Numerical Study of Parameters Effect on Thermo-flow Characteristics of up Bubbling Fluidized Bed Particle Solar Receiver

Kaijun Jiang,¹ Qiang Zhang,¹ Yanqiang Kong,¹ Chao Xu,¹ Xing Ju¹ and Xiaoze Du^{1, 2*}

¹ Key Laboratory of Condition Monitoring and Control for Power Plant Equipment (North China Electric Power University), Ministry of Education, Beijing 102206, China.

² School of Energy and Power Engineering, Lanzhou University of Technology, Lanzhou 730050, China.

* Email: [email protected] (X. Du)

Abstract

Concentrated solar power (CSP) plant integrated with thermal energy storage system has the potential to scale up to megawatt scale. A new concept using dense particle suspension flow in up bubbling fluidized bed (UBFB) solar receiver for solar energy capture is promising. The characterization of the dense particle suspension was rarely investigated when operating conditions and tube structure changed. In this paper, a 3D two-fluids model is established to investigate the behaviour of dense particles suspension in the UBFB receiver. The impacts of the superficial gas velocity, tube diameter and wall temperature on the dense particle suspension hydrodynamics and wall-to-bed heat transfer process are comprehensively studied. The results show that the stirring action of bubbles due to bubble wake drift is enhanced when the superficial gas velocity increased and the wall-to-bed heat transfer coefficient is increased with the tube diameter under a certain range. Moreover, the radiation heat transfer becomes significant when the wall temperature exceeds 1000 K. The results obtained can provide a valuable reference for the design and operation of the UBFB receiver in CSP plant.

Table of Contents

Keywords: Concentrated solar power; Dense particle suspension; Solid particle solar receiver; Two-fluids model; Heat transfer coefficient.

1. Introduction

Nowadays, the main ways of renewable energy utilization are wind power generation, nuclear power generation, biomass energy generation and solar energy generation.^[1] Among all the renewable energy applications, solar energy with the merits of huge abundance, availability, unlimitedness and wide distribution has attracted great attention.^[2] Concentrated solar power (CSP) is a potential electricity generation technology that concentrates solar irradiance reflected by heliostats into the receiver where the heat transfer fluid (HTF) is heated.^[3] The CSP plant with energy storage system can operate steadily at night. The solar power tower is favored by many researchers since its advantages of higher efficiency, lower operating costs and potential of scale-up. Direct steam, air and molten salts (NaNO₃-KNO₃) are mainly used as HTF

in solar power tower currently. However, direct steam as HTF has some drawbacks such as the temperature at the exit is low (<525 ℃), steam can not be directly used as the heat storage medium and so on. Pressure air receiver suffers from low heat flux and low heat transfer coefficient (HTC).^[4] The biggest problem with molten salts as HTF is chemical instability when the temperature exceeds 565 ℃ and solidification when the temperature is lower than 220 ℃.^[5] Recently, a novel concept using dense particle suspension (DPS) as HTF is proposed by Flamant et al. ^[6,7] It has advantages of higher temperature, lower costs, higher HTC, and the solid particle can be directly used to store heat. These characteristics give the opportunity to scale-up the CSP plant.^[8]

Many research teams proposed different kinds of solid particle receivers. According to the methods that HTF absorbs solar irradiance, solid particle receivers can be mainly divided into two categories: direct absorption and indirect absorption.^[9] Direct absorption solid particle receivers are very appealing due to its high solar flux density (~1 MW/m²). However, from the view of engineering operation, the solid mass flow rate is difficult to control and the convection heat losses may be enormous. Indirect absorption solid particle receivers obtain lower radiative flux density compared with direct absorption system, but indirect absorption solid particle receivers have the merits of better control of solid mass flow rate and low particle losses during operation.^[10] Flamant et al. ^[6] proposed a novel concept of solid particle receiver which uses the dense particle suspension (DPS) composed of fine solid particles (SiC, d_p = 64 μm) flow in the metallic tubes, named up bubbling fluidized bed (UBFB) receiver. The solid volume fraction in the riser is under the range of 30–40 % and the HTF outlet temperature can exceed 750 ℃. Besides, the requirements for particle materials are lower and no particle losses when solid flux circulated in the UBFB receiver compared with free-falling particle receiver. Fig. 1 shows a general diagram of a next-generation CSP plant using solid particles as HTF and heat storage medium.^[10] The particles are heated up to a certain temperature and then transported to a hot pot which in turn feeds a heat exchanger where the thermal energy is transmitted to the working fluid. After that, the hot working fluid is sent to the power block. Cold particles are transported to a cold pot that feeds the receiver. Therefore, it has the potential to drive the high-efficiency power cycles, such as supercritical steam Rankine cycle or supercritical carbon dioxide Bretton cycle.^[11]

