Convection heat loss from cavity receiver in parabolic dish solar thermal power system: A review

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Solar Energy 84 (2010) 1342–1355 www.elsevier.com/locate/solener

Convection heat loss from cavity receiver in parabolic dish solar thermal power system: A review Shuang-Ying Wu a,*, Lan Xiao a, Yiding Cao b, You-Rong Li a b

a College of Power Engineering, Chongqing University, Chongqing 400044, China Department of Mechanical and Materials Engineering, Florida International University, Miami, FL 33174, USA

Received 16 June 2009; received in revised form 7 February 2010; accepted 11 April 2010 Available online 18 May 2010 Communicated by: Associate Editor Robert Pitz-Paal

Abstract The convection heat loss from cavity receiver in parabolic dish solar thermal power system can significantly reduce the efficiency and consequently the cost effectiveness of the system. It is important to assess this heat loss and subsequently improve the thermal performance of the receiver. This paper aims to present a comprehensive review and systematic summarization of the state of the art in the research and progress in this area. The efforts include the convection heat loss mechanism, experimental and numerical investigations on the cavity receivers with varied shapes that have been considered up to date, and the Nusselt number correlations developed for convection heat loss prediction as well as the wind effect. One of the most important features of this paper is that it has covered numerous cavity literatures encountered in various other engineering systems, such as those in electronic cooling devices and buildings. The studies related to those applications may provide valuable information for the solar receiver design, which may otherwise be ignored by a solar system designer. Finally, future development directions and the issues that need to be further investigated are also suggested. It is believed that this comprehensive review will be beneficial to the design, simulation, performance assessment and applications of the solar parabolic dish cavity receivers. Ó 2010 Elsevier Ltd. All rights reserved. Keywords: Cavity receiver; Convection heat loss; Mechanism and model; Nusselt number correlation; Wind effect; Parabolic dish solar thermal power system

1. Introduction One of the most cost efficient ways of generating electricity from solar radiation is through a solar thermal power plant, which converts solar heat into electricity. Direct solar radiation can be concentrated and collected by a solar collector system to provide a high-temperature heat source. The heat from the high-temperature source is then converted into mechanical work through an engine, such as a steam turbine or Stirling engine operating in conventional power cycles. Three major solar-thermal systems

*

Corresponding author. Tel.: +86 23 65112284; fax: +86 23 65102473. E-mail address: [email protected] (S.-Y. Wu).

0038-092X/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.solener.2010.04.008

are trough/steam turbine, tower/steam turbine, and dish/ heat engine systems (Mancini et al., 1994). Of these three solar thermal technologies, dish/engine systems have demonstrated the highest efficiency, producing a concentration ratio of more than 3000 and operating at temperatures of 750 °C at annual efficiencies of 23% (Stine and Diver, 1994; Stine, 1993). Dish/engine systems, in general, comprise a parabolic dish concentrator, a thermal receiver, and a heat engine/generator located at the focus of the dish to generate power. As the thermal receiver plays a role of transferring the solar heat to the engine, and heat loss of the thermal receiver can significantly reduce the efficiency and consequently the cost effectiveness of the system, it is important to assess and subsequently improve the thermal performance of the thermal receiver.

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Nomenclature Aap Acav AR a B Dap Dcav DR d Gr H L1 L2 L3 La

La,eff Lc

area of the receiver aperture (m2) area of the inner surface of receiver (m2) aspect ratio of cavity opening height of cubical or rectangular cavity (m) width of cubical or rectangular cavity, or depth of cylindrical cavity (m) diameter of aperture (m) diameter of cavity (m) opening displacement ratio of cavity distance from the centerline of the opening to the base of cavity (m) Grashof number height of cubical or rectangular cavity (m) dimension of wall 1 of cavity (m) dimension of wall 2 of cavity (m) dimension of aperture of cavity (m) projected vertical height between the deepest point of the cavity (in the sense of gravitation) and the upper lip of the aperture (m) effective buoyancy height for the main flow (m) characteristic dimension of the cavity receiver (m)

For capture of punctually concentrated sunlight, cavity receivers are preferred in parabolic dish solar thermal power system because of their many advantages over other types of receivers, such as, less thermal and optical losses, reduced direct heat flux density on the absorber, a nearly uniform internal wall temperature, steady thermal performance, high solar absorption efficiency, low cost and maintenance fee. The thermal losses of a solar cavity receiver include convection and radiation heat losses to the air in the cavity and conduction heat loss through the insulation. The radiation heat loss is dependent on the cavity wall temperature, the shape factors and emissivity/absorptivity of the receiver walls, while conduction is dependent on the receiver temperature and the insulation material. With parabolic dish cavity receivers, conduction and radiation can readily be determined analytically (Holman, 1997). On the other hand, the determination of convection heat loss is rather difficult due to the complexity of the temperature and velocity fields in and around the cavity and usually relies on semi-empirical models. There are too many factors that influence the convection heat loss of cavity receivers, such as the air temperature within the cavity, the inclination of the cavity, the external wind conditions and the cavity geometries. The literature survey shows that the types of receivers investigated both experimentally and numerically are cubical, rectangular, cylindrical, and hemispherical. Fewer studies focus on the convection heat loss mechanism. On the contrary, numerous semi-empirical models for predicting the convection heat loss have been obtained from the

Lcount

Le Ls

Nu OR Pr Ra Tb Tprop Tw T1

counterflow length, corresponding to the vertical projected height between the under lip of the aperture and the cavity wall (m) the ensemble cavity length (m) length of the central section of the cavity wall between the deepest point and the beginning of the stagnant zone (m) Nusselt number opening ratio of cavity Prandtl number Rayleigh number bulk temperature of air (K) temperature at which the air properties are evaluated (K) average operating temperature of receiver wall (K) ambient temperature (K)

