## 1 Introduction

Parabolic troughs are one of main concentrating solar power (CSP) technologies in the current market [13], as illustrated in Fig. 1(a). Total parabolic trough-installed capacity has reached about 6.3 GWe and dominates the global CSP market as of the end of 2018 [4]. Thermal energy storage (TES) can be coupled naturally with parabolic trough and other CSP technologies, enabling valuable dispatchability of CSP in the future grid with high penetration of non-flexible renewable energy [5]. The state-of-the-art TES adopts molten salt as the working media because of its low cost and high heat capacity. Many currently operating parabolic trough plants use an organic heat-transfer fluid (HTF) with maximum operating temperatures around 400 °C to transfer the heat from the solar field to the salt storage tanks [68]. This indirect approach for TES requires heat exchangers to move heat between the storage tanks and organic HTF, increasing the cost per kilowatt hour stored [7,8].

Fig. 1
Fig. 1
Close modal

A direct approach currently being considered entails using the molten-salt storage media as the HTF in the solar field, significantly reducing the cost of TES [9,10]. This direct approach will also boost the solar field–operating temperature to increase the power-cycle efficiency and – combined with the cost reduction for TES – reduce the levelized cost of energy of the plant. Salt mixtures operate within a limited temperature range due to high freezing temperatures (90–220 °C) and relatively low stability limits (≤560 °C) [11,12]. The freezing temperatures being higher than ambient mean that there is a possibility for the salt to freeze in the solar field during its operational lifetime (25–30 years), and the plant must be designed to recover from such an occurrence [13,14]. A traditional approach for thawing the salt frozen in the collector loop is heat tracing in the header piping and using impedance heating in the receiver tubes [13,15]. Although this is a technically effective method, such a freeze protection system is estimated to cost 80 $m−2, adding up to 47% to the installed solar field cost for a large CSP plant [16,17]. A new approach using controllable solar flux heating is proposed to thaw salts for freeze recovery in direct molten-salt parabolic trough receivers. This method tailors the tracking of parabolic trough collectors to control the level of solar flux on the receiver with frozen salt during the day, so that the frozen salt can be melted without harming the receiver tube due to induced thermal stress. The melting and solidification of multi-component salt mixtures is a complex phenomenon with many applications in the energy industry [11,18,19]. Common salt mixtures used for TES include solar salt and HITEC [11,12]. Neither of these nitrate salts is a eutectic composition, meaning the melting process occurs over a temperature range as opposed to a single melting temperature. The thermal properties and phase-change thermodynamics have a significant effect on heat transfer, and thus, on thermal stresses induced in the receiver tube [20,21]. However, the large-density variations and phase change requires a computationally expensive model with three phases (frozen salt, molten salt, and void space) to capture all pertinent physics [2224]. Initial modeling efforts rely on several simplifications to reduce the complexity of the thermal-fluid model, allowing the simulation of several different heating conditions for comparison. This work investigates the melting of frozen solar salt in a receiver tube using controllable solar flux heating with comparison to the impedance heating method, exploring the potential technical and economic benefits of the proposed approach. Section 2 provides details of the geometry, governing equations, and boundary conditions for the computational fluid dynamics (CFD) model. Section 3 describes the methodology used to calculate thermal stresses in the receiver tube. Grid independence of the solution is demonstrated in Sec. 4. In Sec. 5, the computational model is verified by comparing with an analytical solution for the melting of a pure substance. Results of this modeling endeavor – including timescales for melting and allowable thermal stresses – are presented in Sec. 6. Finally, the concluding discussion can be found in Sec. 7. ## 2 Model Description ansys fluent 18.2 is adopted to perform a two-dimensional (2D) model on a typical cross section of the receiver tube in a parabolic trough collector, as illustrated in Fig. 1(a). The 2D model solves equations for the conservation of momentum and energy considering laminar flow and melting of the two-phase salt mixture in the fluid domain. In the wall domain, the energy equation is also solved in 2D and coupled to the fluid domain. At the outer wall surface, the boundary heat flux can vary as a function of space and time. Due to the complexity of the problem being investigated, several assumptions help to make this modeling effort affordable: the salt is assumed to have constant density and to completely fill the receiver tube, thus neglecting density changes and subsequent formation of void spaces; species transport is neglected in the salt, and average properties are input to the model for the solar salt binary mixture, as opposed to a mixing law. The computational resources required to solve the problem are significantly reduced by considering the conduction limited heat transfer problem with a constant density salt. Furthermore, the constant