1.Introduction

Current heat transfer fluids (HTF) for solar concentrating systems are: synthetic oil,steam,molten salt and air. At temperature higher than 565℃ air is the only available HTF,but the poor heat transfer properties of air are well known. Consequently,researches on alternative HTF for the conversion of concentrated solar energy at high temperature (high Carnot efficiency) is an important R\&D topic for improving actual technologies. A review of thermodynamic cycles and working fluid has been published in [1] but for low-grade heat. Carbon dioxide appears to be a good candidate because it is non-flammable and non-toxic fluid. The CO₂ supercritical state (s-CO₂) is observed at 73.8 b and 304.5 K consequently favorable heat transfer and viscous supercritical properties may be built on designing innovative conversion systems. Some works have been done in the field of low and medium temperature solar heat conversion. For example,solar-driven carbon dioxide transcritical power system using evacuated tube type solar collectors was studied in [2] whereas supercritical Rankine cycle was examined in [3] and demonstrated in [4]. In this latter paper evacuated CO₂-based solar collectors showed 65-70\ solar heat collection efficiency and the measured power conversion efficiency was in the range 8.78-9.45\. At high temperature,it was pointed out in [5] that s-CO₂ recompression Brayton cycle can be as efficient as helium Brayton cycle with lower inlet turbine temperature (

$$550\,^{\circ}{\rm C}$$ for s-CO₂ vs. $$850\,^{\circ}{\rm C}$$ for He) but higher inlet pressure (20 MPa for s-CO₂ vs 8 MPa for He). In addition,its high working pressure makes the installation more compact and reduces the investment cost. Due to the relatively low inlet turbine temperature,this cycle was mainly studied for power generation in nuclear power plants [5,6,7]. However,high temperature s-CO₂ cycles may be used to produce power from concentrated solar energy systems. A molten salt solar tower using s-CO₂ Brayton cycle to produce electricity was described in [8] and the integration of heat storage to a supercritical CO₂ cycle was presented in [9]. The great potential of advanced s-CO₂ power cycles for concentrated solar energy conversion was demonstrated in [10]. It is shown that cycle configurations such as recompression cycles combined with intercooling and/or turbine reheat are able to achieve efficiency higher than 50\ for turbine inlet temperature larger than $$700\,^{\circ}{\rm C}$$ . Concerning heat transfer characteristics,experimental investigation of heat transfer in tubes have been carried out by [11] and [12] using 18.4 mm and 4.4 mm i.d. tubes respectively with s-CO₂ at pressures slightly higher than the supercritical one. Fully developed turbulent flow was studied in [12]. This overview of published works indicates the great potential of s-CO₂ for the conversion of concentrated solar energy but that radiation effect on heat exchange performances have been neglected although carbon dioxide is a well known absorbing gas in the IR region. Consequently,the objective of the present article is to show the influence of radiation in a pure CO₂ turbulent pipe flow as a function of pressure. A numerical approach is used to simulate the flow and heat transfer accounting for radiation transport since the pipe wall is at high temperature and CO₂ participates in radiation. First,the model is presented,then the simulation is described and,finally,the results are discussed in the last section.

2. Mathematical models

2.1 Mass transfer model

The mass balance equation and the momentum balance equation for the turbulent flow are solved. The turbulent flow is modeled with a steady Reynolds-averaged Navier-Stokes model which is the standard k-epsilon model. Therefore,the flow dynamics is modelled using the following steady state balance equations for mass (Eq. 1) and momentum (Eq. 2):

$$\begin{equation} \label{eq:mass} \frac{\partial} {\partial x_i} (\rho u_i) = 0 \end{equation}$$

$$\begin{align} \frac{\partial}{\partial x_j} (\rho u_i u_j) = &-& \frac{\partial p}{\partial x_i} + \frac{\partial}{\partial x_j} \left [\mu \left ( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} - \frac{2}{3} \delta_{ij} \frac{\partial u_l}{\partial x_l}\right ) \right] \nonumber \\ &+& \frac{\partial}{\partial x_j} \left [(\mu_t \left ( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right ) - \frac{2}{3} \left ( \rho k + \mu_t \frac{\partial u_k}{\partial x_k} \right ) \delta_{ij} \right] \label{eq:mom} \end{align}$$

