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Energy Procedía 49 (2014) 354 - 362

SolarPACES 2013

Thermal stresses analysis of a circular tube in a central receiver

O. Flores*a , C. Marugán-Cruz b, D. Santana b, M. García-Villalba a

aDepartment of Bioengineering and Aerospace, Universidad Carlos III de Madrid bDepartment of Fluids and Thermal Engineering, Universidad Carlos III de Madrid, Avda Universidad 30, Leganés, 28911. Madrid (Spain).

*Phone number: +0034 91624 8217, email: oflores@ing.uc3m.es

Abstract

The tubes of central receiver power plants are designed to work under very demanding conditions, with heat fluxes varying between 0.2 to 1 MW/m2, depending on both the weather conditions and the operating regime of the plant. Also, the heat flux is localized along the circumferential coordinate of the tube, on the outward facing side of the tube. This results in a non-uniform temperature distribution with strong gradients both in the heat transfer fluid and the pipe walls. Predicting this distribution is important since it can induce high thermal stresses on the pipe walls and it is a key factor for the fluid decomposition.

In this work we have studied different configurations to analyze the effects of tube diameter and wall thickness in the temperature distributions in the fluid and solid, for a typical concentrated solar heat flux distribution. HITEC salt and 316L stainless steel have been used as the heat transfer fluid and the material of the tubes in the receiver, respectively. A numerical method has been used to calculate the temperature distribution of the fluid and the convection coefficient along the circumference. Finally the Von-Mises thermal stresses of the tube receiver have been calculated.

© 2013 O. Flores. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/3.0/).

Selectionandpeerreviewbythescientificconference committeeofSolarPACES2013underresponsibilityofPSEAG. Final manuscript published as received without editorial corrections.

Keywords: External receiver; Temperature distribution; Thermal stresses

1. Introduction

External receivers are one of the key components in central solar power plants. Due to the high cost of these type of receivers, it is of paramount importance to assure a high efficiency and endurance of these receivers. Circular thin-walled metal tubes, assembled into panels, form the external tubular-type receivers. The panels include the tubes, the inlet header, the inlet and outlet nozzles, the tube clips, and other panel support structures. The heliostats located around the tower, concentrate the solar irradiation onto the tubes. The heat transfer fluid flows through the

1876-6102 © 2013 O. Flores. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http:// creativecommons. org/licenses/by-nc-nd/ 3.0/).

Selection and peer review by the scientific conference committee of SolarPACES 2013 under responsibility of PSE AG.

Final manuscript published as received without editorial corrections.

doi:10.1016/j.egypro.2014.03.038

tubes increasing its temperature by convection. Molten nitrate salts, such as HITEC, are often used as heat transfer fluid, because of their heat storage capacity and low conductivity.

Generally, the configuration of the external receivers is similar. To ensure symmetry the salt flow is divided into two paths; in each circuit the salt passes through half of the panels. The salt enters the receiver at a low temperature (slightly above the melting point), through the northern panel, which receives the highest radiation flux. As the salt moves through the rest of the panels, the temperature of the salt increases and the radiation flux on the panels decreases. The operation of the plant must ensure that the temperature of the salt never reaches the decomposition temperature nor the melting temperature by increasing or reducing the mass flow rate.

Since the tubes are irradiated only on their outward facing side, they are subject to highly non-uniform heat fluxes, and therefore they have to bear high temperature gradients in their walls. This could result in large thermal stresses, so it becomes essential to determine the optimum configuration of these tubes and panels [1].

Greater understanding of the temperature distribution is also essential to predict the degree of decomposition of the salts and the maximum film temperature. In numerous studies the temperature of the salt is predicted using the correlations from the literature: for instance, in [2] Kolb predicts the film temperature using the Gnielinski correlation and Jianfeng et al. [3] use the Dittus and Boetter correlation to estimate the heat absorbed by the heat transfer fluid. However, since these correlations have been found imposing a uniform heat flux along the wall, they are not appropriate for the determination of the temperature distribution under non-homogeneous heating (see Yang et al. [4] among others).