density assumption excludes natural convection heat transfer resulting in conservative melt time estimates. This model is used to investigate several different flux profiles and gain understanding of trends in melt time and thermal stresses in the receiver tube. Once a flux profile is down-selected, future modeling studies will investigate the effects of species transport and variations in density on the thawing and freezing process. ### 2.1 Geometry. An example of a parabolic trough collector and the 2D receiver geometry modeled in ansys fluent is illustrated in Fig. 1. The receiver tube is stainless steel 321, and the flow area is filled with Solar Salt. In a collector loop, the receiver tube will be enclosed in an evacuated glass casing; this casing is not modeled because the temperature profile of the glass is not a concern, and heat losses can be accurately quantified with an empirical expression for a similar receiver [25]. ### 2.2 Governing Equations. The computational domain includes both the salt mixture and the receiver wall. In the fluid domain, a single-phase model includes both frozen and molten-salt states. The governing equations described in this section are adapted from the ANSYS Fluent Theory Guide [26]. An enthalpy-porosity technique is used for the phase-change region, where the solid/liquid interface is not tracked explicitly. Instead, the fraction of molten salt in each cell is tracked as an independent variable, i.e., the liquid fraction β. For each cell, β is calculated based on the cell temperature and salt phase diagram as shown in Eq. (1). $β=0ifT≤Tsolidusβ=1ifT≥Tliquidusβ=T−TsolidusTliquidus−TsolidusifTsolidus (1) Cells where 0 < β < 1 are referred to as the mushy zone; these cells are modeled as a porous medium where the porosity is related to the liquid fraction. For fully solid regions, the porosity and thus the velocity are zero. The single-phase formulation means that a single mass conservation expression is required, as written in Eq. (2). For all equations in the model description, density is considered an independent variable because eventually, the goal is relaxing the assumption of constant density $∂ρ∂t+∇⋅(ρν)=0.$ (2) Similarly, a single energy equation is written for the salt mixture in Eq. (3), including temperature-dependent thermal conductivity, γ, and enthalpy, H. To account for phase transition energetics, a latent heat term is included in H that is proportional to β and the heat of fusion Δhf. The full expression for enthalpy is shown in Eq. (4) $∂∂t(ρH)+∇⋅(ρνH)=∇⋅(γ∇T)$ (3) $H=href+∫TrefTcpdT+βΔhf$ (4) For conservation of momentum, shown in Eq. (5), the enthalpy-porosity model calls for an additional source term to account for the momentum sink in regions with where some solid exists. $∂∂t(ρν)+∇⋅(ρνν)=−∇p+∇⋅[μ(∇ν+∇νT−23∇⋅νI)]+ρg+(1−β)2(β3+ε)Amushν$ (5) In the last term in Eq. (5), Amush is the mushy zone set constant to 105 for all simulations discussed, and ɛ is a small number (0.001) to prevent dividing by zero. The receiver tube wall is also meshed and modeled using ansys fluent. Due to the relatively slow heat, conduction dominated, transfer between the inner tube wall and enclosed salt, as well as the nonuniform boundary heat fluxes considered, both radial and circumferential conduction within the tube wall are important. In the receiver wall domain, the energy equation reduces to Eq. (6) $ρwcp,w∂Tw∂t=∇Δ(γw∇Tw)$ (6) where the subscript w indicates a wall property. Accurately capturing the wall temperature profile is critical for thermal stress calculations. These equations are solved using ansys fluent 18.2 with the SIMPLE pressure–velocity coupling. The spatial discretization schemes are as follows: least squares cell based for gradients, second order for pressure, and second-order upwind for momentum and energy. Stringent convergence criteria are required in solidification and melting problems, i.e., 10−5 for continuity and velocity residuals, and 10−8 for the energy equation [23]. ### 2.3 Boundary Conditions. The boundary heat flux is a combination of concentrated solar heat, heat losses to the environment, and impedance heat depending on the simulation, as shown in Eq. (7). Solar heat concentrated using the parabolic mirrors varies spatially and temporally, eliminating any planes of symmetry. Although impedance heat is generally constant around the receiver tube, simulations with only impedance heating are solved using the same domain and mesh. $qin=qsolar+qimp−qloss$ (7) $qloss(Wm−1)=2Ro0.07(0.141T∘C+6.48×10−9T∘C4)$ (8) The heat- loss equation used in this work is an empirical correlation determined from indoor testing of a similar receiver of smaller diameter [25]. Equation (8) is scaled by the receiver diameter (in m) to account for the different geometry. The expression has a fourth-order term attributed to radiation as well as a linear term attributed to convection. The cell temperature should be in Celsius for determining heat losses using Eq. (8), as indicated by the °C subscript. The equation for qloss is also plotted in Fig. 2; at higher temperatures, the plot is dominated by radiation effects assuming vacuum conditions between the receiver tube and the glass casing [25,27]. This parameter has significant effects on simulation results for both heating and cooling of the receiver and should be chosen carefully based on the receiver geometry and ambient conditions. Using the expression for qloss, it is trivial to calculate the minimum heat required to melt salt within the