface would favour thermal radiation in a radiation-dominant configuration due to the increase of the effective thermal emittance (or equivalent hrad ). These observations were done per surface area, which should be taken into account in the design process for obtaining a total heat transfer rate, as that shown in the inset of Figs. 11 and 13. Furthermore, it is worth mentioning that in most applications the blades are non-isothermal and the exact heat exchange rate may vary from those reported in this study. For example, bladed receiver designs [10] have pre-determined flow paths that set the blade tip temperature to be colder than the back wall to further reduce thermal losses. In cooling applications, the fin tips are also colder as there is no heat generation in the material, e.g. similar to the cold blades observed in the wind tunnel experiments in this study. Nonetheless, trends of convection heat transfer as the bladed structure orientation, convection flow regime, and bladed geometry are varied should hold for non-isothermal surfaces where internal temperature differences are not too large.

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Fig. 13. The effect of varying the number of blades on the heat transfer when T w ¼ 300  C. For convection, RBS ¼ 1; U ¼ 6 m=s and / ¼ 0 (headwind) for two pitch angles: h ¼ 30 (M) and 60 ( ). The radiation heat transfer coefficient hrad () and the surface area (right ordinate; dotted line) are also plotted. The total heat transfer rate Q_ ¼ ADTðhconv þ hrad Þ for both inclinations is plotted in the inset.

Fig. 14. Flow patterns for increasing number of blades N b when RBS ¼ 1; U ¼ 6 m=s, T w ¼ 300  C, h ¼ 30 and / ¼ 0 (headwind). (a) Velocity contour on the vertical midplane of symmetry. The inset of N b ¼ 31 highlights the low values of velocity within the blades and the lid-driven-like behaviour of convection. (b) Close-up of the temperature field between the blades for N b ¼ 31.

4. Conclusions In this study, a combined model incorporating computational fluid dynamics (CFD) and Monte Carlo ray tracing (MCRT) was devised to predict thermal losses from a finned or ‘bladed’ structure. The combined modelling approach was compared with experimental data acquired in a wind tunnel and a good agreement was found. Convection heat transfer was investigated for a varying bladed structure orientation (pitch h and yaw / angles), convection flow regime (wind speed U and wall temperature T w ), and geometry of the bladed structure (blade length to spacing ratio RBS and blade number N b ). The isothermal structure had fixed back wall dimensions (300  300 mm), horizontal blade tips with a fixed blade thickness (3 mm) and a uniform blade spacing and blade length for a given geometry. The main findings for each varied parameter are summarised as follows. 1. The agreement between experiments and simulations showed that the prediction of the total heat transfer (convection and radiation) from the bladed structure was within
8% of the experimental error. However, each model (CFD and MCRT)

was not validated independently against convection-only or radiation-only experiments, which is challenging given the wind conditions and high surface temperatures. Despite this apparent shortcoming, the experimentally determined convective heat loss was deemed accurate enough given the small uncertainties in the radiation model and the estimated large ratio of convection to radiation heat transfer, about 10 times for the experimental case. 2. For an increasing pitch angle (Figs. 6(b) and 7), the convection heat transfer rate (as well as the heat transfer coefficient due to constant surface area) gradually increased until a maximum value at
45 , and then significantly decreased up to
53% from the maximum due to a flow behaviour where vortices between the blades formed but then disappeared. This transition was associated with varying flow reattachment length. 3. For wind direction (Fig. 8), the convection heat transfer rate (and heat transfer coefficient) for the crosswind configuration was less than for a headwind. This was due to the thickening of the boundary layers from the blade edges and a flow separation from the supporting structure. 4. For wind speed (Fig. 9), the convection heat transfer rate (and heat transfer coefficient) was found to vary almost linearly.

J.F. Torres et al. / International Journal of Heat and Mass Transfer 147 (2020) 118906

Buoyancy effects were marginal at pitch angles of 30 but significant at 60 . In relation to the threshold between free and forced convection, the wind speed was
1  2 m=s for the investigated geometry at a pitch angle of 30 when the wall temperature was less than 300 C. 5. For an increasing wall temperature (Fig. 10), it was found that the bladed structure increased the convection heat transfer coefficient but reduced the radiation heat transfer coefficient. The temperature at which both modes of heat transfer balanced each other increased, e.g. changing from
340  C in the flat case to
680  C in the bladed configuration for the investigated wind condition and geometry. 6. For an increasing aspect ratio (Figs. 11 and 12), a transition from a single vortex to a vortex pair between the blades was confirmed at RBS  1:5, slightly increasing the convection heat transfer coefficient. This coefficient strongly depended on the pitch angle, which introduced a blocking/cavity effect at steep angles. The total heat transfer rate increased due to a larger surface area. 7. For an increasing number of blades (Figs. 13 and 14), a transition to a recirculating flow between the blades was observed for N b > 7 (similar to lid-driven convection), decreasing the heat transfer coefficient due to a larger temperature in a weaker airflow between the blades. A minimum radiation heat transfer coefficient was found at N b ¼ 17 and a maximum at N b ¼ 3. The heat transfer rate significantly decreased due to a reduction in both convection heat transfer coefficient and surface area. These results provide a basis for the design of bladed structures in mixed convection regimes, taking into account both convection and radiation heat transfer modes for a non-participating airflow. Importantly, this study showed that both modes can be controlled by tuning the bladed geometry and orientation for known wind conditions and wall temperatures. For example, in the case of solar thermal receivers, a large number of blades keeping the same aspect ratio seems beneficial because thermal losses were reduced while maintaining the same expected optical (light-trapping) properties. A further reduction of thermal losses may be expected for more realistic conditions where the blade tips are colder than the back wall, or for grey body emission. These results encourage further investigation of heat transfer control for the case of nonisothermal wall temperature profiles considering grey body wall emission and other geometric shapes. Declaration of Competing Interest There are no conflict of interest in this work. Acknowledgements This research was funded by the Australian Renewable Energy Agency (ARENA) for the project Bladed Receivers with Active Airflow Control (2014/RND010). Simulations were undertaken with the assistance of resources and services from the National Computational Infrastructure (NCI), which is supported by the Australian Government. The authors would also like to acknowledge an anonymous reviewer whose rigorous critique resulted in a substantial improvement of the article.

Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.ijheatmasstransfer. 2019.118906.

15

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