Greek symbols u tilt angle of receiver (degree/radian) e emissivity of the receiver surface

experimental and numerical results. However, all these correlations are based on receivers with particular shapes and dimensions. They do not have a universal range of applicability and also sometimes show large discrepancy in heat loss predictions for the similar receiver geometry. On the other hand, there are two distinct cavity orientations that are of interest in problems involving convection heat transfer from cavities, one being cavities with downward-facing inclined orientation and the other with upward-facing inclination. Studies are mostly concerned with the convection from cavities having vertical or downward-facing aperture planes. Less attention has been given to develop and verify expressions to quantify the heat loss from upward-facing receivers. In this paper, a detail review on the research investigations and activities for cavity receiver convection heat loss is performed. The present study differs from the previous ones due to the facts that: (a) the convection heat loss mechanism of the cavity receiver for both no-wind conditions and wind conditions are addressed in detail, (b) this paper provides a comprehensive review on the correlation development for predicting cavity receiver convection heat loss, which are systematically summarized with tables, (c) many fundamental research works on the convection heat loss of cubical and rectangular cavities which can be applied in the solar cavity receivers as well as other engineering systems are also included, and (d) the progress in predicting convection heat loss with wind effect is outlined. Accordingly, the structure of the paper is organized as follows: the first section is the general introduction of the sub-

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ject. Section 2 deals with convection heat loss mechanism; Section 3 deals with natural convection heat loss for cavity receivers in three shapes, while Section 4 discusses the convection heat loss with wind effect. Finally, Section 5 concludes the paper with suggestions for the research directions and issues on solar parabolic dish cavity receivers. One of the most important features of this paper is that it has covered numerous cavity literatures encountered in various other engineering systems, such as those in electronic cooling devices and buildings. The studies related to those applications may provide valuable information for the solar receiver design, which may otherwise be ignored by a solar system designer. 2. Convection heat loss mechanisms for the solar parabolic dish cavity receiver It is important that convection heat loss mechanisms be understood and be predicted with accuracy. However, few investigations have engaged in exploring the convection heat loss mechanisms for cavity receivers. Clausing (1981, 1983) and Clausing et al. (1987) established a correlation based on the understanding of the physics of the convection heat loss associated with a large central cubical cavity. The cavity receiver was divided into two zones — the stagnant zone and the convective zone. The boundary between the zones is formed by a horizontal plane cutting through the cavity at the upper lip of the aperture. Leibfried and Ortjohann (1995) gave an excellent description of the flow configuration in spherical and hemispherical cavities. 2.1. Natural convection heat loss mechanisms When the air in a hot, open cavity is heated, the buoyancy force, initiated by the density gradient, will force a streaming. The mechanisms of convection heat loss of a cavity can be found based on the understanding of the streaming effects. Fig. 1 shows the streamlines, observed in the cavity receivers. We can divide the air flow patterns into the following regimes of streaming (Leibfried and Ortjohann, 1995):  cold and hot air, entering and leaving the cavity, respectively;  a rising main air flow, heating up along the hot wall;  a counterflow clogging the cold air entering the cavity;  a quasi-isotherm bulk air or central eddy, and  a stagnant zone, with a stable stratification of hot air. The heat transfer happens mainly in the convective zone. As the internal air flow exhibits completely different behaviors for a cavity receiver operating at varying tilt angles, the mechanisms of convention loss can be understood by analyzing the streaming effects during its rotation process from downward-facing position of 90° to upwardfacing position of 90°. The tilt angle of cavity receiver u is

Fig. 1. Streamlines in the cavity receiver: (a) streamlines in the downwardfacing tilted cavity receiver and (b) streamlines in the upward-facing tiled cavity receiver.

defined as the angle between the normal direction of the aperture plane and the horizontal line; downward-facing corresponds to positive angle, and upward-facing corresponds to a negative angle. In the downward-facing position of 90°, the cavity is almost dominated by the stagnant zone, thus convection heat loss out of the receiver is negligible. The hot air will rise up, resulting in a hot temperature of the interior air, close to the wall temperature. If there is a gradient in temperature, this occurs in a stable stratification. When the aperture is turned anticlockwise, as shown in Fig. 1a, the extent of the stagnant zone is decreasing with decreasing tilt angles. A part of the hot air can leave the cavity; cold air is streaming into it, which induces a rising main air flow, heating up along the hot wall. A quasi-isotherm bulk air or central eddy can be observed. On the bottom of the cavity, air is heated up and streams in a counterflow to the main circulation directly up to the aperture. The magnitude of this

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counterflow, at a given tilt angle, is dependent on the inner geometry of the cavity. When the cavity receiver is turned upwards, as depicted in Fig. 1b, the convective zone becomes dominant. The boundary layer between the hot wall and the streaming air takes up most of the cavity wall, while the stagnant zone becomes very small. The three effects: increasing main circulation, decreasing stagnant zone, and increasing counterflow result in a maximum of convection heat loss at a tilt angle of 60° to 30°, depending on the geometry of the cavity. When the aperture is facing upward of 90°, no main circulation can take place, due to the counterflow rising from all hot walls to the aperture. The temperature of the hot air in the cavity rises again. 2.2. Free-forced convection heat loss mechanisms (with wind effect) For cavity receivers operating in practical dish/engine systems, the wind can considerably affect the convection heat loss. The resulting convection heat loss is determined by the combined effects of receiver geometries, tilt angles, as well as wind speed and direction. The wind effect can be explained as follows: when no external wind would exist, the circulation of natural convection would be turned upwards. If the direction of external wind strengthens the circulation of natural convection, the combined free-forced convection heat loss would be more than the natural convection heat loss. Otherwise, the heat loss will be reduced below the natural convection value. For sideward and downward-facing cavities (0° 6 u 6 90°) with external head-on wind (wind normal to the aperture plane), natural and forced convection have the same trend of turning upward which results in a direct increase of heat loss. But for external side-on wind (wind

