Numerical simulation of the heat flux distribution in a solar cavity receiver

Abstract

In the solar tower power plant, the receiver is one of the main components of efficient concentrating solar collector systems. In the design of the receiver, the heat flux distribution in the cavity should be considered first. In this study, a numerical simulation using the Monte Carlo Method has been conducted on the heat flux distribution in the cavity receiver, which consists of six lateral faces and floor and roof planes, with an aperture of 2.0 m×2.0 m on the front face. The mathematics and physical models of a single solar ray’s launching, reflection, and absorption were proposed. By tracing every solar ray, the distribution of heat flux density in the cavity receiver was obtained. The numerical results show that the solar flux distribution on the absorbing panels is similar to that of CESA-I’s. When the reradiation from walls was considered, the detailed heat flux distributions were issued, in which 49.10% of the total incident energy was absorbed by the central panels, 47.02% by the side panels, and 3.88% was overflowed from the aperture. Regarding the peak heat flux, the value of up to 1196.406 kW/m² was obtained in the center of absorbing panels. These results provide necessary data for the structure design of cavity receiver and the local thermal stress analysis for boiling and superheated panels.

Keywords

solar cavity receiver / Monte Carlo method / heat flux distribution

Introduction

Extensive research on the solar tower power plant principle has been conducted in America, Germany, Switzerland, Spain, France, Italy, the former Soviet Union, and Japan ever since it was put forward by Alvin Hildebrandt and Lorin Vant-Hull of Houston University in the early 1970s. So far, this technique, whose laboratory investigation has been completed, has been put into commercial use, and more than 10 solar tower power plants have been built.

The receiver is one of the main components of efficient concentrating solar collector systems. Its configuration selection and layout pattern is not only playing an important role in the energy distribution of the collecting system but also is of great significance for thermoelectric conversion efficiency of the whole system. Due to the low heat loss, easy controlling, and high heat capacity, the cavity receiver becomes a feasible design for the heat collecting system of solar thermal utilization. The solar tower plant hands down the routine technique of fossil fuel plant in which high temperature and high pressure are presented. Now, in this heat collecting system, the cavity receiver that has a higher design point efficiency and a higher thermal capacity is the focus.

In order to design the structure of the receiver, precise study should be conducted to study the heat flux distribution in the cavity receiver. The objective of this study is to set up a numerical simulation using the Monte Carlo Method (MCM) of the heat flux density distribution in the cavity receiver. MC approaches have been used extensively in the last decade for photon, electron, phonon, and neutron transport problems. An important research area where the MCM has been fully exploited to study the transport phenomena is the radiative transfer [1-5]. Radiative transfer applications of MCM have been well-documented in Refs. [6,7]. By simulation using MCM, the heat flux distribution of cavity receiver is shown at the conditions when the reradiation of the wall is considered and omitted, respectively.

Monte Carlo model for thermal radiative transfer

Monte Carlo Method is a statistic method whose basic thought is that the radiation process is divided into some independent processes, such as reflecting, absorbing, and scattering process. Tracking many rays whose paths are decided by a series of random numbers, a stabilize statistic result could be obtained. Applying the MCM in radiative transfer, there are two approaches [8]: one is the early MCM, in which a ray is supposed to carry energy, and the other is the improved MCM in which a ray does not carry energy.

Radiation transfer using the early MCM

In the early MCM, a ray carries certain energy. By tracking plentiful of rays, a serial of parameters can be obtained. The tracking process for a ray is shown in Fig. 1.

The paths of rays are tracked until they are absorbed or escaped. Then, the numbers of the rays that are absorbed are counted by the given cell i. The radiative heat transfer of cell i is written as

(1)Q_{Vi}^{\mathrm{r}}=\delta Q_{Vi}^{\mathrm{r}}N(Vi)-\mathrm{4}{k}_i{V}_i\sigma T_{Vi}^{\mathrm{4}},

(2)Q_{Si}^{\mathrm{r}}=\delta Q_{si}^{\mathrm{r}}N(S_i)-{ϵ}_i{S}_i\sigma T_{Si}^{\mathrm{4}},

where subscriptions V_i and S_i denote the volume cell and area cell, respectively; Q^{\mathrm{r}} is the radiative energy; \delta Q^{\mathrm{r}} is the energy of a single ray; N is the amount of rays absorbed by calculation cell; and T is the cell temperature.

When the temperature field is uncertain, the temperature solution must be an iterative process. Moreover, the energy of a single ray changes along with the cell temperature. Therefore, the number of rays cannot be very large. It is disadvantageous for improving simulation precision.