In this study we have focused on determining the time-averaged temperature profile of the salt, the time-averaged temperature distribution on the walls of the tube, and the thermal stresses in different configurations. The effects of the velocity of the salt inside the tubes, the tube diameter and thickness have been studied. On the other hand, the number of panels and the number of tubes per panel has been varied too. The results are obtained using a numerical model that assumes constant properties for the salt, and a simple eddy diffusivity model for the turbulent heat fluxes.

2. Cases studied

The object of this study is to investigate the heat transfer and absorption characteristics of an external receiver tube. Two different tube diameters have been analyzed, both with the same wall thickness. Increasing the external diameter of the tubes implies the allocation of fewer tubes in the external receiver. Reducing the number of tubes can be done by keeping the same number of panels and reducing the number of tubes per panel, or by reducing the number of panels. In other words, assuming a constant number of panels and assuming that the mean temperature of the salt at the inlet and outlet of the receiver remains constant; then selecting larger tubes (tubes with larger diameter) implies that the number of tubes per panel has to be smaller.

We have analyzed the operating conditions of the tubes of three configurations, listed in table 1 and sketched in Fig. 1 a). Configurations A and B correspond to configurations with the same number of panels (Np), while C has twice that number (2Np). Configurations B and C have tubes with twice the diameter of the case A. Therefore, configuration A with Np panels has Nt tubes per panel. Configuration B has also Np panels and 2Nt tubes per panel. Since configuration C has 2Np panels, they must be half the size of the panels in A and B, and therefore each panel in configuration C can only have Nt/4 tubes.

In order to establish a meaningful comparison, we impose that the increase in averaged temperature per area of receiver is the same in all three cases. That results in an increase in average temperature per unit length of tube in configurations A and B that is twice the increase in average temperature per unit length in a tube in configuration C. Since the diameter in A is smaller than in B, the velocity in A is twice as large as in B to keep the same overall mass flux on each panel. For configuration C, the velocity has to be twice as in configuration B, since the overall length of the circuit is twice as long in C than in B.

Table 1. Parameters of study.

Parameters

Configuration A

Configuration B Configuration C

Inlet panel

Number of panels Number of tubes per panel Outer diameter, Do (mm) Tube thickness, e (mm) Material of the pipes Mean temperature, (°C) Maximum Heat Flux, (MW/m2) Heat transfer fluid Velocity of the salt, Vi (m/s) Reynolds number, Re = Prandtl number

Np Nt 21.3 1.65

Stainless Steel 316L 315 0.704 HITEC 1.76 ; 3.52 2 104 ; 4 104 8

Np Nt/2 42.2 1.65 316L 315 0.704 HITEC 0.81 2 104 8

2Np Nt/4 42.2 1.65 316L 315 0.704 HITEC 1.63 4 104 8

Outlet panel Number of panels

Number of tubes per panel Outer diameter, Do (mm) Tube thickness, e (mm) Material of the pipes Inlet temperature, (°C) Maximum Heat Flux, (MW/m2) Heat transfer fluid Velocity of the salt, Vi (m/s) Reynolds number, Re = -Prandtl number

Np Nt 21.3 1.65

Stainless Steel 316L

1.76 ; 3.52

2 104 ; 4 104

Table 2. Properties of heat transfer fluid and the tube walls.