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parallel to the aperture plane), the convection heat loss is determined by the combined effects of two factors: the tilt angle and the wind speed. When both the tilt angle (0° 6 u 6 45°) and the wind speed are relatively small, the total convection heat loss is actually reduced below the natural convection value. Only when wind is strong enough to determine the air flow through the cavity, an increase in heat loss is expected. This is because a small speed wind acts as a barrier at the aperture preventing the hot air to flow out of the receiver with slightly downward inclinations. However, when the tilt angle of receiver is larger (in the range of 45° < u 6 90°), even a small sideon wind will cause the original dominant stagnant zone to become smaller, while the convective zone becomes larger. As a result, the convection heat loss increases with the increasing wind speed. For upward-facing cavities (90° 6 u < 0°), head-on air blowing into the cavity would be turned downwards. This means an external head-on wind causes an air flow in the opposite sense to the air circulation in the cavity driven by buoyancy forces. Thus, external head-on wind is not amplifying but clogging the natural convective circulation. Only when wind is strong enough to determine the air flow through the cavity, an increase in heat loss is expected. For the case of side-on wind, similarly, the combined convection heat loss is dependent on the wind speed as well as the tilt angle. 3. Advances in natural convection heat loss (under no-wind conditions) During the past two decades, both experimental and numerical investigations on natural convection heat trans-

Fig. 2. The main features of a tilted partially open cavity.

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fer in cavities have been carried out by researchers. Fig. 2 illustrates the main features of a tilted partially open rectangular or cylindrical cavity. The rate of heat transferred from the cavity to the surrounding is influenced by the following geometrical parameters: the tilt angle u, the aspect ratio (AR = L1/L2 = H/B or D/B) which is the ratio of cavity height (or diameter) to width (or depth), the opening ratio (OR = L3/L1 = a/H or Dap/Dcav) which is the ratio of the opening height (or diameter) to the cavity height (or diameter), and the opening displacement ratio (DR = d/L1 = d/H or d/Dcav) which is the ratio of the distance from the centerline of the opening to the base of cavity, to the height (or diameter) of the cavity. The configuration and number of the heated walls are also important parameters in the determination of convective heat transfer in cavities. 3.1. Convection heat loss from cubical and rectangular cavities Le Quere et al. (1981a,b) numerically and experimentally investigated thermally driven laminar natural convection in an open cubical cavity with isothermal sides, one of which facing the opening. They used primitive variables and finite difference expressions suitable for treating problems with large temperature and density variations. The computational domain was an enlarged domain comprising a square open cavity and a far field surrounding it. The convection heat loss was found to be strongly dependent on the cavity inclination and the correlation for each inclination was established. Penot (1982) studied a similar problem using stream function-vorticity formulation. He also used an enlarged computational domain similar to that of Le Quere et al. (1981a) with approximately the same

boundary conditions. Clausing (1981, 1983) and Clausing et al. (1987) developed a model to calculate the convection heat loss of a large cubical cavity based on the hypothesis that two factors governed the convection heat loss, i.e. (i) the ability to transfer mass and energy across the aperture and (ii) the ability to heat air inside the cavity. An analytical method was developed based on the above assumption. It is concluded that the latter factor was of greatest importance. Based on the work by Clausing et al. (1987), Leibfried and Ortjohann (1995) developed a more generalized model that can be used for both downward and upwardfacing cavities with various geometries. The definitions of dimensions used in the modified Clausing model (Leibfried and Ortjohann, 1995) are shown in Fig. 3. A simple model for the convective heat transfer from a solar cavity receiver based on the results of experimental studies from cubical cavities was presented by Siebers and Kraabel (1984). They reported a correlation that accounts for the physical property variations present in a strongly heated cavity with elevated wall temperature. The correlations obtained in the aforementioned studies have been widely applied for predicting solar cavity receiver convection heat loss. A summary of the correlations based on cubical cavity is presented in Table 1. Table 2 gives the empirical correlation coefficients and exponents in Le Quere et al. model (Le Quere et al., 1981b). It should be pointed out that the heat transfer mechanisms of cavity solar thermal receiver are similar to the cavities which are encountered in various engineering systems, such as those in electronic cooling devices and in buildings. A literature review shows that numerous studies have been published on cubical and rectangular open cavities. In this paper, we will present a review for the convection heat loss investigations in these cavities. Table 3 is a summary of

Fig. 3. Definitions of the dimensions used in the modified Clausing model (Leibfried and Ortjohann, 1995).

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Table 1 A summary of the Nusselt number correlations based on cubical cavity. Models Le Quere et al. model (Le Quere et al., 1981b)

Nusselt number correlations b

7

Comments 9

Nu = aGr , 10 6 Gr 6 5  10 , 90° 6 u 6 90°

The effects of varying receiver temperature and tilt angle are included, while the effect of varying aperture sizes is not included

The coefficient ‘a’ and exponent ‘b’ are presented in Table 2 Lc = a, Tprop = T1 Clausing model (Clausing, 1981, 1983; Clausing et al., 1987)

Nu = g f  b, 0° 6 u 6 90°

An implicit model developed for large central receivers; the cavity is divided into a convective zone and a stagnant zone, thus more complicated but best fit for all the models

Laminar regime: Ra < Ral = 3.8  108, gl = 0.63Ra1/4, fl = 1 Turbulent regime: Ra > Rat = 1.6  109, gt = 0.108Ra1/3 ft = 0.2524 + 0.9163(Tw/T1)  0.1663(Tw/T1)2 b is an implicit function defined as b = (Tw  Tb)/ (Tw  T1) Modified Clausing model (Leibfried and Ortjohann, 1995)

The Nusselt number correlation is in the same form with Clausing model, but the Lc and b is defined as follows: Lc = (1/2)(La,eff + Ls)

An extension of the original Clausing Model; the definitions of characteristic length Lc, buoyancy height La as well as the wall area of the convective zone Acz were generalized for geometries with varied directions, including upward-facing orientations

where La,eff = La  0.94Lcount, Lcount = (Dcav  Dap)/ (2cos u) and the definitions of La, Ls, Lcount are shown in Fig. 3 b = c/{ ln [1/(1  c)]} where c can be found in Leibfried and Ortjohann (1995). Siebers and Kraabel model (Siebers and Kraabel, 1984)

Nu = 0.088Gr1/3(Tw/T1)0.18, 102 6 Gr 6 105

Simple, but has a large degree of uncertainty

Lc = cavity dimension; Tprop = T1

Table 2 Empirical correlation coefficient and exponent in Le Quere et al. model (1981b). Receiver tilt angle, u (degree)