Fig. 1 Schematic diagram of solar power tower plant. Reproduced with the permission from Ref. [10]. [email protected].

The wall-to-bed HTC is a critical parameter in the design of the UBFB receiver.^[12,13] The wall-to-bed HTC is determined by many factors, including the superficial gas velocities, tube diameter, tube wall temperature. Various empirical correlations to predict HTC have been proposed by many scholars. The commonly used models are Molodstof-Muzyka model and Goriz-Grace model.^[14,15] However, the result determined by Molodstof-Muzyka model is not precise, and Goriz-Grace model only predicts HTC precisely within a narrow range of operating conditions. Besides, Zhang et al. ^[16] proposed a modified surface renewal model to predict HTC in different operating conditions, but it needs to measure too many parameters to compute. In a word, this empirical correlation models do not yield detailed information about DPS dynamics and heat transfer process.

With the development of the computer, computational fluid dynamics (CFD) is an effective way to obtain detailed information about DPS dynamics and heat transfer mechanism. There are two general methods of simulation for granular flow: the discrete element method (DEM) and the two-fluids model (TFM).^[17] The main difference between DEM and TFM is the method to solve the solid phase. The DEM is an Euler-Lagrangian method which tracks the behaviour of each particle. The TFM is also called Euler-Euler method which treats solid phase as a continuum basing the Kinetic theory for granular flow (KTGF) to enclose the conservation equations. Although the tracking of every particle is more precise, the computational cost may be very high and the number of tracking particles is limited (<10⁶). In this study, the number of particles in the rising tube is more than 10⁹. As a result, the Euler-Euler model is more suitable in this study. Many scholars used TFM to simulate gas-solid behaviour in the tubes, but most of them focused on traditional gas-solid system and only hydrodynamics is investigated. Wang et al. ^[18] used TFM to study the minimum bubbling velocity of A-type particles. Hosseini et al. ^[19] successfully used TFM to study the influence of various drag models on bubbles formation and break up. For the UBFB receiver, scarce simulations are carried out in the field. Ansart et al.^[20] established a single tube 3D model in NEPTUNE_CFD, but the heat transfer process is not considered. Marti et al. ^[21] successfully established a 2D model in OpenFoam to investigate the DPS heat transfer mechanism while the simplified 2D model cannot capture the details of the DPS in the tube. Urrutia et al. ^[22] used TFM to simulate the behaviour of the DPS in the tubes, but the influence of operating parameters and the radiation heat transfer is not further investigated.

As mentioned above, the effects of operating conditions and the tube structure under high-temperature on the UBFB receiver need to be further investigated. To our best of knowledge, no CFD model has been established to study the influences of operating conditions and structural parameters on the thermo-flow of the UBFB receiver. Moreover, most of the studies focused on the hydrodynamics of DPS in the rising tube. Based on reviewing prior work, the main aims of this paper are (1) establishing the TFM model to simulate the behaviour of the particles in the rising tube under the transient condition and predict the wall-to-bed HTC; (2) investigating the effects of superficial gas velocity, tube diameter, and wall temperature on the behaviours of DPS and the wall-to-bed HTC; (3) considering the contribution of the radiation heat transfer. Therefore, the results in this paper can provide a useful reference for the UBFB receiver design in the future and explain the behaviour of the DPS under the high temperature.