receiver, qmelt = qloss(Tsolidus). For the selected Tsolidus and heat-loss equation, qmelt = 74.8 W m−1, as seen in Fig. 2. Fig. 2 Fig. 2 Close modal Inside the receiver tube, a no-slip condition is imposed at the inner wall boundary. Heat transfer at this inner wall boundary is inherently calculated by Fluent based on the properties and fluid dynamics in the salt domain. ### 2.4 Material Properties. The properties input to Fluent for the Solar Salt are described in Table 1. Species diffusion is neglected in this model, so properties are input as if the mixture were a single component. Significant literature exists on measuring properties of molten solar salt [12,2830], whereas there are much less data available for the frozen state [31]. Most properties are assumed constant in the frozen state except for viscosity, μ, and heat conductivity, γ. Properties only documented for the pure components were averaged on a mass basis, including the enthalpy of fusion Δhf and ρ. Since ρ is assumed constant, a single value must be selected; in this study, the solid-phase value for ρ is chosen to study the case with the most thermal mass. This approach should yield conservative estimates for the time required to melt salt within the receiver tube tmelt. Table 1 Properties of solar salt mixture input to ansys fluent model PropertyValueSource Tsolidus (K)498[28] Tliquidus (K)520[28] Δhf (J kg−1)146,537[29] ρ (kg m−3)2232[30] cp (J kg−1 K−1)1553[12,30,37] μ (kg m−1 s−1)7.551(10−2) − 2.775(10−4) T + 3.488(10−7)T2 − 1.474(10−10)T3[12,30] γ (W m−1 K−1)0.56415 − 1.55527(10−4)T[30] PropertyValueSource Tsolidus (K)498[28] Tliquidus (K)520[28] Δhf (J kg−1)146,537[29] ρ (kg m−3)2232[30] cp (J kg−1 K−1)1553[12,30,37] μ (kg m−1 s−1)7.551(10−2) − 2.775(10−4) T + 3.488(10−7)T2 − 1.474(10−10)T3[12,30] γ (W m−1 K−1)0.56415 − 1.55527(10−4)T[30] The receiver wall is made of stainless steel 321 (UNS S32100); properties of the wall material are detailed in Table 2. Properties required for the thermal stress analysis, described in the following section, are also included in Table 2. Table 2 Properties of stainless steel 321 (UNS S32100) used to model the receiver wall PropertyValueSource ρ (kg m−3)7,900[38] cp (J kg−1 K−1)500[38] γ (W m−1 K−1)10.41175 + 0.01525 · T[38] E (GPa)193[38] α (K−1)17.0 · 10−6[38] ν$0.27$[38] PropertyValueSource ρ (kg m−3)7,900[38] cp (J kg−1 K−1)500[38] γ (W m−1 K−1)10.41175 + 0.01525 · T[38] E (GPa)193[38] α (K−1)17.0 · 10−6[38] ν$0.27$[38] ## 3 Thermal Stress Calculation Approach Once the temperature profile in the receiver wall is known from solving the CFD model described in Sec. 2, thermal stresses are approximated using a biharmonic approach for non-axisymmetrically heated tubes [21]. First, the average surface temperatures for the inner and outer wall boundaries, $T¯i$ and $T¯o$, are calculated as follows: $T¯i=12π∫02πTdθforr=Ri$ (9) $T¯o=12π∫02πTdθforr=Ro$ (10) These temperatures are used in the definition of the circumferentially varying temperature fluctuation expression, $Tθ$ referenced to the average outer surface temperature $Tθ=T−(T¯i−T¯o)lnRo/rlnRo/Ri−T¯o$ (11) $Tθ=∑n=1∞(Anrn+Bnr−1)cosnθ+(Cnrn+Dnr−1)sinnθ$ (12) Referring to Eq. (6), a harmonic Fourier series exists with radial-dependent functions that can satisfy the energy equation in the wall domain; $Tθ$ is represented by such an equation in Eq. (12). A least-squares fitting algorithm is used to fit values of $Tθ$ calculated with Eq. (11) to the expression in Eq. (12), yielding the coefficients An, Bn, Cn, and Dn. The fitted parameters B1 and D1 are important for thermal stress calculations. Expressions for the radial and circumferential components of stress in a non-axisymmetrically heated cylinder are written in Eqs. (13) and (14), respectively $σr=KαE2(1−ν)[−lnRor−Ri2Ro2−Ri2(1−Ro2r2)lnRoRi]+KθαE2(1−ν)(1−Ri2r2)(1−Ro2r2)$ (13) $σθ=KαE2(1−ν)[1−lnRor−Ri2Ro2−Ri2(1+Ro2r2)lnRoRi]+KθαE2(1−ν)(3−Ri2+Ro2r2−Ri2Ro2r4)$ (14) where E is Young’s modulus for the receiver wall, α is the linear coefficient of thermal expansion, and υ is Poisson’s ratio. The coefficients K in front of each term are defined in Eqs. (15) and (16) and represent contributions from n = 0 and 1, respectively: $K=T¯i−T¯olnRo/Ri$ (15) $Kθ=rRi2+Ro2(B1sinθ+D1cosθ)$ (16) The shear component of thermal stress is calculated with Eqs. (17) and (18) $σrθ=KταE2(1−ν)(1−Ri2r2)(1−Ro2r2)$ (17) $Kτ=rRi2+Ro2(B1sinθ−D1cosθ)$ (18) To estimate the axial component of shear stress from this 2D geometry, an assumption regarding the constraints of the receiver tube must be made. A commonly used reference case is the plane strain assumption, where the ends of the receiver tube are constrained from moving in the axial direction; a more appropriate assumption is that of net zero axial force in the receiver tube, referred to as a generalized plane strain state. Manipulating expressions of Hooke’s law and assuming external mechanical loads are zero, Logie et al. (2018) derived an expression for axial stress, σx, assuming generalized plane strain, as shown in Eq. (19) $σx=KαE2(1−ν)[1−2lnRor−2Ri2Ro2−Ri2lnRoRi]+KθαEυ(1−ν)(2−Ri2+Ro2r2)−αETθ$ (19) Once all the components of stress are calculated, the effective von Mises stress in the receiver tube is determined using Eq. (22). The von Mises stress is considered an appropriate value to compare with maximum allowable stresses in the receiver tube to understand viable operating conditions. $σVM=12[(σr−σθ)2+(σθ−σz)2+(σz−σr)2]+3σrθ$ (20) In addition to thermal effects, internal pressure can also contribute to