Coefficient, a

Exponent, b

90 75 60 45 30 15 0 15 30 45 60 75 90

0.0570 0.0470 0.0545 0.0465 0.0480 0.0465 0.0925 0.0810 0.0640 0.0605 0.0685 0.0330 Not applicable

0.353 0.360 0.360 0.370 0.369 0.368 0.330 0.331 0.332 0.316 0.292 0.302 Not applicable

work done on fully/partially open cubical and rectangular cavities. Table 4 explains the various types of wall boundary conditions (B.C.s) considered in Table 3. Some major activities undertaken or observations are outlined as follows:  Experimental investigations: The velocity profiles were measured by Laser Anemometer, while the heat transfer rates were measured by an interferometer. The local Nusselt numbers and heat transfer coefficients were

determined from the experimental results (Hess and Henze, 1984; Chan and Tien, 1986; Sernas and Kyriakides, 1982; Chakroun et al., 1997).  Two-dimensional numerical investigations: (a) Natural convection for fully or partially open square cavities was numerically investigated using an enlarged computational domain (Chan and Tien, 1985a,b; Miyamoto et al., 1989) or considering a restricted computational domain (Mohamad, 1995). (b) The inclusion of the outside domain into the computations has a minimal effect on the heat transfer results for those cavities where one wall is isothermal and other two walls are adiabatic (Angirasa et al., 1992). (c) Using the finite-volume-based power law (SIMPLER) algorithm; the open cavity Nusselt number approaches the flat plate solution when either Grashof number or tilt angle increases (Elsayed et al., 1999). (d) For inclined open shallow cavities, flow and heat transfer are governed by Rayleigh number, aspect ratio and the inclination (Polat and Bilgen, 2002). When considering the effect of conduction heat transfer in the convection heat loss prediction, the heat transfer through the massive wall is shown to be an important parameter affecting natural convection in the cavity and hence its heat transfer characteristics (Polat and Bilgen, 2005).

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Table 3 A summary of work done on fully/partially open cubical and rectangular cavities. References

AR

OR

DR

B.C.

u (degree)

Pr

Gr or Ra

Hess and Henze (1984) Chan and Tien (1986) Sernas and Kyriakides (1982) Chakroun et al. (1997) Chan and Tien (1985a,b) Miyamoto et al. (1989) Mohamad (1995)

1 0.143 1 0.25, 0.5, 1 1, 0.143 0.5, 2, 1 1 1 1 1 0.125, 0.5, 1 0.125, 0.5, 1 1 1 1 2, 1, 0.5, 0.25, 0.125 1 0.25, 0.5, 1 1 1 1 1 1 1

0.5, 1 1 1 0.25, 0.5, 1 1 1 0.5, 1 0.5, 1 1 1 1 1 1 1 0.25, 0.5, 0.75 1 1 1 1 1 0.25, 0.5, 0.75 1 1 1

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.25, 0.5, 0.75 0.5 0.5 0.5 0.5 0.5 0–1 0.5 0.5 0.5

C C B E C C A A C C E E C C, E C C C A C A E E F G

0 0 0 90–90 0 80–0 45–80 0 0 60–90 45–0 45–0 90 to 10 60–0 90–30 0 90–90 90–30 0 0 90–90 90–90 90–90 90–90

7 8.7 0.7 0.7 1,7 0.7 0.7 0.7 0.1–1 0.71 1 1 0.7 0.71 0.71 0.71 0.71 0.7 0.7 0.7 0.7 0.7 0.7 0.7

3  1010 6 RaH 6 2  1011 106 6 RaB 6 108 GrH = 107 GrH = 5.5  108 103 6 RaB 6 107 103 6 RaH 6 107 7  103 6 RaB 6 7  104 105 6 RaB 6 7  105 102 < RaB 6 108 102 6 RaH 6 105 103 < RaB 6 1010 106 < RaB 6 1012 105 6 RaH 6 1010 103 6 RaH 6 106 103 6 RaB 6 106 103 6 RaH 6 106 104 6 RaB 6 107 104 6 RaB 6 5  105 RaB = 1010,1011 104 6 RaB 6 5  105 GrH = 5.5  108 GrB = 1.3  108 GrB = 1.3  108 GrB = 1.3  108

Angirasa et al. (1992) Elsayed et al. (1999) Polat and Bilgen (2002) Polat and Bilgen (2005) Nateghi and Armfield (2004) Saha et al. (2007) Bilgen and Oztop (2005) Sezai and Mohamad (1998) Hinojosa et al. (2006) Showole and Tarasuk (1993) Lin and Xin (1992) Angirasa et al. (1995) Elsayed and Chakroun (1999) Chakroun (2004)

Table 4 Explanation for the types of boundary conditions given in Table 3. B.C. types

A B C D E

F

G

Boundary condition on walls Wall 1

Wall 2

Wall 3

Wall 4

Th Th Th Tc q00 a a q00 q00 a q00

Th Th a a a q00 a q00 a q00 q00

Th T1 a a a a q00 a q00 q00 q00

Th NA a a a a a a a a a







Note: Tc < T1 < Th, a = adiabatic, NA = not applicable, and q00 = constant heat flux on the wall.

(e)

(f)

The flow is steady for the low Rayleigh number at all tilt angles and becomes unsteady for the high Rayleigh number at all tilt angles. The critical Rayleigh number decreases when the tilt angle increases (Nateghi and Armfield, 2004). Streamlines and isotherms were produced and heat and mass transfer was calculated (Saha et al., 2007; Bilgen and Oztop, 2005). Through a parametric study, it is found that the volume flow rate and Nusselt number are increasing functions of Rayleigh number, aperture size and generally aperture position.