where velocity

$$u_i$$ and pressure $$p$$ are Favre's mean variables (mass-averaged values) and where the Boussinesq approach is used to model the Reynolds stresses. The turbulent viscosity $$\mu_t$$ is modeled by a $$k-\epsilon$$ approach,therefore it depends on the turbulent kinetic energy $$k$$ and its dissipation rate $$\epsilon$$ following the relationship $$\mu_t = \rho C_{\mu} \frac{k^2}{\epsilon}$$ . Thus,two additional equations for $$k$$ and $\epsilon$ are solved:

$$\begin{equation} \label{eq:k} \frac{\partial}{\partial t} (\rho k) + \frac{\partial}{\partial x_i} (\rho k u_i) = \frac{\partial}{\partial x_j} \left[\left(\mu + \frac{\mu_t}{\sigma_k} \right) \frac{\partial k}{\partial x_j}\right] + \mu_t 2S_{ij} S_{ij} - \rho \epsilon \end{equation}$$

$$\begin{equation} \label{eq:eps} \frac{\partial}{\partial t} (\rho \epsilon) + \frac{\partial}{\partial x_i} (\rho \epsilon u_i) = \frac{\partial}{\partial x_j} \left[\left(\mu + \frac{\mu_t}{\sigma_{\epsilon}} \right) \frac{\partial \epsilon}{\partial x_j} \right] + C_{1 \epsilon}\frac{\epsilon}{k} \mu_t 2S_{ij} S_{ij} - C_{2 \epsilon} \rho \frac{\epsilon^2}{k} \end{equation}$$ $$\begin{equation} \mu_t = \rho C_{\mu} \frac{k^2}{\epsilon} \end{equation}$$

where the following constants are used:

$$C_{1 \epsilon} = 1.44$$ , $$C_{2 \epsilon} = 1.92$$ , $$C_{\mu} = 0.09$$ , $$\sigma_k = 1.0$$ , $$\sigma_{\epsilon} = 1.3$$ ; and $$S_{ij}$$ represents the mean rate-of-strain tensor.

The density,specific heat capacity,thermal conductivity and viscosity of CO₂ were extracted from the work of [13].

2.2 Heat transfer model

Turbulent heat transport is modeled using the concept of Reynolds' analogy to turbulent momentum transfer. The energy balance equation is thus given by:

$$\begin{equation} \label{eq:nrj} \frac{\partial}{\partial t} (\rho E) + \frac{\partial}{\partial x_i} [u_i (\rho E + p)] = \frac{\partial}{\partial x_j} \left( \lambda_{\rm eff} \frac{\partial T}{\partial x_j} + u_i \mu_{t} \left(\frac{\partial u_j}{\partial x_i} + \frac{\partial u_i}{\partial x_j} \right) - u_i \frac{2}{3} \mu_{t} \frac{\partial u_k}{\partial x_k} \delta_{ij} \right) + S_{\rm rad} \end{equation}$$

where $E$ is the total energy and

$$\lambda_{\rm eff}$$ is the effective thermal conductivity,\]\lambda_{\rm eff} = \lambda + \frac{c_p \mu_t}{{\rm Pr}_t} $$(with \[\lambda$ the thermal conductivity and$$ {\rm Pr}_t=0.85$).

The energy balance equation includes a radiative source term,

$$S_{\rm rad}$$ ,to account for CO₂ participation in radiation transport inside the pipe. Indeed,carbon dioxide is a participating medium that strongly absorbs infrared radiation mainly in the 2.7 $$\mu$m bands,4.3 \[\mu$m bands and 15 \[\mu$$ m bands. Carbone dioxide is a semi-transparent gas that emits radiation and mainly absorbs radiation emitted from the high temperature pipe wall. Thus,radiation absorption can play a significant role in heat transfer balance.