stresses in the receiver tube. During normal operation, the glass casing enclosing the receiver tube is under vacuum to limit convection cooling effects; with this setup, all flow pressure within the tube contributes to the overall stress state [21]. For a generalized plane strain state, the stress due to internal pressure can be calculated as follows: $σr=pRi2Ro2−Ri2(1−Ro2r2)$ (21) $σθ=pRi2Ro2−Ri2(1+Ro2r2)$ (22) $σx=pRi2Ro2−Ri2$ (23) where p indicates the internal pressure relative to the outside of the tube. During normal operation, p will be very similar to the absolute pressure and could be significant depending on the pressure drop through the collector loop. ## 4 Grid Independence The receiver tube and enclosed fluid are discretized using a conformal mesh of triangular and quadrilateral elements. The two parameters considered for mesh independence are the time required to completely melt the salt, tmelt, and the maximum von Mises stress recorded in the receiver wall during the simulation, σVM,max. Both parameters are important to ensure that the mesh is refined in both the wall and fluid domains. The mesh independence simulation used pure solar heating at the outer boundary (qimp = 0), and results for various mesh sizes are plotted in Fig. 3. In Fig. 3(a), tmelt is plotted as a function of mesh size in the fluid domain. This parameter remains relatively constant for different grid sizes, suggesting that the coarsest mesh tested (1 mm) could be appropriate in the fluid domain. In Fig. 3(b), σVM,max is plotted as a function of mesh size in the wall domain. The results indicate that a much finer mesh is required for grid-independent results in the wall compared with the fluid domain. Fig. 3 Fig. 3 Close modal The grid independence study plotted in Fig. 3 suggests that the wall domain requires a significantly smaller element; however, meshing the entire domain with a small enough grid size for accurate thermal stress calculations significantly increases the computational expense of the model. To achieve grid-independent results for both tmelt and σVM,max, the wall and fluid domains are meshed with different element sizes. The mesh remains conformal such that the nodes within the fluid domain match the wall domain nodes at the inner wall interface and grow toward the center of the receiver tube. The fluid domain mesh size was not reduced past 0.5 mm; in Fig. 3(b), using a finer mesh required for grid-independent results, the fluid domain mesh size remains fixed at 0.5 mm. The selected mesh has elements of 0.125 mm in the wall domain and 0.5 mm in the fluid domain for a total of 74,722 nodes. ## 5 Model Verification The model is verified by comparing with the analytical solution of the two-phase Stefan problem solved on a semi-infinite domain in 1D, also known as the Neumann solution [32]. The semi-infinite domain is approximated in Fluent as a long rectangle, and the appropriate independent variables are sampled along the centerline for comparison. The Neumann solution yields the location of the solid/liquid interface and the temperature profile in both solid and liquid regions for a pure substance that undergoes a phase change. The fluid is initially frozen at a given temperature below the melting temperature Tmelt. A constant temperature boundary condition is imposed at x = 0; this provides heat to melt the initially frozen fluid. For this case, the domain is initialized at 561 K and the wall is set to 600 K; the fluid considered is pure NaNO3 with Tmelt = 581 K. Results for the benchmark case are shown in Fig. 4. The numerical simulation does an excellent job of capturing both the temperature profile plotted as in Fig. 4(a) and the location of the solid/liquid interface as shown in Fig. 4(b). In Fig. 4(a), the transition between liquid and solid phases is clearly visible where each temperature profile intersects the dotted line indicating Tmelt. This simple benchmark case serves as verification of the solidification and melting module adopted for this work. Fig. 4 Fig. 4 Close modal ## 6 Results and Discussion Using the CFD model described in Sec. 2 and thermal stress model in Sec. 3, several solar heat flux profiles are used to simulate the melting of solar salt within the receiver tube. Results for controlled solar flux heating are compared with those of currently used impedance heating systems, and the possibility of hybrid systems is explored. ### 6.1 Cost Analysis. Before looking at model results, the economics motivating the work is analyzed. The freeze protection system for a molten-salt parabolic trough plant involves feeder and header piping as well as the solar field. Insulated and stationary sections of piping will be heat traced, while impedance heating is expected for components of the solar field including collectors and interconnects. The estimated capital cost for freeze protection, including just the impedance heating system, is compared with the installed solar field cost in Table 3. The freeze protection system can add up to 47% to the installed solar field cost for a 100 MWe CSP plant [17]. The impedance heating system accounts for 67.5% of the cost for freeze protection. This work is focused on reducing the cost of the impedance heating system, taking advantage of the negligible capital cost required for controllable solar flux heating of the collectors. Reducing the cost of the impedance heating system by 50% would decrease total installed cost for the solar field and freeze the protection system from 220 to 193$ m−2.