 Three dimensional numerical simulation: The two dimensional results are valid for aspect ratio equal to and greater than unity, and for Rayleigh numbers equal to



and less than 105 (Sezai and Mohamad, 1998). For high Rayleigh numbers, the Nusselt number changed substantially with the tilt angle of the cavity (Hinojosa et al., 2006). Combination of experimental and numerical studies: experimental study using a Mach–Zehnder interferometer and numerical study by a finite difference technique. The results showed that the average heat transfer increased significantly for all Rayleigh numbers for the case of upward-facing cavity (Showole and Tarasuk, 1993). Numerical simulation of transient natural convection: flows at moderate to high Rayleigh numbers are periodic or unsteady (Lin and Xin, 1992; Angirasa et al., 1995). The effect of the aperture geometry: Four different geometrical arrangements for the opening were investigated: (a) high wall slit, (b) low wall slit, (c) centered wall slit, and (d) uniform wall slots (Elsayed and Chakroun, 1999). The effect of wall conditions and tilt angles: tilt angle, wall configuration, and the number of heated walls are all factors that strongly affect the convective heat transfer coefficient between the cavity and the ambient air (Chakroun, 2004).

3.2. Convection heat loss from cylinder cavities

 Early models: (a) A model proposed by Koenig and Marvin (1981) cited in Harris and Lenz (1985) was based on cylindrical cavity receiver.

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(b)

(c)

Based on Siebers and Kraabel model (1984), Stine and McDonald (1989) proposed a extended correlation of the Nusselt number for a cylindrical shaped frustum receiver incorporating aperture size, surface temperature and receiver tilt angle. Using the Stine and McDonald model, the natural convection heat losses for two different types of receivers, which have conical and dome shapes, were predicted by Seo et al. (2003). Leibfried and Ortjohann (1995) extended the model of Stine and McDonald (1989), and the newly generalized model could be used for both downward and upward-facing cavities with various geometries. Comparing the calculating results of convection heat loss from upward-facing cavity receivers with the measured data, it was concluded that the simpler explicit modified Stine and McDonald correlation gave slightly better results than the implicit modified Clausing model.

 Recent models: (a) The researchers in Centre of Sustainable Energy System, Department of Engineering, Australian National University made great efforts to develop more general correlations for predicting the receiver convection heat loss. Taumoefolau and Lovegrove (2002) as well as Paitoonsurikarn and Lovegrove (2002) experimentally and numerically investigated the natural convection heat losses from a 70 mm cylinder receiver (model cavity receiver) with cavity temperatures ranging from 350 °C to 500 °C. It was reported that the experimental and numerical results obtained were in good agreement qualitatively with those predicted by various correlations proposed by previous researchers. (b) Lovegrove et al. (2003) have attempted to develop a correlation that can reliably predict natural convection heat losses from cavity receivers employed in solar parabolic dishes at all tilt angles. A correlation was developed using the concept of the ensemble cavity length Ls as the characteristic length to account for the combined effect of the cavity geometrical parameters and the inclination. (c) Paitoonsurikarn and Lovegrove (2003) undertook the numerical investigation of natural and combined convection heat loss from cavity receivers. A new correlation in the form Nu = CRanf(Pr) was developed for prediction of heat transfer coefficients. The ensemble cavity length Ls was modified to include the aperture geometry. Later, they (Paitoonsurikarn et al., 2004) carried out a parametric study of several relevant parameters in natural convection heat loss from open cavity receiver in solar dish application. The previously proposed correlation model in Paitoonsurikarn and Lovegrove (2003) has been modified to take into

(d)

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account the variation of additional parameters. Moreover, a correlation based on the modified Stine and McDonald model was developed. Both models are quite promising in the natural convection heat loss prediction in most cases. Based on the numerical simulation results of three different cavity geometries and the previous works (Paitoonsurikarn and Lovegrove, 2003; Paitoonsurikarn et al., 2004), an improved version of correlation was presented by Paitoonsurikarn and Lovegrove (2006a).

 Experimental studies: (a) Yeh et al. (2005) presented an experimental method of modeling natural convective flows in solar cavity receivers using water as a working fluid with density differences due to salt concentration. The flow was visualized for a range of cavity receiver orientations of 0–90°, and flux Grashof numbers 7  104–1010. It was found that the flows observed inside the model cavity receiver are laminar over the range of flux Grashof numbers (of order 107) applicable to actual receivers. However, a transition to turbulent flow happens when the Grashof number exceeds 107. (b) Taumoefolau et al. (2004) experimentally investigated the natural convection heat loss from an electrically heated model cavity receiver for different inclinations varying in 90–90° with temperature ranging from 450 to 650 °C. It was found that the Clausing model showed overall the closest prediction for both numerical and experimental results with downward-facing angles despite its original use for big scale central receivers. For upward-facing angles, the modified Stine and McDonald model showed the closest agreement to the experimental results. The inclination, for which maximum convection heat loss occurs, increases as the opening ratio decreases, which was also observed by Leibfried and Ortjohann (1995). (c) Most recently, an experimental and numerical study of the steady state convection heat losses occurring from a downward-facing cylindrical cavity receiver having the ratio of aperture diameter to the cavity diameter greater than one (OR > 1) has been carried out. Nusselt number correlations based on the receiver aperture diameter were proposed for the convection heat losses under nowind conditions (Prakash et al., 2009). (d) A cylindrical cavity receiver containing a tubular ceramic absorber has also been considered related to a solar chemical reactor for performing thermochemical processes using concentrated solar radiation (Melchior et al., 2008). (e) Table 5 presents the correlations obtained based on cylindrical receivers as discussed above.