Properties

Values

Hitec Salt

Density, p ( kg/m3) Specific heat, Cp (J/kg K) Thermal conductivity, k (W/m K) Viscosity, n (kg/ms) Stainless Steel 316L Density, p ( kg/m3)

Specific heat, Cp (J/kg K) Thermal conductivity, k (W/m K) Modulus of elasticity, E (GPa) Thermal expansion coefficient, a (°C-1) 140 10 Poisson ratio, V 0.305

1855 (315°C)-1740 (473°C)

2.9 10-3 (315°C) - 1.3 10-3 (473°C)

Maximum Heat Flux, (MW/m2)

Rodríguez-Sánchez et al. [5] showed that the tubes that were subject to higher stresses were those in the inlet panel and in the western panel (fourth panel in their study). Hence, we have analyzed cases A, B and C for the mean temperature and irradiation of these two locations, named inlet panel and outlet panel in Table 1. We have also taken

into account that the properties of the fluid depend on the temperature, and therefore on the position of the tube. Table 2 shows the properties of the HITEC salt (7% NaNO3, 53% KNO3, 40% NaNOi), together with the properties of the material of the tubes (Stainless Steel 316L).

Fig. 1. (a) Sketch of the three configurations of pipes and panels. The sketch shows one panel with Nt tubes for configuration A, one panel with Nt/2 tubes for B, and two panels with Nt/4 tubes for C. (b) Heat flux absorbed by the tubes. The solid line represents the heat flux on the inlet panel, while the dashed liner represents the heat flux on a tube of the outlet panel.

As was previously noted, the heat flux irradiating the tubes is not uniform. The external side of the tubes receives an incident heat flux that is often modeled with a cosine function [2], while the rear side can be considered as adiabatic. The heat flux absorbed by the tubes, shown in Fig. 1b, has been calculated using a model proposed by Rodríguez-Sánchez et al. et al [5], that uses a radiation map from the heliostats and considers all the surfaces involved in the problem as gray surfaces.

3. Numerical method

3.1. Formulation

We consider the heat transfer problem of a fluid flowing inside a circular pipe that is subject to homogeneous heating along the axial/streamwise direction (x) and non-homogeneous heating in the circumferential direction (d ). In this section we briefly describe the mathematical formulation of the problem, which is similar to that used in [6,7], except that we consider the conjugate heat transfer problem to analyze the flux and temperature distributions in both on the fluid and the solid (circular pipe). Hence, a non-uniform heat flux q(0) is applied at the external solid surface r =Do/ 2 = D¡/ 2 +e, while the solid-fluid interface is at r =D¡/ 2, where D¡ =Do - 2e is the inner diameter of the pipes.

For simplicity, we assume that the fluid has constant properties and that the flow is fully developed, both in terms of velocities and temperature. Under these conditions, the hydrodynamic problem is decoupled from the thermal problem, so the mean velocity U(r) and the Reynolds stresses of the flow can be determined independently from the heat flux applied to the pipe. Note that the hydrodynamic problem is then homogeneous in the axial and circumferential directions, so that both mean velocity and Reynolds stresses are functions of the radial coordinate only. For convenience, we take for U(r) and for the Reynolds stresses the analytic expressions given by Cess [8], although similar results are obtained when profiles obtained from detailed Direct Numerical Simulations (DNS) are used instead.

For a fully developed thermal problem, it is possible to show that the temperature of the fluid T(x,r, 6,t) can be decomposed in

T( x, r, 0, t) = Tb ( x)+0(r, 0)+0 '( x, r, 0, t ) = T + xdjb + 0(r, 0)+0 '( x r, 0, t),

where the bulk temperature Tb(x) is the time and cross-plane (r-6) averaged temperature, &(r,ff) is the time averaged temperature fluctuation in the cross-plane, and ®'(x,r,0,t) is the remaining fluctuation. For the case considered here, the bulk temperature increases linearly with x, and the bulk temperature gradient dxTb is univocally determined by the total heat flux received through the fluid-solid interface,

pCpübdxTbnDi/2= /kdr0(Di /2,0)d0,

where Ub =A -1 JU(r)dA i s the bulk velocity, equal to the time and cross-plane averaged velocity. The fluid properties appearing in equation (2) are defined in Table 2.