Table 3

Estimated costs for relevant components of molten salt parabolic trough plants

ComponentCost ($m−2)Source Solar field170[17] Freeze protection system80[16] Impedance heating system55[16] ComponentCost ($ m−2)Source
Solar field170[17]
Freeze protection system80[16]
Impedance heating system55[16]

Note: The impedance heating system cost is also included as part of the freeze protection system cost.

### 6.2 Impedance Heating.

Salt frozen in the receiver tube can be melted via impedance heating of the tube itself or by using concentrated solar heat from the parabolic mirrors. Impedance heating is modeled as a uniform heat flux at the outer wall surface. The goal of this study is to assess the feasibility of using mirrors to thaw salt frozen in the receiver tube and also to understand what benefits this method of heating can offer over impedance heating in terms of cost and melting time. All melting simulations assume a uniform initial temperature of 10 °C (283 K) for both the salt and the receiver tube. This initial condition is a worst-case scenario assuming enough time has passed for thermal equilibration with the environment.

Impedance heating systems used in operational parabolic trough plants generally provide between 100 and 250 W m−1 spread uniformly around the receiver tube [13]. Results for melting simulations with pure impedance heating (qsolar = 0), including the time and energy required to completely melt the salt mixture (tmelt and Pmelt), are summarized in Fig. 5. Note that these results are heavily dependent on the selected heat-loss equation, as plotted in Fig. 2. In this case, qmelt = 74.8 W m−1; as discussed in Sec. 2.3, this value becomes a vertical asymptote in Fig. 5 because any lower heat inputs will not completely melt the salt mixture. As qimp is increased from qmelt, the time and energy required to melt decrease sharply. tmelt continues to decrease in an exponential fashion as the heat input is increased; however, Pmelt is relatively constant for qimp > 200 W m−1, reaching a minimum value of 2.64 kWh m−1 at qimp = 395 W m−1. For these higher impedance heating levels, a slightly faster tmelt does not compensate for the extra heat flux input to the receiver tube, resulting in a relatively constant Pmelt.

Fig. 5
Fig. 5
Close modal
The plots in Fig. 5 are plotted with fitted lines determined through a least-squares fitting algorithm. As discussed in the previous paragraph, the parameter qmelt, determined from the heat-loss equation, is of significant importance to the results plotted in Fig. 5. Mathematically, a vertical asymptote is represented by a term in the denominator, suggesting that the results for impedance heating simulations may be represented by a rational expression. For both tmelt and Pmelt, excellent fits with R2 > 0.99 were achieved with the second-order rational expression shown below
$Pmelt=a2qimp2+a1qimp+a0qimp−qmelt$
(24)
where a0, a1, and a2 are fitting parameters, and qmelt is determined using the selected heat-loss equation.

### 6.3 Controllable Solar Flux Heating.

Melting salt with controllable solar flux heating could be a low-cost alternative to impedance heating in the event of a freeze. Controllable solar flux heating takes advantage of the parabolic trough mirrors to concentrate solar radiation onto the receiver tube, thus providing heat for thawing. Generally, the parabolic mirrors are oriented north–south and track the sun from east to west throughout the day. During normal plant operation, the mirrors focus on the receiver tube to heat solar salt from 300 deg to 550 °C. Due to the lower receiver temperatures and inefficient heat transfer to the stagnant salt during freeze recovery, holding the mirrors on-focus results in high thermal gradients across the receiver. For this reason, the solar heating flux profile, qsolar, must be adjusted to prevent receiver failure.

Two examples of the spatially varying solar heat flux boundary condition are plotted in Fig. 6(a). In this manuscript, the bottom of the receiver $(θ=180deg)$ refers to the side closest to the parabolic mirrors. These are the two extreme conditions with the mirrors on-focus (high heat input) and the mirrors off-sun (low heat input). For the case when the mirrors are off-sun, the plotted values assume 1 sun (1000 W m−2) at 0-deg incidence angle irradiating the top of the receiver. This heat flux is also included in calculations for the on-focus profile as seen by the matching values for $θ<45deg$ and $>315deg$; however, in the on-sun case, this heat input is small compared to the concentrated heat from the mirrors. When the mirrors are on-focus, the bottom of the receiver tube heats very rapidly compared to the top, resulting in large temperature gradients and thermal stresses. The low conductivity of the salt mixture relative to the wall material does not allow heat to dissipate from the receiver tube into the salt before reaching critical stress levels. To mitigate excessive heating, the mirrors are cycled between on-focus and off-sun positions, resulting in the spatially averaged transient flux profile shown in Fig. 6(b). Different flux profiles are achieved by controlling how long the mirrors are held on-focus, ton, how long the mirrors are held off-sun, toff, and the mirror tracking speed to move between these two positions, $θ˙$.

Fig. 6
Fig. 6
Close modal

Simulation results for several different solar heating flux profiles are detailed in Table 4. Flux profiles 13–15 are reference cases where the mirrors jump between on-focus and off-sun positions; these simulations represent the theoretical limit where the mirrors track infinitely fast. Results suggest that there are multiple combinations of ton, toff, and $θ˙$ that will result in a similar ΔTw,max; in these cases, faster $θ˙$ appears to melt the salt quicker. For slower values of $θ˙$, a shorter ton and longer toff are required to allow the heat to dissipate and maintain reasonable receiver temperatures gradients. As $θ˙$ is increased, it is possible to decrease toff while maintaining similar temperature differences, resulting in significantly faster cycle times.