Radiation transfer using the improved MCM

In the improved MCM, a ray does not carry energy. The Radiative Transfer Factor \mathrm{R}{\mathrm{D}}_{ji} [9] whose function is similar to view factor, is conducted to separate the radiative character and cell temperature. The subscriptions i and j denote two arbitrary microcontrolled cells. The factor is a ratio that the energy absorbed by the cell i emitting from the cell j is divided by the total one emitting from cell j in a radiation transfer system.

For a closed body made up of M volume cells and N area cells, the energy equation of volume cell V_i and area cell S_i is written as

(3)\mathrm{4}{k}_i{V}_i\sigma T_i^4={\displaystyle \sum _{j=1}^N{ϵ}_j{S}_j\sigma T_j^{\mathrm{4}}\mathrm{R}{\mathrm{D}}_{ji}}+{\displaystyle \sum _{k=1}^M\mathrm{4}{e}_k{S}_{k}s{T}_k^{\mathrm{4}}\mathrm{R}{\mathrm{D}}_{ki}},

(4)e_i{S}_{i}s{T}_i^{\mathrm{4}}={\displaystyle \sum _{j=1}^N{e}_j{S}_j{T}_j^{\mathrm{4}}\mathrm{R}{\mathrm{D}}_{ji}}+{\displaystyle \sum _{k=1}^M\mathrm{4}{k}_k{V}_k{T}_k^{\mathrm{4}}\mathrm{R}{\mathrm{D}}_{ki}}-q_i{S}_i,

where {ϵ}_i is cell emissivity; s_i is the area of cell i; T_i is the temperature of cell i; q_i is the heat flux of cell i transferred from the boundary; and \mathrm{R}{\mathrm{D}}_{ji} is Radiative Transfer Factor.

By educing \mathrm{R}{\mathrm{D}}_{ij}, the radiative character of cells is separated from the temperature. \mathrm{R}{\mathrm{D}}_{ji} can be kept constant or slightly altered in energy equation iterative process. Therefore, a large number of rays can be introduced into the simulation.

The calculation chart is described, as shown in Fig. 2.

Parameterization of cavity receiver

Cavity receiver structure and essential parameter

The solar receiver is a cavity receiver with a polygonal shape. Solar radiation enters through one of the faces and is absorbed by the interior walls of the cavity. The three faces of the cavity interior face absorbing panels are formed by a set of tubes welded together. The other two faces are coated with insulation, which decreases the absorbing energy. The profile of the cavity receiver is shown in Fig. 3.

The front face of the cavity is a 2.0 m×2.0 m aperture through which solar radiation enters. The aperture is bordered by a refractory material, resistant to the high temperature produced in these areas. The boiling zone has three panels of horizontal tubes welded together to form a membrane wall. Two panels are at the sides of the cavity and one is at the center.

The detail parameters of the receiver are shown in Table 1.

In order to simplify the simulation, the following assumptions are adopted:

1) The cavity walls and the boiling panel surfaces are gray bodies. Their spectral character is ignored.

2) To decrease the difficulty of the radiation simulation, a hypothetical surface is located before boiling panels. In the radiation simulation, the hypothetical surface is considered as the radiation surface. In addition, the real heat flux of panel surfaces can be obtained by making corresponding corrections.

3) The effect of air natural convective and forced convective in the cavity are ignored.

4) The absorbing and scattering effect of the air in the cavity receiver are ignored.

Based on these assumptions, the receiver becomes a closed cavity made up of N plane surfaces including the aperture. In the radiation transfer simulation, the domain for Monte Carlo simulations in this model consists of a 3D structure composed of quadrilateral grid cells, each of which is considered to represent a homogeneous and isothermal area.

Threshold condition

The heliostat of a solar power plant is displayed in the Fig. 4, and the simulation is conducted at a certain time. Figure 5 shows the distribution of aperture energy gathered by the heliostat field. These data come from Changchun Optical and Fine Machinery Institute. By those data, it can be supposed that more than 95% of energy is limited in the area of 2.0 m×2.0 m.

Radiation transfer boundary

In the radiation transfer, the heat flux and the wall temperature calculation is coupled, so that they must be iterative derivate simultaneously. The cells temperature is a bridge between radiative transfer and boundary. The calculation process of cells temperature is given in the following step.

In this study, the radiation boundary is a nontransparent surface and is coupled with thermal conductivity problem. A fictitious surface is put before boiling panels, and the fictitious surface is taken as the radiation boundary, and for the cavity wall without locating boiling panels, the cavity wall is taken as radiation boundary.