2 Physical and numerical model

2.1 Mathematical model

In this research, a modified Euler-Euler model integrating the KTGF for solid phase is used to simulate the behaviour of DPS in the UBFB receiver under a wide range of operating conditions. The modifications include a model to calculate the effective conductivities according to solid volume fraction and a model to estimate the radiation heat transfer. All cases are performed on ANSYS Fluent 15.0.

2.1.1 Conservation equations

Euler-Euler model contains the set of mass, momentum and energy conservation equations for both phases, which are coupled by the pressure and interphase exchange coefficients. For the sake of brevity, only the governing equations of the solid phase are represented due to the governing equations of the gas phase are similar. The mass conservation equation for the solid phase without any mass transfer between two phases is determined by

+ = 0 (1)

The momentum conservation equation is given by

(2)

The energy conservation equation is given by

(3)

where S_rad represents the radiation heat transfer through the solid phase, h_g,p is the interfacial HTC.

2.1.2 Closure equations

In the Euler-Euler model, a series of constitutive equations is needed to balance the conservation equations for the solution because the solid phase is treated as the continuous phase. In this paper, for solid phase, the constitutive equations derive from KTGF. The KTGF is a significant tool to describe the movement and fluid property of the solid phase. First of all, it needs to set a fundamental parameter that is the granular temperature (=/3) which describes the variations in the velocity of the particles. Table 1 summarizes the relevant constitutive equations. ^[13,18]

Table 1 Constitutive equations ^[13,18]

Constitutive equations

Solid kinetic viscosity

Collisional viscosity

Frictional viscosity doesn’t need to consider because always less than the , besides =0.63

Solid bulk viscosity

Table 2 Drag model equations ^{[19, 24]}

Drag model equations

,, and, which is determined by minimum fluidized gas velocity.

2.1.3 Define the effective diameter

Because particles are non-spherical actually, it needs to define effective diameter d_p,eff that satisfied the Ergun equation.^[23] The effective diameter d_p,eff is refer to the mean Sauter diameter d_p according to the following equation:

(4)

whereis the shape factor of particles. In this case, SiC particle is used, =0.77, n=2.53.

2.1.4 Modified drag force model

Gas-solid drag force is the main force in granular flow, especially in vertical flow, which determines bubble behaviour, entrainment and transport of solid particles. The corresponding model is the key to describe two-phase movement in numerical simulation accurately. There are two types of drag force models for gas-solid interaction: one is empirical or semi-empirical models based on pressure drop, superficial gas velocity and bed expansion such as Syamlal-O’Brien, Gidaspow. Another one is derived from gas-solid interaction theory such as Koch-Hill. In this paper, a modified Syamlal-O’Brien drag force model is applied and relevant equations can be seen from Table 2.^{[19, 24]}

2.1.5 Radiation Model

The radiation model needs to be selected according to optical thickness. In this case, DPS is composed of small particles and optical thickness is greater than 1. As a result, P-1 radiation model is suitable in this case.^[25] P-1 radiation is based on the spherical-harmonics method. Radiation flux q_r is given by

(5)

where , is the scattering coefficient, is the linear-anisotropic phase function, is the incident radiation. The incidence radiation is calculated from

(6)

As mentioned above, the radiation source term is given by

(7)

2.1.6 Effective conductivity

The heat transfer process may be underestimated if the gas and solid phase conductivity directly be used in the simulation. The reason is that the DPS is composed of the gas phase and solid phase. Besides, the wall-to-bed HTC is determined by two phases. Therefore, it is necessary to modify the correlations of conductivity. Kuipers et al. ^[26] proposed an expression comprehensively consider the effects of the gas phase and solids phase as follows.

(8)

(9)

(10)

where, and.