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Table 5 A summary of the Nusselt number correlations based on cylindrical cavity. Models Koenig and Marvin (1981)

Nusselt number correlations Nu ¼

0:52P ðuÞl1:75 c

Comments

1=4

ðGrPrÞ

Based on higher temperature 550 °C 6 Tw 6 900 °C

2.2 P(u) = (cos u)3.2 for 0° 6 u 6 45°, P(u) pffiffiffi = 0.707(cos u) for 45° 6 u 6 90°, lc = Dap/Dcav, Lc ¼ ð 2=2ÞDcav , 3 T prop ¼ 11 16T w þ 16 T 1

Stine and McDonald (1989)

Nu = 0.088Gr1/3(Tw/T1)0.18( cos u)2.47(Dap/Lc)s

An explicit model accounts for the combined effects of operating temperature, tilt angles and aperture size

s = 1.12  0.98(Dap/Lc), 0° 6 u 6 90° Lc = average internal dimension of the cavity; Tprop = T1 Modified Stine and McDonald model (Leibfried and Ortjohann, 1995)

Nu = 0.0176Gr1/3(Tw/T1)0.18(4.256Aap/Acav)sh(u, umax, ustag)

Inconvenient to use due to the implicit function h(u, umax, ustag)

s = 0.56  1.01(Aap/Acav)1/2 h(u, umax, ustag) is an implicit function for the tilt angle dependence, refers to (Leibfried and Ortjohann, 1995), Tprop = (Tw + T1)/2 Lovegrove et al. model (Lovegrove et al., 2003)

Nu = 0.004Ra0.44(Dap/Dcav)0.03 Pr 0.25

Using Le as the characteristic length to account for the combined effect of the cavity geometrical parameters and the inclination

Lc = Le = [2.05(cos u)3.27  0.06(sin u)0.66]Dcav + [6.04(cos u)5.27 + 0.33(sin u)0.13]B Paitoonsurikarn and Lovegrove model (Paitoonsurikarn and Lovegrove, 2003)

Nu = cRan, c = 0.00324, n = 0.447, Tprop = (Tw + T1)/2

Including the effect of aperture geometry into the ensemble cavity length Le

Lc = Le = [4.79( cos u)4.43  0.37( sin u)0.719]Dcav + [1.06( cos u)3.24  0.0462( sin u)0.286]Dap + [7.07( cos u)5.31 + 0.221( sin u)2.43]B Paitoonsurikarn et al. model (Paitoonsurikarn et al., 2004)

Modified Paitoonsurikarn and Lovegrove model (Paitoonsurikarn and Lovegrove, 2006a)

Nu = cRan

The constant c and the exponent n have been modified with additional parameters; valid for AR < 2

c = 8.2066  106(Tw/T1)2.5837, n = 0.67824(Tw/T1)0.064548 Lc is the same as Paitoonsurikarn and Lovegrove model (2003) For modified Stine and McDonald model, minor modification was The modified h(u) is more convenient to use; made, Nu = 0.016Gr1/3(Tw/T1)0.18(4.256Aap/Acav)s)h(u) valid for AR < 2 h(u) = 1.1677  1.0762 sin (u0.8324) where u is in radian P Still using the concept of ensemble length scale Nu = 0.0196Ra0.41 Pr 0.13, Lc ¼ Le = j 3i¼1 ai cosðu þ wi Þbi Li j Le, but more easier to use

where u is in radian. The constants ai, bi, and wi can be found in Paitoonsurikarn and Lovegrove (2006a). Prakash et al. model (Prakash et al., 2009)

For 50 °C 6 Tw 6 75 °C: Nu = 0.21Gr1/3(1 + cos u)3.02(Tw/T1)1.5 Non-uniform wall temperature; the opening ratio OR > 1, which is different from other studies. 1.08 6 Tw/T1 6 1.14, 3.3  108 6 Gr 6 4.9  108, 0° 6 u 6 90°, Lc = Dap For 100 °C 6 Tw 6 300 °C: Nu = 0.246Gr1/3(1 + cos u)2.03(Tw/ T1)0.58 1.2 6 Tw/T1 6 1.9, 0° 6 u 6 90°, Lc = Dap

3.3. Convection heat loss from hemispherical cavities Fig. 4 shows the main features of a typical tilted hemispherical cavity. The rate of heat transferred from the cavity to the surrounding is influenced by the following geometrical parameters: the tilt angle u, the opening ratio (OR = Dap/Dcav) which is the ratio of the opening diameter to the cavity diameter, and the opening displacement ratio

(DR = d/Dcav) which is the ratio of the distance from the centerline of the aperture to the base of the cavity, to the diameter of the cavity. Natural convection in a hemispherical enclosure heated from below was investigated by Yasuaki et al. (1994), and an experimental correlation was obtained. Khubeiz et al. (2002) carried out an experimental analysis of laminar free convection heat transfer from an isothermal hemispherical

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(CPC) and trumpet reflectors was performed (Reddy and Sendhil Kumar, 2009). The receiver with the trumpet reflector has shown better performance as compared to other configurations. The Nusselt number correlations obtained based on hemispherical cavities as discussed above are summarized in Table 6. 4. Advances in combined free-forced convection heat loss (under wind conditions)

Fig. 4. The main features of a tilted partially open hemispherical cavity.

cavity. A simplified analytical solution, numerical calculations and experimental studies of laminar natural convection heat transfer from an isothermal hemispherical cavity were presented. Two distinct receivers: semi-cavity and modified cavity were introduced for solar dish collector system (Kaushika, 1993). The modified cavity receiver (hemisphere with aperture plate) was found to be more efficient than the semicavity. The thermal performance characteristics and optimizations of cavity receiver of a low cost solar parabolic dish were presented, and it is concluded that the conventional cavity receivers are inadequate for fuzzy focal dish concentrator (Kaushika and Reddy, 2000). The modified cavity receiver is so designed to capture maximum reflected solar radiation at focal region of fuzzy focal dish concentrators with minimum heat loss. A comparative study was performed to predict the natural convection heat loss from the cavity, semi-cavity and modified cavity receivers. Among the three receivers, the modified cavity receiver was found to be the preferred receiver for a fuzzy focal solar dish collector system (Sendhil Kumar and Reddy, 2008). Sendhil Kumar and Reddy (2007) used a two dimensional model to investigate the approximate estimation of the natural convection heat loss from an actual geometry of the modified cavity receiver of a fuzzy focal solar dish concentrator. The total heat loss from the receiver has been estimated for both the configurations ‘‘with insulation’’ (WI) and ‘‘without insulation’’ (WOI) at the protecting aperture plane of the receiver. Also, they (Reddy and Sendhil Kumar, 2008) presented a numerical study of combined laminar natural convection and surface radiation heat transfer in a modified cavity receiver of solar parabolic dish collector. The influence of operating temperature, emissivity of the surface, orientation and the geometry on the total heat loss from the receiver has been investigated. Additionally, a numerical analysis of solar dish modified cavity receiver with cone, compound parabolic concentrator