Table 4

Summary of results for thawing of frozen salt in the receiver tube from an initial temperature of 10 °C using only controllable solar flux heating

Flux profile ID$θ˙$ (deg/s)ton (s)toff (s)tcycle (s)$q¯$ (W m−1)tmelt (min)ΔTw,max (K)
10.35210451.3401.15389.961.17
20.35120271.3608.44274.695.35
30.310210461.3531.75306.388.27
40.33120267.3521.39316.478.04
50.31120263.3431.70379.360.20
60.15210494.0688.55250.4132.42
70.11210486.0594.10285.0114.22
80.11300666.0457.50359.691.27
90.55210442.8337.03472.341.78
100.55120262.8507.13320.972.10
110.510210452.8471.51339.973.57
120.53120258.8415.64387.753.18
135210215.0236.07633.823.76
145120125.0342.18426.540.92
1556065.0576.18262.782.28
Flux profile ID$θ˙$ (deg/s)ton (s)toff (s)tcycle (s)$q¯$ (W m−1)tmelt (min)ΔTw,max (K)
10.35210451.3401.15389.961.17
20.35120271.3608.44274.695.35
30.310210461.3531.75306.388.27
40.33120267.3521.39316.478.04
50.31120263.3431.70379.360.20
60.15210494.0688.55250.4132.42
70.11210486.0594.10285.0114.22
80.11300666.0457.50359.691.27
90.55210442.8337.03472.341.78
100.55120262.8507.13320.972.10
110.510210452.8471.51339.973.57
120.53120258.8415.64387.753.18
135210215.0236.07633.823.76
145120125.0342.18426.540.92
1556065.0576.18262.782.28
Table 4 includes the average heat input per unit length, $q¯$, calculated for each solar heating flux profile. $q¯$ is the average function value as shown in Eq. (25); this parameter depends on the three control variables $θ˙$, toff, and ton. The averaged heat input during controllable solar flux heating results in a value easily comparable with impedance heating system power levels. Impedance heating systems provide uniform heat, so qimp is not included in the circumferential integral and $q¯=qimp$ for cases with only constant impedance heating.
$q¯=1tcycle∫0tcycle(qimp+Ro∫02πqsolar)$
(25)

Results for solar heating are compared with the previously discussed impedance heating results in Fig. 7. The combined results of impedance and solar heating show similar trends for both heating methods, allowing for a single fit on all the simulation data using Eq. (24). With the added mirror heating simulation data points, the minimum melt energy is 2.59 kWh m−1 at 349 W m−1, which is similar to results reported from pure impedance heating data points (2.64 kWh m−1 at 395 W m−1) in terms of energy; but $q¯$ to achieve this minimum energy is lowered significantly. With the added data points, a more distinct minimum followed by a linear increase is observed for Pmelt, compared to impedance heating simulations where Pmelt appeared to plateau after the minimum point. For Pmelt, the added data points result in a slightly worse fit with the rational expression (R2 lowered from 0.995 to 0.984), which appears to be caused by the variance in the solar heating data. For tmelt, Eq. (24) does a good job of capturing trends in the data for $q¯≤600Wm−1$, and at higher heat inputs, the fitted curve decreases linearly as opposed to the more asymptotic behavior observed in simulations.

Fig. 7
Fig. 7
Close modal

The results plotted in Fig. 7 show that faster melt times were achieved with controllable solar flux heating compared to impedance heating. Much higher heat inputs are possible using the parabolic mirrors due to cost and mechanical constraints on pure impedance heating systems, which limit these systems to about 300 W m−1 [13]. The higher heat inputs possible with solar heating results in significantly faster tmelt compared to impedance systems; however, heating with mirrors also results in much higher thermal stresses, as shown by the wall temperature differences plotted in Fig. 8. For comparison, results for a 300 W m−1 impedance system indicate tmelt = 8.8 h with ΔTw,max = 0.1 °C, whereas flux profile 1 $(q¯≈400Wm−1)$ results in tmelt = 6.5 h with ΔTw,max = 61.2 °C. The impedance heating system provides low levels of uniform heat resulting in low temperature gradients across the wall and slow tmelt. The concentrated solar heat is nonuniform, with high peak values as shown in Fig. 6(a), resulting in faster melting times with large temperature gradients. For these reasons, it is very important to monitor the temperature difference across the receiver wall, ΔTw, during solar flux heating.

Fig. 8
Fig. 8
Close modal

The maximum temperature difference across the receiver wall, ΔTw,max, observed during melting simulations with solar heating is plotted in Fig. 8. Results suggest that ΔTw,max varies significantly with the mirror tracking speed, $θ˙$, as well as $q¯$. For each value of $θ˙$, ΔTw,max increases linearly as a function of $q¯$. In some cases, significant differences in ΔTw,max are evident comparing simulations with similar $q¯$, e.g., flux profiles 8 and 11 with a 3% difference in $q¯$ compared to a 19% change in ΔTw,max. In these cases, lower temperature differences are observed for flux profiles with faster $θ˙$. The reference cases show the limit of infinitely fast $θ˙$ such that mirrors jump from on-focus and off-sun positions; these simulations indicate the minimum possible ΔTw,max for a given $q¯$. Results at different mirror tracking speeds suggest that there is no benefit in moving the mirrors faster than 0.5 deg s−1. For $θ˙≥0.5degs−1$, the heat input during mirror tracking is minimal and ΔTw,max can be well described as a function of $q¯$ only. A more comprehensive thermal stress analysis on the wall temperature profiles is performed in Sec. 6.4 to understand which values of ΔTw,max can be supported by the receiver tube. If possible, it is desirable to melt using solar heat and save on costs associated with impedance heating.