2.1.7 Interphase heat transfer

Assuming particles are sphere, the interphase HTC in Eq. (3) is determined by (11)

The Nusselt number for the gas-solid system is determined by Gunn correlation ^[27] as follows

(12)

where Re_s is Reynolds number of particles in the solid phase, Pr is Prandtl number of the gas phase determined by

(13)

Table 3 SiC and Air thermophysical properties ^[28-30]

SiC physical properties

(kg/m³) 3210

(μm) 64

0.95

(J/kgK)

(W/mK)

(m/s)

correlation from Singh et al. ^[28]

correlation from Marti et al. ^[29]

Air physical properties

ideal gas law

(J/kgK) 1020

(W/mK)

(Pa∙s) correlation from Perry et al. ^[30]

2.2 Details of the simulations

2.2.1 Phases properties

The solid particles used in this study are SiC and fluidized gas is air, detailed information about the particle properties can be

seen from Table 3.^[28-30]

2.2.2 Boundary and initial conditions

1) Inlet and outlet boundary conditions

The inlet is set as a velocity inlet condition. Both particles and air enter the tube with the same temperature which is determined by experiments. The solid phase enters the tube with an interstitial velocity and volume fraction corresponding to the average values measured by the experiment. The velocity difference between the gas and solid is slip velocity (U_slip). Therefore, the gas phase velocity at the entrance is set as U_g=U_s+U_slip. These settings applied to the simulations except investigating the effect of superficial gas velocity. The gas phase velocity at the entrance is set as 3U_mf, 5U_mf, 7U_mf and 9U_mf for investigating the impacts of superficial gas velocity.

The outlet is set as a pressure outlet and zero-gauge pressure is fixed at the outlet.

2) Tube wall conditions

The circumferential temperature of the rising tube is convenient to measure compared with the heat flux. Besides, the circumferential temperature of the rising tube can be view as uniform due to the drastic heat conduction in metallic tubes.^[31] As a result, in the heated tube section, the wall temperature condition is applied. Based on average experimental results, the wall temperature profile is assumed a polynomial change temperature function T(z). The functional relationship is changed with the operating condition. The reduction at the top of the tube is assumed heat insulation.

The no-slip boundary condition is assumed for the two phases at the walls. ^[32]

3) Initial conditions and other settings

At the simulation beginning, a fixed bed with a porosity of 0.4 and a height of 0.25 is assumed. The temperature of the initial bed is equal to the gas inlet temperature.

Due to the fluidized bed fluctuates, all simulations performed under the transitory state. The SIMPLE algorithm is used for the pressure-velocity coupling. The second-order upwind discretization scheme is used for discretizing density, momentum and energy conservation equations. Quick discretization scheme is applied to discretize volume fraction.

2.3 Experimental model and calculate domain

In order to understand the dynamics and heat transfer process of DPS in the UBFB receiver, a single rising tube experimental setup was constructed by Flamant et al.^[6,7] This study was conducted on a solar furnace (1WM) in France CNRS-Promes. On the premise of minimizing the amount of calculation and keeping the results accurate, only the section of the tube, length = 0.5 m, exposed to concentrated radiative flux is simulated. The detail information about the rig and experimental results can refer to the references.^[6,7,21]

Fig. 2 3D schematic of the tube with mesh and the schematic of the upper zone used to determine h_wb and cross-sectional view of the tube with mesh.

2.4 Date process method

To avoid impacts of the initial conditions and boundary, the upper section of the tube is used to determine the wall-to-bed HTC, as shown in Fig. 2. The time-averaged power transmitted to the solid phasewhen simulation reaches quasi-steady state is determined by

(14)

whereis the time-averaged solid mass flow rate, is solid specific heat capacity, T_s,m is the average middle cross-sectional solid particle temperature, T_s,o is the average outlet solid particle temperature.

The wall-to-bed HTC is given by

(15)

where theis upper section tube internal area,is logarithmic mean temperature difference given by

(16)

where T_w,m is the middle tube wall temperature, T_w,o is the outlet tube wall temperature.^[22]

3 Results and discussion

3.1 Model validation

3.1.1 Mesh independency validation

In this study, a hybrid mesh is applied, as shown in Fig. 2. The boundary layer grid is applied in the near-wall region because the temperature intensely changes in the region. As shown in Fig. 2, a reduction is added at the top of the rising tube to improve the simulation convergence.

Fig. 3 Outlet solid temperature and time-averaged solid volume fraction as a function of the number of cells.