All investigations discussed above are under the no-wind conditions. However, it is important to note that operating conditions for real receivers tend to have some wind effects. Compared to numerous investigations on natural convection heat loss of cavity receivers for no-wind condition, much fewer studies have been undertaken on convection heat loss with wind effect. The Clausing model (1983) derived from a large cubical central receiver included the effect of wind speed. Ma (1993) has reported the experimental investigations on the convection heat loss under wind conditions for a cylindrical receiver having an aperture diameter smaller than the receiver diameter (OR < 1). The tests have been carried out at wind speeds greater than 3 m/s. The trends obtained from the study are similar to those of the no-wind cases. Leibfried and Ortjohann (1995) investigated the influence of wind on receiver losses for an upward-facing cavity (u = 45° by simulation with a fan. Results showed that ventilation reduces the loss by more than 11%. The reduction is a function of the wind direction. They pointed out that for small wind speeds, the dependence on wind for upward-facing cavities is much smaller than for sideward and downward-facing cavities, and it can even have a reducing effect. For typical windy conditions, both upward and downward-facing cavity receivers are expected to have the same magnitude of convection heat loss. The combined free-forced convection heat loss study has been undertaken by Paitoonsurikarn and Lovegrove (2003). However, their study was limited only to the cases of side-on and head-on wind.  For the case of side-on wind, the relationship of wind speed and convection heat loss for three receivers with varied dimensions was investigated. It was found that the heat loss is actually reduced below the natural convection value by wind speeds up to about 7 m/s. At higher wind speeds, the wind increased losses as expected. The magnitude of wind speed that results in the minimum heat loss tends to decrease with increasing receiver dimension.  For the case of head-on wind, similar result was obtained. Initially, the sharp increase of heat loss was found up to its local maximum at wind speed of about 3 m/s. After that heat loss decreases to its local minimum at wind speed of about 6 m/s, where it starts to unboundedly increase with increasing wind speed.

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Table 6 A summary of the Nusselt number correlations based on hemispherical cavity. Models Yasuaki et al. model (Yasuaki et al., 1994) Khubeiz et al. model (Khubeiz et al., 2002)

Nusselt number correlations 0.25

6

Comments 10

, 10 6 Ra 6 6  10 , 6 6 Pr 6 13; 000, Lc = Dcav

An experimental correlation for hemispherical receiver

Theoretical correlation: Nu = 0.296Ra1/4, 105 < Ra < 109, Lc = Dcav

The comparison of theoretical and numerical solutions with experimental results shows good agreement

Nu = 0.185Ra

Numerical correlation: Nu = 0.340Ra1/4, 105 < Ra < 109, Lc = Dcav Experimental correlation: Nu = 0.316Ra1/4, 1.7  105 < Ra < 3.4  105, 376 < Pr < 1140, Lc = Dcav Sendhil Kumar and Reddy model (Sendhil Kumar and Reddy, 2007)

For with insulation:

Correlations developed from a modified cavity receiver (hemisphere with aperture plate)

Nu = 0.0303Gr0.315(1 + cos u)3.551(Tw/T1)0.086(d/D)0.878, Lc = Dcav For without insulation: Nu = 0.503Gr0.222(1 + cos u)1.231(Tw/T1)0.165(d/D)0.304, Lc = Dcav 106 6 Gr 6 107 ; 0° 6 u 6 90°, 0.3 6 Dap/Dcav 6 0.4, 1.9 6 Tw/ T1 6 3.0, Pr = 0.69. Sendhil Kumar and Reddy model (Reddy and Sendhil Kumar, 2008)

The convection Nusselt number (NuC):

For both natural convection and surface radiation; natural convection accounts for the emissivity of the receiver surface, while surface radiation considers the effect of the receiver orientation

NuC ¼ 0:534Gr0:218 ð1 þ cos uÞ0:916 ð1 þ eÞ0:473 ½N rc =ðN rc þ 1Þ1:213 T R0:082 ðDap =Dcav Þ0:099 ; Lc = Dcav The radiative Nusselt number (NuR): ð1  T 4R Þ8:768 NuR = 9.650Gr0.068(1 + cos u)0.0010.546, N 0:478 rc ðDap =Dcav Þ0:493 N rc ¼

rT 4w ðDcav =2Þ ðT w T 1 Þk ; 7

TR = T1/Tw, Lc = Dcav

106 6 Gr 6 10 ; 0° 6 u 6 90°, 0.3 6 Dap/Dcav 6 0.4, 0.33 6 TR 6 0.53, 0 6 e 6 1 38 6 Nrc 6 175

Later, Paitoonsurikarn et al. (2004) performed a numerical investigation of natural and combined convection heat loss from cavity receivers employed in solar parabolic dishes to represent the case of the cylindrical model cavity receiver. The results of combined convection study were shown to illustrate the wind effect for the cylindrical model receiver.  The influence of wind speed on the convection heat loss was found to be similar to that in Paitoonsurikarn and Lovegrove (2003).  The effect of receiver tilt angle on the combined convection heat loss for the side-on wind case was also identified. The local minimum of heat loss at wind speed of 5 m/s was observable at inclinations less than 45°. However, this characteristic disappeared at higher inclination where heat loss seemed to always increase with wind speed.  A contour plot of heat loss versus both wind speed and incident angle for model receiver at zero inclination was obtained. Chen et al. (2006) presented the modeling results for the radiation and convection heat losses though the aperture of a cavity solar receiver at the CSIRO National Solar Energy

Centre, Australia, using Fire Dynamics Simulator (FDS) package. It was shown that,  In general, if there is no wind effect, convection heat loss through the aperture is much less than the radiation heat loss (5–15% of the radiation heat loss) for small receivers, while this ratio increases for larger apertures.  Ambient wind may have a substantial impact on the overall heat loss from the receiver. It is suggested that a small receiver should have wind guards installed to reduce the wind effect, thus achieve high thermal efficiency. To the author’s best knowledge, less study was carried out on the interaction between the wind and the dish structure. Paitoonsurikarn and Lovegrove (2006b) have investigated the relation between the magnitude of the free stream wind and that of the wind immediate to the cavity aperture at various wind directions, while the interaction between the wind and the dish structure was considered. It was found in the numerical simulations that the magnitude and the direction of the wind can greatly affect the amount of convection heat loss. In most cases, the magnitude of the