If we model the turbulent heat flux using the concept of turbulent eddy conductivity, we can write the Reynolds-Averaged energy conservation equation for the cross-plane temperature fluctuations

d0 1 1 d (1

pCpü(r)dxTb = V-kTV® = ~I rkT(r)^ I+ 1 -kT(r)^-1,

dr J r d0\r

where kT is the turbulent eddy conductivity. In principle, kT is a function of r and d. However, it has been shown that the effect of the circumferential variation of kT is small compared to its radial variation [9,10]. Hence, in the present paper we assume kT =kT (r). Moreover, we assume that the turbulent Prandtl number PrT = pCpvT/kT is equal to 1. Schwertfirm et al [11] show that this approximation is appropriate in the range Pr=0.1- 10. Since in the present case the highest Prandtl number is Pr=8 (see table 1), we conclude that the approximation PrT=1 is reasonable.

With these approximations, the turbulent viscosity model of Cess [8] leads to the following model for the turbulent eddy conductivity:

= Pr V = PL

^(1 -n2)2(1 + 2n2)2

1 - exp

(n- 1)ReT . A

where n = 2r/Dt . The tunable parameters of the model are k=0.42 and A=27, whose values are taken from [8]. The friction Reynolds number Rej- appearing in (4) is a known function of the Reynolds number (Re).

The temperature field in the solid Ts\x,r,0,t) can also be decomposed in bulk (Tb(s> = Tb), cross-averaged temperature (0(s)) and fluctuation (0'(s)). Again, the energy conservation equation yields a linear variation of the bulk temperature with the axial coordinate, and the energy conservation in the cross plane yields the following equation for the cross-averaged temperature fluctuations,

0 = V-ksV0(s) =1— r dr

( s) ^ 1 d r +--

1 . d0(s)

r S d0

In order to solve the conjugate heat transfer problem, we solve equation (3) for 0(r,6) in 0<r <Di/ 2 and equation (5) for ©(s)(r,6) in Di/ 2 <r < Do/ 2. At the outer boundary of the solid pipe, the non-homogeneous heat flux conditions yields

ksdr&s)(D0/2,6) = q(6). (6)

At the solid-fluid interface, the following coupling conditions apply

ksdr&(S)(Di/2,6) = kT(Di /2)dr0(D,/2,6) and 0(s)(D,/2,6) = 0(D,/2,6). (7)

3.2. Numerical method

The coupled system of equations (3-7) is solved using a spectral method. We use a collocation method in which a Fourier expansion is used in the circumferential direction, and a Chebychev expansion in the radial direction. The method is implemented in MATLAB, using the formulation described by Trefethen [12]. The code is validated using the date of a DNS of forced convection in a pipe [13], with uniform heating at Re=5300 and Pr=0.7. Using 20 modes in the circumferential direction and 51 modes in the radial direction, the error between the DNS data and the present method in the mean temperature distribution is smaller than 1%. We have also computed the Nusselt number (Nu) over a broad ranges of Re and Pr. Despite the strong assumptions in our method (constant fluid properties, simple model for the turbulent eddy diffusivity), the values obtained for Nu are in good agreement with the usual correlations found in the literature: Gnielinski, Ditus-Boetter and Sieder-Tate. This result gives us confidence in the method. In addition, some of the cases computed by Reynolds [6] for non homogeneous heating have been recomputed using the present method, obtaining good agreement.

For the present calculations we have used 60 collocation points in the circumferential direction, and 120 collocation points in the radial direction. Further refinement of the grid did not result in any appreciable variation of the results. Typically, each case runs in less than 15min in a workstation with 12GB of RAM, which shows the potential of the numerical implementation used here for industrial applications.