Ensuring that the numerical results are independent with the grid size before the simulation is important. Grid independence of the simulation results is tested for four different grid sizes by comparing the solid phase temperature at the tube outlet and the average solid volume fraction in the tube. The solid particles temperature at the tube outlet and the average solid volume fraction as a function of the number of cells is illustrated in Fig. 3. As the grid size reduced, the solid phase temperature at the tube outlet is increased and the solid volume fraction is decreased. The temperature difference between the last two cases of grids is less than 0.5% and the solid volume fraction difference between the last two sets of grids is less than 1%. The simulation result is almost unchanged. Therefore, the authors conclude that 334196 cells are enough and grid size with 334196 cells is used in the following analysis.

Table 4 Simulation conditions and results compared with experiment results.

(kg/h)	D (mm)	(K)	(K)	(K)	(%)	(W/m^2K)	(W/m^2K)	(%)	T(z), 0<z<0.5m
42.35	36	455.78	577.36	548.83	4.94	489.71	449.66	8.18	-414z^2+302z+559
64.29	36	421.76	532.67	516.86	2.97	630.75	596.35	5.45	-501z^2+290z+538
70.59	36	485.40	582.14	564.26	3.07	656.20	637.70	2.82	-229z^2+165z+585
87.81	36	389.70	528.15	495.85	6.11	666.53	628.56	5.70	-442z^2+334z+505

3.1.2 Simulation results compared with experiment results and basic analysis

The simulation is deemed as convergence when the mean solid temperature at the tube exit and mean solid fraction reached relative stability. The solid mass flow rate is negatively related to the simulation time. It needs approximate 24 days to compute with lower solid mass flow rate and it needs approximate 14 days to finish the job with higher solid mass flow rate. In Table 4, the numerical results of the time-averaged outlet solid particles temperature and the wall-to-bed HTC with the experimental results are compared. The temperature of solid particles is convenient to determine by thermocouples in the experiment, so the temperature difference between simulation and experiment is the main comparison. The temperature was much overestimated with low solid mass flow rate. The reason may be that the solid phase recirculation at low solid mass flow rate is overestimated. Fig. 4 displays the solid particles velocity vector line at m_s=42.35kg/h, t=60s. It can see from the figure that solid move downward at the vicinity of the wall and wall-slugging is observed. The downward solid flux moves to the center region due to lower pressure at the bubble wake. The maximum relative error of the outlet solid phase temperature between simulation and experiment is 6.50%, which verifies that the numerical model can correctly simulate the DPS behaviour in the rising tube.

Fig. 4 Typical picture of solid particles velocity vector line.

3.2 Basic analysis

Fig. 5 shows the XZ-plane temperature contour of the solid phase in different operating conditions. As shown in Fig. 5, the solid mass flow rate significantly impacts the solid temperature profile and T_s,o. The result shows the rapid increase of the solid phase temperature with height and the radial profile distribution. In the radial direction, a thermal penetration thickness exists near the wall area where heat is transferred mainly by convection and conduction. As a result, the heat transfer between the wall and DPS is acutely and the temperature gradient is high. Fig. 6 shows the solid volume fraction contours at different time. The solid concentration in the near-wall region and rising bubbles at the top of the tube are obvious. The solid concentration in the near-wall region is beneficial for the wall-to-bed heat transfer. The differences in solid volume fraction, tube wall temperature profile and superficial gas velocity between different operating conditions are the main reason for the phenomenon. The factors will be further investigated in the following sections.

Fig. 5 XZ-plane solid phase temperature contours in different solid mass flow rate (a: m_s=42.35kg/h; b: m_s=64.29kg/h; c: m_s=70.59kg/h; d: m_s=87.81kg/h, t=73s).

3.3 The effects of superficial gas velocity

The parameter of superficial gas velocity significantly influences the DPS dynamics and heat transfer process. In order to investigate the effect of the superficial gas velocities (U_g=3U_mf, 5U_mf, 7U_mf and 9U_mf), a constant set of wall temperature condition and inlet solid phase velocity is used.