2.3 Radiative transfer model

Radiation transport is considered in an emitting,absorbing,non-scattering and non-gray medium surrounded by gray walls. The monochromatic RTE (Radiative Transfer Equation) for an absorbing and emitting medium,at position

$${\vec r}$$ and in the direction $${\vec s}$$ ,can be written as:

$$\begin{equation}\label{eq:etrnu} \frac{\textrm{d}I_\nu }{ \textrm{d}s}=\kappa_\nu (I_{b\nu}-I_\nu) \end{equation}$$

with the associated boundary condition for a gray surface that emits and reflects diffusely:

$$\begin{equation}\label{eq:wallnu} I_{w\nu}(\vec{s})=\epsilon_w I_{b\nu}+ \frac{1-\epsilon_w }{ \pi} \int_{\vec{n}\cdot \vec{s'}<0} | \vec{n} \cdot \vec{s'}|I_\nu(\vec{s'})\textrm{d}\Omega' \end{equation}$$

The radiative property model chosen in this study is a multi-gray approach and the RTE for the $j$th gray component is expressed as [14]:

$$\begin{equation}\label{eq:etr} \frac{\textrm{d}I_j }{ \textrm{d}s}=\kappa_j (a_j \frac{\sigma T^4 }{ \pi}-I_j) \end{equation}$$

where $a_j$ and $\kappa_j$ are,respectively,the emission weighting factor and absorption coefficient for the $j$th gray component. The quantities $a_j$ and $\kappa_j$ are temperature dependent. The boundary conditions for gray walls thus become:

$$\begin{equation}\label{eq:wall} I_{w}(\vec{s})=\epsilon_w a_j \frac{\sigma T^4 }{ \pi} + \frac{1-\epsilon_w }{ \pi} \int_{\vec{n}\cdot \vec{s'}<0} | \vec{n} \cdot \vec{s'}|I_j(\vec{s'})\textrm{d}\Omega' \end{equation}$$

The radiative source term (in Eq. 5) is computed with a finite volume model for spatial and directional integrations. The radiative transfer equation (Eq. 8) is solved using the Discrete Ordinates (DO) radiation model (as it is called in the ANSYS Fluent commercial software) for a finite number of discrete angles. This DO model corresponds to the finite-volume method developed by Chui and Raithby [15] which is different but presents many similarities with the classical discrete ordinate method. In two-dimensional simulations,only four octants are solved due to symmetry. In the presented two-dimensional simulations,16 directions are adopted for the angular discretization of each octant. For spectral integration the global spectral ADF (Absorption Distribution Function) model [16] is chosen for the description of global radiative properties of CO₂ at high temperature (ADF-CO₂). These two models (DO and ADF) are approximate models but they are suitable for coupled CFD simulations and do not involve huge computer time.

2.4 Radiative properties of carbon dioxide

The participating medium is pure carbon dioxide at high temperature (up to 1100 K) and from low (0.1 MPa) to high pressure (20 MPa). At high temperature and Therefore,the radiative properties of CO₂ should be computed over a large temperature and pressure ranges. At high temperature,the radiative properties of CO₂ can be computed based on high temperature spectroscopic databases such as HITEMP-2010 [17] or CDSD-4000 [18]. The spectroscopic database selected is HITEMP-2010 because CO₂ temperature does not reach $4000 K$ in the present calculation so we do not need to involve transitions occuring around 4000 K. For low pressure,CO₂ spectra can be computed within the limits of the impact approximation based on the isolated line concept. The line shape is considered Lorentzian and accordingly to the isolated line concept,each line shape is not modified by other transitions. However,as pressure increases the gas density increases and the collision-induced transfers of population become important [19]. Spectral transitions can overlap significantly with each other and the isolated line assumption breaks down. Thus,the spectral shape should be modeled with the line mixing process. A line-by-line model [20] was used to obtain synthetic spectra of pure CO₂ assuming a Lorentz line profile corrected by a

$$\chi$$ factor [21] that accounts for line mixing effects and results in a transfer of population from line wings to line core. Carbon dioxide spectra were computed for different pressures and temperatures. To ensure short computation time while keeping a sufficient accuracy on the radiative source term computation,the high resolution CO₂ radiative properties should be approximated by a global spectral model requiring only a few number of parameters. Indeed,high resolution spectra requires to discretize absorption coefficient spectra with tens of millions of points whereas a global spectral model requires only few tens of points. High resolution spectra are also necessary to compute the parameters of approximate global spectral models. In this study,spectra were used to create a set of radiative properties based on the Absorption Distribution Function (ADF) global spectral model. The parameters of the ADF-CO₂ model are the gray gases absorption coefficients $$k_j(T)$$ and its correponding wheights $$a_j(T)$$ . It is assumed that radiation transport in the simulated CO₂ pipe flow is well described using nine grey gases. These parameters were computed after rearranging the global high resolution spectra into k-distributions [16]. The ADF spectral radiative properties of CO₂ were computed based on the high temperature spectroscopic data of HIPTEMP-2010 for CO₂ in the spectral range between 330 and 10000 cm-1 (1-30 $$\mu$$ m). Four sets of ADF-CO₂ parametres were computed for different working pressures (0.1 MPa,1 MPa,5 MPa and 20 MPa) and each set of ADF-CO₂ parameters is valid for the temperature range 300-1500 K.