4. Results and discussion

4.1. Temperature distribution

Fig. 3 shows the temperature distribution of the molten salt and the tube wall at the inlet panel for two different Reynolds numbers for the Configuration A. The wall temperature distribution is uneven due to the non-uniform incident heat flux, in both cases. These figures show in both cases that the maximum wall temperature is located on the outside wall, where the incident heat flux is maximum. The temperature difference between the front and rear sides is very important, as well as the temperature difference between the inner and the outer wall of the pipe. The effect of the Reynolds number is clearly visible comparing both figures: the heat transfer is enhanced by the velocity, and thus, the maximum value of the wall temperature decreases as the Reynolds number increases. Furthermore, the salt temperature distribution is much more uniform except for the region close to that location where the temperature increases very rapidly.

Fig. 3. Temperature distribution at the inlet panel for configuration A (a) Re = 2 104 and (b) Re = 4 104.

Fig. 4 a) and b) reflect the effect of the non-uniform heat flux distribution on the film temperature. The temperature differences between the front and rear side are smaller for the tubes with smaller diameter (Configuration A), than for larger diameter tubes (especially for Configurations B). In this sense it can be noticed that the results predicted by the numerical model show that the maximum value of the film temperature at the outer panel for Configuration B is close to 1000°C, and therefore could result in salt decomposition. It is also worth noticing the effect of the Reynolds number on the temperature distribution: as it was expected, the temperature profile is more homogenous in the cases with larger Reynolds numbers.

Fig. 4. Film temperature profile at the inlet panel (4.a) and at the outlet panel (4.b).

4.2. Thermal stresses

The incident solar flux on the receiver produces temperature gradients in the tube wall large enough to develop significant thermal stresses in the axial, circumferential and radial directions. The radial, axial and circumferential stresses are calculated using the expressions proposed by Faupel et al. [14] (p. 919). The effective stresses, according to Von-Mises theory, can be expressed as follows:

(°eff ) + °2r - (°x°r + + ) (8)

Fig. 5 presents the Von Mises Thermal stresses for Configuration A and Reynolds number, Re =20000 in the inner and outer panel. It can be seen that the maximun stress coincides with the maximum heat flux. The values of the stress at the rear side of the tube is almost negligible. The effective stress is maximun at the inner wall, it decreases with the thickness and it increases again. The Von Mises stress at the outer wall is slightly smaller than at the inner wall, and since the ultimate tensile stress decreases with temperature, it is the outer wall that has more chances of cracking. This result is consistent with the theory of thin cylindrical shells with linear temperature distribution [14] (p.921).

Fig. 5. Effective stresses distribution in configuration A 5.a) inlet panel 5.b) outlet panel.

Finally, fig.6 shows the distribution of the effective stresses at the outer radius in the inlet and the outlet panel for the different configurations analyzed in this paper. It can be noticed that the values of the Von-Mises stresses are very similar in all the cases. This can be explained because qualitatively the temperature distribution is similar in all the cases, and the ratio e/Do is always small. It is also remarkable that maximum stress appears at the inlet panel. Nevertheless, it is important to highlight that, although the values of the effective stress are slightly larger in the inlet panel than in the outlet panel, since the ultime tensile strength decreases with temperature, the pipe wall at the outlet panel are subject to higher stress to yield strength ratios.

Fig. 6. Effective stresses at the outer radius a) inlet panel b) outlet panel.

5. Conclusions

We have studied the conjugate heat transfer problem in several configurations of central receivers. In particular for three possible configurations we have studied the temperature distribution, both in the heat-transfer fluid and in the solid annular cylinder, as well as the thermal stresses in the later. A simple mathematical model has been used, in which the fluid velocity problem is decoupled from the heat transfer problem. In the model, constant fluid properties and a simple eddy diffusivity model have been employed.

The effect of the different configurations, diameter and velocity on the effective stress is small. From a thermal point of view, this study reveals that configuration A (that has the smallest diameter of the tubes) is the best of the three configurations considered here, since it has the lowest maximum temperatures (in the solid and the film temperature).

Acknowledgments

The authors would like to thank the financial support from Ministry of Economy and Competitivenss (Project ENE2012-34255). The support from CDTI and S2m Solutions for the project MOSARELA (IDI-20120128) is also acknowledged.

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