Fig. 7 presents the time-averaged wall-to-bed HTC and outlet

solid phase temperature as a function of U_g. The value of HTC tends to progressively increase to a maximum value and then gradually decrease when the U_g increased. For the bubbling fluidized bed, the stirring action of bubbles due to bubble wake drift is enhanced when the U_g increased. As a result, the renewal frequency of fresh solid particle in the near-wall region is improved and wall-to-bed HTC is enhanced. However, the solid phase concentration is decreased and the fraction of time that the heating surface is covered by gas phase increased, and this eventually outweighs the renewal effect and leads to the decrease of the wall-to-bed HTC. Fig. 8 presents the contour of the solids holdup for various U_g at 3.5s. The results show that the number and frequency of bubbles are increased with the U_g. Therefore, the stirred affection of gas bubbles is obviously increased with U_g increased.

Fig. 6 XZ-plane solid volume fraction at different time (m_s=87.81kg/h, t=68s).

Fig. 7 The wall-to-bed HTC and outlet solid phase temperature as a function of U_g.

Fig. 8 Solid volume fraction contours for different U_g at 3.5s.

Fig. 9 Solid phase temperature contours for different U_g at 60s (XZ-plane).

Fig. 9 presents the different cases instantaneous solid phase temperature contours at 60s. The result shows that the value of superficial gas velocity significantly affects the solid phase temperature profile by improving the stirring action of bubbles and solid particles concentration distribution. It is found that the increase of solid particles temperature in different working conditions is different in the Z-axis direction, and the temperature of solid particles swiftly increased when superficial gas velocity is 5U_mf. Fig. 10 presents the distributions of time-averaged solid particles temperature along X-axis direction and Y-axis direction. In order to avoid reflux influence, the value is got from the height of 0.4m. The distributions of time-averaged solid particles temperature along X-axis direction and Y-axis direction present similar variation tendency that maximum temperature occurs in the vicinity of the wall and decrease toward the center. As shown in Fig. 10, the temperature of the solid phase quickly decreases in the vicinity of the wall. The results indicate that the main heat transfer process of solid particles is concentrated near the wall by convection and conduction and the heat transfer between wall and DPS is acutely in the near-wall region. The maximum temperature of particles occurs in the near-wall region. This is mainly due to that the solid particles concentrated in the near-wall region and the maximum temperature gradient also occurs in the near-wall region. Fig. 10 also presents the temperature gradient near the wall is different in each case. The most intense heat transfer occurs when U_g=5U_mf and the poorest heat transfer occurs when U_g=3U_mf. As a result, the inference above manifests that there is the effective circulation of solid particles between the wall and the center tube that maintains the driving force supporting the heat transfer from the wall to DPS. Besides, the heat transfer between the wall and DPS is weakened for lower U_g. These results are consistent with previous inference.

Fig. 10 The distributions of time-average solid phase temperature (a) along X-axis and (b) Y-axis at Z=0.

3.4 The effects of tube diameter

The cost of investigating the influence of tube diameter through the experiment is too high. Besides, the tube diameter significantly impacts the DPS behaviour in the rising tube. In order to scrutinize the influences of tube diameter on the DPS dynamics and heat transfer process in the rising tube, four different tube diameters (D=24mm, D=36mm, D=48mm and D=60mm) is set with a constant boundary condition.

Fig. 11 shows time-averaged wall-to-bed HTC and outlet solid phase temperature as a function of tube diameters. The result indicates that the wall-to-bed HTC is increased with the tube diameter for a certain range. The total amount of heat power transmitted to the solid phase is increased due to an increasing solid mass flow rate, 39.05kg/h to 156.03kg/h, when the riser diameter increased. Therefore, the wall-to-bed HTC is increased with the tube diameter. However, the outlet solid phase temperature is negatively correlated with the tube diameter. The main reason is that the total solid heat capacity rate C_s=C_p,s∙is swiftly increased with the increase of solid mass flow rate and the increase rate of C_s is significantly higher than the increase rate of power transmitted to the solid phase. The instantaneous temperature profile of solid phase under different tube diameters is presented in Fig. 12. Moreover, these findings may provide references to control the DPS temperature by changing the solid mass flow rate.

Fig. 11 Time-averaged the wall-to-bed HTC and outlet solid phase temperature as the function of tube diameters.