### 6.4 Thermal Stress Analysis.

The thermal stresses are calculated based on the temperature profile output from the ansys fluent simulations using an analytical solution method [21], described in Sec. 2. The 2D temperature profile is used to calculate radial and circumferential components of stress, while the axial component is determined by manipulating Hooke’s law assuming a generalized plane strain state. Each component of stress contributes to the effective von Mises stress, σVM, at any given location. A typical temperature profile and the resulting von Mises stress are plotted in Fig. 9; these results represent the largest recorded temperature difference for flux profile 2 a pressure of 1 bar. The temperature profile shown in Fig. 9(a) is representative of a typical profile at a high stress point when the receiver tube is heated purely with qsolar. Solar heat is focused toward the bottom part of the receiver with the mirrors before allowing time for heat to dissipate, thus maintaining a reasonable temperature gradient across the tube. This heating method results in large circumferential temperature gradients around the receiver tube, but relatively low gradients in the radial direction. Such a temperature profile results in the effective von Mises stress distribution shown in Fig. 9(b). The highest effective stress levels are at the top and bottom of the receiver, toward the outer radius.

Fig. 9
Fig. 9
Close modal

The individual components of stress at 1 bar for flux profile 2 are also plotted in Fig. 10. The axial and circumferential components of stress, σx and σr, dominate the von Mises stress state. The radial and shear components, σr and σ, are orders of magnitude lower than σx and $σθ$. All three components of stress ($σθ$, σx, and σr) vary sinusoidally as a function of θ; each component has a similar period, but σx has the largest amplitude. Comparing different radial positions, a relatively constant shape is observed for all σx plots; the magnitude of this component is highest at the inner wall surface in Fig. 10(a) and steadily decreases to the outer radius plotted in Fig. 10(c). On the other hand, σr and $σθ$ both vary significantly in the radial direction peaking at different angles around the tube. Interestingly, the sinusoidal plot representing $σθ$ appears to shift through the radial positions: where there is tension at the outer radius, there is compression at the inner radius, and vice versa. All three sinusoidal curves are included in the calculation for σVM, plotted with a solid line in Fig. 10, resulting in two peaks near the top and bottom of the receiver for all radial positions. It is important to note the stress state described is heavily dependent upon the selected constraints.

Fig. 10
Fig. 10
Close modal

The maximum observed von Mises stress during each simulation, σVM,max, is plotted as a function of ΔTw,max for each flux profile at three different pressures in Fig. 11. The values in this plot were calculated assuming a generalized plane strain condition as discussed in Sec. 3. The plotted results indicate that σVM increases linearly with more aggressive temperature differences across the receiver tube for all pressures; fit parameters for each pressure are detailed in Table 5. Also plotted with a dotted line is the maximum allowable stress for stainless steel 321 at relevant temperatures for melting of solar salt [33]; the temperature where the fitted exponential curve intersects each dotted line is considered the maximum allowable ΔTw during melting. Values for allowable stress at different temperatures and pressures are tabulated in Table 5. Based on these results with an internal pressure of 1 bar, the maximum ΔTw should be kept below 78.7 °C and 72.6 °C for heating the receiver wall up to 100 °C and 300 °C, respectively. Based on the results in Figs. 7 and 8, to maintain ΔTw ≤ 72.6°C using only solar flux heating with $θ˙=0.5degs−1$, the heat input should be $q¯≤495Wm−1$, resulting in tmelt ≥ 5.6 h.

Fig. 11
Fig. 11
Close modal
Table 5

Allowable ΔTw during melting with controllable solar flux heating determined through the thermal stress analysis of CFD results

p (bar)11040
σVMTw) fit (MPa)2.09ΔTw − 17.352.15ΔTw − 22.552.02ΔTw + 8.20
R20.9930.9910.969
Average Tw (°C)100300100300100300
Allowable σVM (MPa)138127138127138127
Allowable ΔTw (°C)78.772.678.973.065.057.6
p (bar)11040
σVMTw) fit (MPa)2.09ΔTw − 17.352.15ΔTw − 22.552.02ΔTw + 8.20
R20.9930.9910.969
Average Tw (°C)100300100300100300
Allowable σVM (MPa)138127138127138127
Allowable ΔTw (°C)78.772.678.973.065.057.6

Note: Results are based on the maximum allowable stress from ASME standards [33].