There are some equations for the boiling panel surfaces. For the ith cell, they can be written as

(5)q_{\mathrm{1}}=k_{\mathrm{1}}[T_{\mathrm{p}}-(T_{\mathrm{w}}+T_{\mathrm{0}})/\mathrm{2}],

(6)q_{\mathrm{1}}{s}_{\mathrm{p}}=c_{\mathrm{p}}G(T_{\mathrm{w}}-T_{\mathrm{0}}),

(7)qs=q_{\mathrm{1}}{s}_{\mathrm{p}}+q_{\mathrm{2}}s,

(8)q_{\mathrm{2}}=k_{\mathrm{2}}(T_{\mathrm{p}}-T_{\infty }),

where q_{\mathrm{1}} is the heat flux of boiling panels; q_{\mathrm{2}} is the heat flux of cavity wall; k_{\mathrm{1}} is the overall heat transfer coefficient of boiling panels; k_{\mathrm{2}} is the over-all heat transfer coefficient of cavity walls; s is the acreage of hypothetical face; s_{\mathrm{p}} is the corresponding panels acreage; G is the mass ratio flow of water; c_{\mathrm{p}} is the specific heat capacity of water at constant pressure; T_{\mathrm{p}} is the ektexine temperature of the panels; T_{\mathrm{w}} is the terminal temperature of water in the calculating cell; T_{\mathrm{0}} is the original temperature of water in the calculating cell; and T_{\infty } is environment temperature.

From Eqs. (5)-(7), the value of T_{\mathrm{p}} and T_{\mathrm{w}} can be obtained.

(9)T_{\mathrm{p}}={\displaystyle \frac{sq+s{k}_{\mathrm{2}}{T}_{\infty }+\mathrm{2}{c}_{\mathrm{p}}G{s}_{\mathrm{p}}{k}_{\mathrm{1}}{T}_{\mathrm{0}}/(\mathrm{2}{c}_{\mathrm{p}}G+s_{\mathrm{p}}{k}_{\mathrm{1}})}{\mathrm{2}{c}_{\mathrm{p}}G{s}_{\mathrm{p}}{k}_{\mathrm{1}}{T}_{\mathrm{0}}/(\mathrm{2}{c}_{\mathrm{p}}G+s_{\mathrm{p}}{k}_{\mathrm{1}})+s{k}_{\mathrm{2}}}},

(10)T_{\mathrm{w}}={\displaystyle \frac{s_{\mathrm{p}}{k}_{\mathrm{1}}{T}_{\mathrm{p}}}{c_{\mathrm{p}}G+s_{\mathrm{p}}{k}_{\mathrm{1}}/\mathrm{2}}}.

Results and analysis

Heat flux distribution of the boiling panels when reradiation from walls is omitted

Only considering the absorbability and reflection effect of solar energy on cavity walls and boiling panels, the heat flux distribution of boiling panels can be obtained using the Monte Carlo Method in which a ray is supposed to carry energy. As to the total inlet solar rays, the profile of the heat flux of boiling panels is shown in Fig. 6, in which the absorbing energy is 47.06% for central panels, 48.89% for side panels, and 2.78% for aperture. For easy display of the distribution of the three panels, they are smoothed into a plane. It is found in Fig. 6 that the heat flux distribution has a single peak at the center.

Heat flux distribution of the boiling panels when the wall’s reradiation is considered

Because of the high temperature of cavity walls and boiling panels, the radiation will have an effect on the heat flux distribution of the whole cavity. In order to have a full understanding, the effect of reradiation from walls is considered.

Using the improved MCM in which a ray does not carry energy, the heat flux distribution and panel temperature are obtained, as shown in Figs. 7 and 8, respectively. The peak heat flux, whose value is up to 1196.406 kW/m², is obtained in the center of absorbing panels. The absorbing energy is 49.10% for central panels, 47.02% for side panels, and 3.88% for aperture. It can be found that the distribution is very similar to the one when reradiation from walls is omitted. However, the percentage of boiling panels and aperture energy is increased. This is because the radiation energy of nonabsorbing walls is larger than that of the boiling panels.

Comparison between the simulation results with experimental data of CESA-I

The CESA-I receiver also is a cavity receiver with a polygonal shape [10]. Three faces of the cavity interior have absorbing panels. The experimental heat flux distribution of boiling panels is shown in Fig. 9. Comparing the simulation result with the measuring data, it can be found that they have a similar profile.

Conclusions

1) A model on heat flux distribution for solar cavity receiver has been built in this study. After comparison, it is found that the simulation results of this model agree very well with the experimental data of CESA-I.

2) Regarding the energy loss of the cavity receiver, the energy overflowing from the aperture by reflecting contributes to 2.78% of the total inlet solar energy, and the heat loss is 1.1% due to reradiation of cavity walls and boiling panels.

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