It is evident that the speed of the solid phase temperature rising is faster in the small tube diameter. It is mainly due to that the total solid heat capacity is lower when the same temperature in small tube diameter increases. Fig. 13 presents the instantaneous solids holdup contours of various tube diameter at 65s (XZ-plane). For A-type particles, the dynamic balance of bubble merging and splitting causes the size of bubbles gradually increase when the bubble moves upward in the axial direction and then reaches the maximum size. The slug flow may occur when the size of the bubble equal to the tube diameter. Moreover, the emergence of slug-flow in the rising tube may cause pressure fluctuation between parallel tubes and then reduce the operation reliability of a CSP plant. In the case of small pipe diameter, there will be slug flow, and there is a tendency to form bubbles. The main reason is that the wall constraint effect is more powerful for small tube diameter and the possibility of bubble formation is increased. These results are helpful to find an optimum tube diameter where slug flow doesn’t occur in the UBFB receiver.

Fig. 12 Solid phase temperature for different tube diameter at 71s (XZ-plane).

Fig. 13 Instantaneous solid volume fraction contours of different tube diameter at 65s (XZ-plane).

3.5 The effects of the tube wall temperature

In an actual CSP plant, the tube wall temperature varies during operation and significantly impacts the DPS behaviour. Thus, the DPS behaviour with different tube wall temperature is studied in this section. Four cases with different tube wall temperatures (T_w=600, 800, 1000, 1200K) are introduced and the other boundary conditions are consistent.

Fig. 14 The time-averaged wall-to-bed HTC and outlet solid phase temperature as a function of T_w.

Fig. 15 The time-averaged logarithmic mean temperature difference and solid volume fraction in the tube as a function of the tube wall temperature.

Fig. 14 shows the time-averaged wall-to-bed HTC and outlet solid phase temperature as a function of T_w. The results show that wall-to-bed HTC reaches a maximum and then decreases with the increase of tube wall temperature. However, the outlet solid phase temperature presents a positive correlation with the tube wall temperature. The reason for the outlet solid phase temperature increase is obvious, but the decrease of HTC needs to be further analyzed. To find out the explanation for the results, the calculation method of the wall-to-bed HTC needs to be analyzed in detail. The wall-to-bed HTC is calculated from Eq (15) and depends on the, T_LMTD and A. The internal wall area is constant, so theand T_LMTD are the influence factors. The power transmitted to the solid phase is increased. However, the logarithmic mean temperature is progressively increased and the solid volume fraction is decreased due to gas expansion, as depicted in Fig. 15. The effects of T_LMTD outweigh the effects of. The solid particles have enough time to be heated up in lower and middle tube wall temperature. However, particles do not have enough time to take full advantage of the higher tube wall temperature. The results indicate that higher particle residence time or higher particle heating rate is required to take full advantage of the higher tube wall temperature.

Fig. 16 shows time-averaged quasi radiative HTC determined by and its fraction as a function of the tube wall temperature. The result shows that h_r is increased with tube wall temperature and radiation heat transfer is not a crucial factor for low and middle temperature. Radiation heat transfer should be taken into consideration when tube wall temperature is higher than 1000K.

Fig. 16 The time-averaged quasi radiative HTC and its fraction as a function of tube wall temperature.

4 Conclusions and prospects

In this study, a 3D numerical model is established to investigate the comprehensive DPS behaviour in different operating conditions, including the effects of superficial gas velocity, tube diameter and tube wall temperature on the DPS dynamics and heat transfer process. The feasibility and accuracy of the proposed 3D model have been confirmed against experimental results. Salient findings are summarized as follows:

(1) The modified Syamlal-O’Brien drag force model and defined effective conductivity for gas and solid phase is the key to simulate the DPS behaviour in the rising tube accurately. The former significantly influences the dynamics and the latter ensure simulation of the heat transfer process correctly.

(2) The superficial gas velocity affects the DPS behaviour through improving the stirring action of bubbles and the renewal frequency of fresh solid particles in the near-wall region. The wall-to-bed HTC reaches a maximum and then decreases. The results indicate that there is an optimum superficial gas velocity during operation. Furthermore, these findings give the opportunity to control the DPS temperature by changing the superficial gas velocity.