In this study, no mass flow is assumed so flow pressure is negligible; however, it is interesting to compare the thermal stresses at different internal tube pressures typical during operation. Results shown in Fig. 11 and Table 5 at different pressures suggest that for P ≤ 10 bar, thermal stresses dominate the stress state and the pressure is of minor importance. In Table 5, the allowable ΔTw changes by less than 0.5 °C for all cases as the pressure increases from 1 to 10 bar. As the internal pressure continues to increase, the contribution of pressure becomes significant. At p = 40 bar, the maximum σVM observed during melting increased by an average 10.3% compared to 10 bar, resulting in a lower allowable ΔTw by about 15 °C. The generalized plane strain assumption and 2D formulation provide a conservative estimate of the thermal stresses experienced by the receiver tube, but a three-dimensional (3D) finite-element analysis will be required to accurately capture the constraints of a real system. Such a 3D study is reserved for future work.

Fig. 12
Fig. 12
Close modal

### 6.5 Hybrid Heating Systems.

Results for controllable solar flux heating in the previous sections indicate at least 5.6 h of ideal weather conditions are required to melt salt within the receiver tube while maintaining ΔTw ≤ 72.6 °C. For a more robust freeze recovery process, there could be advantages to a combined mirror/impedance heating method. Such a system would rely on solar heat to provide most of the incident flux, while the added impedance heat prevents cloudy conditions from interrupting the melting process. Coupling with a low-cost impedance heating system (75–125 W m−1) would keep costs manageable compared to pure impedance heating and result in a less weather-dependent system compared to pure solar heating. Results for melting simulation using flux profile 10 with added impedance heating are compared in Table 6.

Table 6

Melting simulation results for mirror heat flux profile 2 augmented with varying levels of impedance heating

Flux profile IDDNI (W m−2)Impedance heat (W m−1)tmelt (min)ΔTw,max (K)% Impedance power
1010000320.972.10.0
10100075281.571.218.0
101000125261.070.326.8
Flux profile IDDNI (W m−2)Impedance heat (W m−1)tmelt (min)ΔTw,max (K)% Impedance power
1010000320.972.10.0
10100075281.571.218.0
101000125261.070.326.8

Note: Increasing the impedance power can reduce tmelt and, to a lesser extent, ΔTw,max.

Results indicate that impedance heating can lower tmelt, but has little effect on the wall temperature difference when axial gradients are neglected. With a combined solar and impedance heating system, tmelt follows similar trends to those in Fig. 7; however, ΔTw,max for the combined value of $q¯$ is lower than the pure solar heating results plotted in Fig. 8. With a minimal-cost 75 W m−1 impedance system, tmelt is reduced by 12.3% from 320.9 to 281.5 min, whereas ΔTw,max is lowered by 0.9 °C. At this low power level, impedance heat accounts for only 18% of the total energy required to melt the salt, indicating that the hybrid system could still result in significant cost-savings compared to pure impedance heating; 75 W m−1 is just enough heat to completely melt the salt, because qmelt = 74.8 W m−1. However, without additional heat input from the parabolic mirrors, the melting process would take well over a day based on results plotted in Fig. 5.

## 7 Conclusions

This paper assesses the viability of using controllable solar flux heating for freeze recovery of molten-salt parabolic trough receivers through computational modeling of the melting process. The modeling approach considers a constant density salt with conduction-dominated heat transfer, resulting in conservative melt times. The study shows that the proposed controllable solar flux heating approach has potential to replace or supplement the impedance heating system to reduce the freeze recovery time and reduce the operating costs.

Thermal stresses associated with nonuniform solar flux during the controllable solar flux heating process are also calculated. The analysis indicates that the temperature difference across the receiver wall, ΔTw, should be maintained below 70 °C for heating up to 300 °C at p ≤ 10 bar to stay below the maximum allowable stress constraints. At these conditions, the heating process will have a minimal impact on the receiver life. To achieve ΔTw ≤ 70 °C using only solar flux heating with $θ˙=0.5degs−1$, the selected heat flux profile should have an average heat input of $q¯≤495Wm−1$. This value of $q¯$ can be achieved with ton = 0 and toff ≥ 117 s, resulting in a total melting time of tmelt ≥ 5.6 h. Comparing with a total melting time of tmelt = 8.8 h with a 300 W m−1 impedance heating system, the controllable solar flux heating method can be more efficient.

The analysis also shows that a hybrid freeze recovery incorporating both controllable solar flux heating and impedance heating may be advantageous. Both methods can be complementary to each other in practical operation. For a molten-salt plant with a low-cost impedance heating system installed, the mirrors can supplement solar heat to the freeze recovery process to save electrical energy for the impedance heating when solar irradiation is available.

By recognizing the potential of the controllable solar flux heating method, a more comprehensive engineering analysis is required to determine an optimal heating system for molten-salt collector loops in practice. Future modeling efforts will assess the impact of temperature-dependent density and void-space formation on thermal gradients during melting, as well as developing a 3D model to account for axial gradients present in a real system.

## Acknowledgment

This work was authored in part by Alliance for Sustainable Energy, LLC, the manager and operator of the National Renewable Energy Laboratory for the U.S. Department of Energy (DOE) under Contract No. DE-AC36-08GO28308. Funding was provided by U.S. Department of Energy Office of Energy Efficiency and Renewable Energy Solar Energy Technologies Office. The views expressed in the article do not necessarily represent the views of the DOE or the U.S. Government. The authors would like to acknowledge Dr. Janna Martinek for many useful conversations regarding model setup and troubleshooting.

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