As an illustration of carbon dioxide radiative properties,Figure 1 presents an example of its pressure and temperature dependencies. Figure 1(a) shows the evolution of narrow band absorption coefficent spectra for different CO₂ pressures at a temperature of 600 K and Figure 1(b) shows high resolution spectra of CO₂ absorption coefficients (at 0.1 MPa) for two temperatures. At high pressure the absorption coefficient reaches high values and becomes opaque in the band center (Fig. 1)(a). At a pressure of 0.1 MPa the average absorption coefficient is almost superimposed on the x-axis. The temperature dependence of the CO₂ spectra is shown in Fig. 1(b). The absorption coefficient of cold lines decreases when the temperature increases. In addition,towards high wavelength (not shown in Fig. 1(b)),hot lines appear when the temperature increases resulting in a slight increase of the absorption coefficient.

3. Computational implementation of the model

3.1. Numerical methods

The ANSYS Fluent commercial finite volume fluid dynamics solver is used to solve the transport Eqs. (1)-(5) and (9). In particular,the pressure based solver is used,employing the SIMPLE pressure velocity coupling method [22]. Gradients are computed using the least squares cell based method,and the standard pressure interpolation scheme is used. The QUICK [23] scheme is used for the spatial discretization of the flow and energy transport equations. The near-wall modeling method selected for the wall-bounded turbulent flow is the enhanced-wall-treatment that can be used with fine and coarse meshes [24]. The finite volume method is also employed for the computation of radiation transport using a $4\times 4$ discretization of the zenith and azimuth angles in each octant. The temperature and composition dependent thermo-physical properties of CO₂ are computed for each cell using Fluent's User-Defined-Functions (UDFs). The radiation absorption coefficient is also computed by an UDF. The solution time is several hours (about 10 hours) on a modern PC workstation (parallel Fluent on a Quad Core processor at 2.93 GHz).

4. Results and discussion

4.1. Computational domain

The pipe is circular in cross section and is modelled using a structured two-dimensional axisymmetric grid. The grid is oriented such that the

$$x$$ -direction lies along the length of the reactor and the $$y$$ -direction lies along the radius of the reactor. The radius and length of the reactor are,respectively,2 cm and 2 m,leading to an aspect ratio $$L/R$$ = 100.

In the

$$y$$ direction,a refined grid (boundary layer) is used close to the wall,starting with $$\Delta y = 10^{-5}$$ m and increasing with a growth factor of $$1.05$$ over the first 50 cells. A constant grid spacing of $$\Delta y = 2 \times 10^{-4}$$ m is used thereafter up to the axis of symmetry. In the $$x$$ -direction,a grid spacing $$\Delta x = 0.5$$ mm is used in the refined boundary layer and another grid spacing $$\Delta x = 1$$ mm is used for the remaining mesh. The total number of cells in the domain is $$3.5 \times 10^{5}$$ . Although not discussed in detail,a grid sensitivity study was performed prior to selecting the present discretization.

4.2.Boundary conditions

The inlet boundary condition is an inlet velocity for pure CO₂ at 400 K and different simulations are conducted with different inlet pressures. The selected pressures cover the range from 0.1 MPa to 20 MPa with intermediate values of 1 MPa and 5 MPa. Similar inlet velocities were chosen for the different flow simulations having different static pressures. The inlet velocity is specified parabolic with a mean value of 1 m/s that leads to 0.0017 kg/s at 0.1 MPa,0.0169 kg/s at 1 MPa,0.0915 kg/s at 5 MPa and 0.4782 kg/s at 20 MPa. The flow is heated by the hot pipe wall at a constant temperature of $1100$ K considered as a blackbody ($\epsilon_w = 1$).