(3) The wall-to-bed HTC increases with the increase of tube diameter. However, the outlet solid phase temperature is negatively correlated with tube diameter. It is mainly due to the significant increase of the total solid heat capacity outweighs the increase of solid mass flow rate. Besides, tube wall diameter significantly affects the bubble size and the slug flow formation. An appropriate tube diameter for the designation of UBFB receiver should be selected.

(4) The parameter study presents that the increase of the tube wall temperature results in a decrease of the wall-to-bed HTC. A strong decrease of the logarithmic mean temperature is the main reason for the result. Moreover, the increased tube wall temperature leads to gas expansion and decrease of the solid fraction in the tube. Lastly, the influence of superficial gas velocity and tube diameter on radiant heat transfer is not significant. Radiation heat transfer presents a positive relationship with the tube wall temperature and DPS temperature. Radiation heat transfer should be taken into account when the tube wall temperature exceeds 1000K.

Although the paper research provided a 3D model based on TFM to investigate the DPS behaviour in the tube, there are still some aspects that need to be further studied. Future works can focus on the aspects as follows. (1) Considering the influence of the particle size distribution; (2) investigating the characterizations of parallel rising tubes in the actual UBFB receiver; (3) considering further reduce the computational cost by using 2.5D model, which attempts to impose flow symmetry in a cylindrical column by adopting the wedge-shape computational domain and it allows the flow to pass through the central axis by incorporating the 2D Cartesian flow assumption in the central region at the same time.

Acknowledgments

This work was financially supported by the National Natural Science Foundation of China (Grant No.: 51676069 and 51821004).

Supporting Information

Not applicable

Conflict of interest

There are no conflicts to declare

Nomenclature

Latin characters
Aint	Heat transfer area (m^2)
Ar	Archimedes number
CD	Drag coefficient
Cp	Specific heat capacity (J/kgK)
D	Tube diameter (m)
dp	Sauter mean particle diameter (μm)
e	Percentage relative error (%)
er	Restitution coefficient
G	Incident radiation
h	Heat transfer coefficient (W/m^2 K)
hs	Solid specific enthalpy (kJ/kg)
hg,s	Interphase HTC (W/m^2 K)
kg,s	The momentum exchange coefficient (W/mK)
kg,o	Conductivity of SiC (W/mK)
ks,o	Conductivity of Air (W/mK)
ks,eff	Effective conductivity of SiC (W/mK)
kg,eff	Effective conductivity of Air (W/mK)
m	Mass flow rate (kg/s)
Nu	Nusselt number
p	Pressure (Pa)
Pr	Prandtl number
q	Heat flux (W/m^2)
Q	Power (W)
Re	Reynolds number
S	Source term
T	Temperature (K)
t	Time (s)
U	Velocity (m/s)
z	Tube height (m)
Greek symbols
α	Volume fraction
β	Extinction coefficient
μ	Dynamic viscosity (kg/ms)
ρ	Density (kg/m^3)
▽	Nabla operator
ω	Scattering albedo
	Optical thickness
σs	Scattering coefficient (m^-1)
σ	Stefan-Botzmann constant (W/m^2K^4)
	Shaper factor of particles
	Granular temperature (m^2/s^2)
Subscripts
col	Collisional
eff	Effective
fr	Frictional
g	Gas-phase
i	Inlet
kin	Kinetic
m	Meddle
max	Maximum
o	Outlet
rad	Radiation
s	Solid-phase
w	Wall
Superscript
exp	Experiment
sim	Simulation
Abbreviations
CFD	Computational fluid dynamics
CSP	Concentrated solar power
DPS	Dense particle suspension
DEM	Discrete element method
HTF	Heat transfer fluid
KFGF	Kinetic theory for granular flow
TFM	Two-fluids model
UBFB	Up bubbling fluidized bed

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Author information

Kaijun Jiang, male, 1995–, PhD candidate. Research field: next-generation CSP technologies especially in solid particle solar receiver; gas-solid two phases flow; CFD.

Xiaoze Du, male, 1970–, PhD. Research field: enhancing heat transfer; CSP; energy storage materials and energy storage technology; hydrogen energy and fuel cells.

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