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heat loss is higher for the side-on wind than that for the case of the head-on wind. This finding is in a good agreement with the conclusion in the experimental work by Ma (1993). Prakash et al. (2009) carried out an experimental and numerical study of the steady state losses occurring from a downward-facing cylindrical cavity receiver of length 0.5 m, internal diameter of 0.3 m and a wind skirt diameter of 0.5. Besides no-wind tests, investigations were also carried out to study the effects of external wind at two different velocities in two directions (head-on and side-on). The head-on wind was found to cause higher convection heat loss than the side-on wind, which seems conflictive to those reported by Ma (1993) and Paitoonsurikarn and Lovegrove (2006b). This may be due to that the cavity geometries studied in Prakash et al. (2009) is quite different from those in Ma (1993) and Paitoonsurikarn and Lovegrove, 2006b. That is, the cylindrical receiver considered in Prakash et al. (2009) has a wind skirt diameter larger than the cavity diameter (OR > 1), while the cylindrical receivers investigated in Ma (1993) and Paitoonsurikarn and Lovegrove, 2006b have a condition of OR < 1. 5. Conclusions The cavity receiver is an important component of solar dish/engine systems. The convection heat loss of the receiver significantly reduces the efficiency and consequently the cost effectiveness of the solar thermal power system. The convection heat loss should be estimated with considerable accuracy, so that the optimal design as well as the selection of the optimal operating parameters can be made to minimize the receiver heat loss. Only a few literatures have attempted to work through the convection heat loss mechanisms of cavity receivers, especially for those under wind conditions. It is believed that the understanding of the convection heat loss mechanism is still incomplete at this time. Four types of receivers shaped in cubical, rectangular, cylindrical, and hemispherical have been investigated both experimentally and numerically. Numerous studies have been published on square and rectangular open cavities due to their wide applications in various engineering systems in addition to those in solar thermal receivers. On the contrary, fewer studies of predicting convection heat loss for cylindrical and hemispherical cavity receivers have been undertaken. Various correlations for free convection prediction have been proposed in the previous works. However, each correlation has a limited range of applicability, which is inherently based on a particular cavity geometry and operating conditions used in the experiment in each of those works. Study on wind effect is still in the early stage; few correlations have been generated for predicting the combined free-forced convection heat loss with wind effect. Based on reviewing the previous work on the cavity receiver convection heat loss, some issues that should be further researched are summarized as follows:

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 In order to minimize heat loss through the aperture, the aspect ratio AR, the opening ratio OR, the opening displacement ratio DR, and the tilt angle u, should be chosen as design parameters. For example, the aperture size should be optimized to minimize the heat loss. The larger the ratio of cavity area to the aperture area, the closer the cavity receiver approaches a blackbody absorber, but at the expense of higher conduction losses through the insulated cavity walls. Smaller apertures will also reduce re-radiation losses but they intercept less sunlight. Consequently, an optimum aperture size would be a compromise between maximizing radiation capture and minimizing radiation losses (Steinfeld and Schubnell, 1993).  The Boussinesq approximation has been used to simplify the modeling of the flow in most investigations up to date, since the cavity receivers operate at relatively low temperature (300–400 °C). However, for a cavity receiver, particularly that used in dish/engine system, the receiver usually operates at a high temperature (>700 °C), which would result in large temperature changes in air as well as significant compressible effects in the flow (air expands significantly when heated). Thus, non-Boussinesq flow effects ought to be investigated in developing a more precise cavity receiver model. Additionally, as indicated in Prakash et al. (2009), the extension of the low temperature analysis to high temperatures for the wind induced convection heat loss is not feasible as the loss values are not linear. Therefore, more attention should be paid to the investigation of the convection heat loss at high operating temperatures.  It was shown that in solar parabolic dish applications, other heat losses, such as the radiation heat exchange could be non-negligible due to the high operating temperature and must be taken into account for an overall improvement in receiver performance. Particularly, the convection heat loss was dominated by the radiation heat loss for higher receiver tilt angle (>45°) Prakash et al., 2009. It seems that it is necessary to investigate the emissivity of surface for total heat loss of receivers in solar dish/ engine systems. However, most investigations on the convection heat losses have been addressed excluding the radiation loss effect.  The present research results for convection heat loss from the cavity receivers in wind conditions have shown some inconsistence or even sometimes conflicting results, such as those in Prakash et al. (2009), Ma, 1993 and (Paitoonsurikarn and Lovegrove, 2006b) as discussed in Section 4. Moreover, it is concluded in Prakash et al. (2009) that the higher the wind speed, the larger the convection heat loss for head-on wind, which is inconsistent with the results in Paitoonsurikarn and Lovegrove (2003), in which a certain value of wind speed exists for the minimum heat loss. These conflicting results signify that the same wind speed may have different impact on the con-

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vection heat loss of receivers having different geometries, especially the configuration of the aperture, i.e. the ratio of aperture diameter to the cavity diameter is equal to or less than one, or greater than one (OR 6 1, or OR > 1). Therefore, the convection heat loss mechanism in wind conditions should be further understood. Furthermore, the wind directions that are currently researched are head-on or side-on wind, which is not complied with practical conditions. Because a real wind in nature is often parallel to the ground, as shown in Fig. 2, a reasonable definition of wind direction is also needed.  To the author’s best knowledge, so far few literatures are available for empirical correlations related to the combined convection heat loss prediction. Taumoefolau (2004) put forward an empirical correlation based on the experimental data of the model cavity receiver. The range of the incidence angle is from 0° to 90°, corresponding to the cases of side-on wind and head-on wind, respectively. A more detailed study of wind effects on cavity receiver convection heat loss would be necessary to provide more quantitative information.

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