5. Results and Discussion

A sketch of the reactor is depicted in Figure 2 with a real length to radius ratio. The reactor size does not allow one to clearly represent fields inside the pipe while keeping the real length to radius ratio. This is the reason why the simulated fields inside the reactor will be represented in figures where the radius scale (y axis) is enlarged. The simulation results are presented below in Figures 3 and 4,and the effects of radiation transport in the pipe are discussed. The fields are depicted using isocontour plots of approximately unit aspect ratio; however,the aspect ratio of the reactor (length to radius ratio) is 100,and,therefore,to correctly interpret the geometry of the intercontour regions it is necessary to consider that the axial distance between contours is 100 times greater than depicted,while gradients with even a small apparent radial component are essentially radial. Concerning the validation of the calculated results,a validation was obtained on a numerical basis by checking the heat and mass balances between inlet and outlet and this step is not shown here. A full experimental validation of calculated results is out of the scope of this article and will be part of a future work since it requires additional efforts. Concerning the calculated results,the flow and energy conservation were checked numerically between the inlet and the outlet and this step is not shown here.

The influence of radiation transport on temperature and velocity fields is studied for CO₂ pipe flow at four different pressures. The effect of radiation is highlighted when comparing two simulation results obtained with and without the radiative heat transfer model. Figures 3(a)-3(d) present isocontours of temperature in the axisymmetric geometry of the pipe and Figures 4(a)-4(d) present isocontours of velocity. Because the pipe wall is at high temperature,one can expect that radiation absorption by the CO₂ flow leads to a fluid temperature increase. This phenomena occurs clearly for flows with 0.1 and 1 MPa (Figs 3(a)-3)(b) but tends to diminish for higher pressures (Figs 3(c)-3(d)). Indeed,when comparing the temperature profiles obtained for different pressures and neglecting or not the radiation transport,the results show a decrease of the influence of radiation with respect to a pressure increase. The cross comparison of velocity and temperature fields (Figs 3-4) show also the effect of heat and flow coupling which leads to an acceleration of the flow as the temperature increases (for the same fixed pressure). Consequently,the lower velocity is find for the cases where the influence of radiation absorption is weaker,i.e. at high pressures.

As the pressure increases,while keeping the same inlet velocities,the simulated results show a decrease of heat transfer from the hot wall to the tube center due to radiation transport. This effect is due to an increase of the CO₂ optical thickness which contributes to stop the radiation transport from the hot wall to the colder fluid at the pipe center. Thus,as the pressure increases one can consider that radiative exchanges between CO₂ and the wall occur more gradually,the radiation from the wall heats only the fluid in its own vicinity as does CO₂ which exchanges by radiation only with its close neighborhood. As the pressure increases (e.g. above 5 Mpa),the pure CO₂ media become optically thick and one can model the radiation transport by a diffusion process and use simple radiative models such as the diffusion approximation (P₁ model) or the Rosseland approximation [14]. Moreover,the presented results clearly show the radiative transfer can be neglected in pure CO₂ above 20 MPa. It is worth noting that these results do not stand for an accurate determination of pressure intervals where a given model can be used,but they show the trend associated to the influence of pressure on radiation transport in a high temperature pipe.

6. Conclusion

A two-dimensional axisymmetric model to solve the turbulent flow and the heat transfer inside a hot pipe is described. The influence of radiation on the flow is discussed. It is shown that the influence of radiation is weak at high pressure for the optical thickness involved in this study. A two-dimensional axisymmetric model to solve the turbulent flow and the heat transfer inside a hot pipe is described. The pipe temperature is assumed constant at 1100 K and CO₂ inlet temperature is 400 K. The influence of radiation on the flow temperature and velocity distribution is discussed. It is shown that the influence of radiation is important at low pressure (less than 5 MPa) but becomes negligible at pressure larger than 5 MPa. In the pressure range 0.1-5 MPa radiation results in CO₂ flow temperature and velocity increases. The influence of radiation on both temperature and velocity decreases with an increase of the working pressure and becomes non significant at about 20 MPa. This means that the influence of radiation absorption is rather weak at high pressure for the optical thickness involved in this study. At high pressure CO₂ becomes optically thick and radiation is absorbed in the very thin layer near the hot wall. Consequently,this shadowing effect limits drastically radiation propagation through the pipe.

Acknowledgements

The authors want to thanks Pr. Jean-Michel Hartmann for his valuable advices concerning high pressure radiative properties of CO₂.