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Conjugate heat transfer model for feedback control and state estimation in a volumetric solar receiver

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Abstract and Figures

Open volumetric solar receivers (VSRs) are a promising technology for concentrated solar power plants due to their capability to provide heat using ambient air as the working fluid operating at temperatures over 700 C. Nevertheless, VSRs are challenged by the unsteadiness and high intensity of the radiation flux, which may cause unreliable or unsafe outflow temperatures, and may compromise the lifetime of the porous ceramic absorbers due to extreme thermal loads, thermal shock or thermal fatigue. We propose a data assimilation framework to address these matters using blower actuation, measurements from sensors located in the outflow stream of air, and a model for the conjugate heat transfer in an open VSR. We formulate said model and compare it against full three-dimensional CFD simulations to show that it captures the relevant dynamics while reducing the computational cost enough to allow for online calculations. A linear quadratic Gaussian (LQG) controller is used with the model to perform simultaneous state estimation and feedback control in three simulated scenarios. Our framework proves capable of stabilizing outflow air temperatures during the passing of a cloud, estimating the radiation flux hitting the absorber during daily operation, monitoring temperature cycling in the solid matrix, and avoiding extreme temperature gradients during start-up procedures. Artificial noise and disturbances are added to the system for all scenarios and the LQG controller proves to be robust, rejecting disturbances and attenuating noise, as well as compensating for model uncertainty.
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Conjugate heat transfer model for feedback control and state estimation in
a volumetric solar receiver
Benjam´ın Herrmanna,b,, Masoud Behzadc, Jos´e M. Cardemild, Williams R. Calder´on-Mu˜nozd,e, Rub´en M.
Fern´andezd
aDepartment of Mechanical Engineering, University of Washington, 3900 E Stevens Way NE, Seattle, WA 98195, USA
bInstitut f¨ur Str¨omungsmechanik, Technische Universit¨at Braunschweig, Hermann-Blenk-Str. 37, 38108 Braunschweig,
Germany
cIndustrial Engineering School, Faculty of Engineering, Universidad de Valpara´ıso, Brasil 1786, Valpara´ıso, Chile
dDepartment of Mechanical Engineering, FCFM, Universidad de Chile, Beauchef 851, Santiago, Chile.
eEnergy Center, FCFM, Universidad de Chile, Av. Tupper 2007, Santiago, Chile.
Abstract
Open volumetric solar receivers (VSRs) are a promising technology for concentrated solar power plants
due to their capability to provide heat using ambient air as the working fluid operating at temperatures
over 700C. Nevertheless, VSRs are challenged by the unsteadiness and high intensity of the radiation flux,
which may cause unreliable or unsafe outflow temperatures, and may compromise the lifetime of the porous
ceramic absorbers due to extreme thermal loads, thermal shock or thermal fatigue. We propose a data
assimilation framework to address these matters using blower actuation, measurements from sensors located
in the outflow stream of air, and a model for the conjugate heat transfer in an open VSR. We formulate said
model and compare it against full three-dimensional CFD simulations to show that it captures the relevant
dynamics while reducing the computational cost enough to allow for online calculations. A linear quadratic
Gaussian (LQG) controller is used with the model to perform simultaneous state estimation and feedback
control in three simulated scenarios. Our framework proves capable of stabilizing outflow air temperatures
during the passing of a cloud, estimating the radiation flux hitting the absorber during daily operation,
monitoring temperature cycling in the solid matrix, and avoiding extreme temperature gradients during
start-up procedures. Artificial noise and disturbances are added to the system for all scenarios and the LQG
controller proves to be robust, rejecting disturbances and attenuating noise, as well as compensating for
model uncertainty.
Keywords: volumetric solar receiver; conjugate heat transfer; reduced-order model; feedback control; state
estimation.
1. Introduction1
The growing awareness of the effect regarding the process of climate change has fostered the deployment2
of renewable energy sources for electricity generation. In that context, the most matures and economically3
viable renewable technologies are wind and photovoltaic (PV), which currently are cost competitive with4
conventional (fossil) power plants, showing a significant increment on the deployment of such plants dur-5
ing the last decade, moving from an installed capacity of 221 GW in 2010, to 941 GW in 2017 (REN216
Secretariat, 2012, 2018). Indeed, these two sources concentrates around 95% of the total investment in Re-7
newable Energy, during 2017 (REN21 Secretariat, 2018).The power generation from wind and solar PV are8
characterized by high variations on their availability, due to environmental factors. However, the inherent9
variability of wind and PV power raises new challenges for power systems operators and regulators (Lott10
Corresponding author
Preprint submitted to Solar Energy December 1, 2019
and Kim, 2014).11
12
The Concentrating Solar Power (CSP) technologies stands out as one of the best options for delivering13
dispatchable electricity, and among the CSP plants Central Receiver Systems (CRS) have received larger14
attention during the last years, since it allows achieving higher operating temperatures, and therefore higher15
conversion efficiencies (IEA, 2010). Currently the most common application of CRS considers molten salts as16
working fluid and storage media, which limits its operating temperature to the chemical stability limit of the17
salt. In order to overcome such limit, the utilization of compressible gases as working fluid have been pointed18
out as a feasible option (Mehos et al., 2017). Indeed, several authors have proposed the use of super-critical19
carbon dioxide as working fluid in closed Brayton Cycles (Atif and Al-Sulaiman, 2017; Reyes-Belmonte20
et al., 2016). Nevertheless, using air as working fluid also presents several advantages, such as non toxicity,21
low cost, high availability, stability at high temperatures and low environmental impact (´
Avila-Mar´ın, 2011).22
23
The use of air as working fluid in CRS requires a special design on the receiver that allows to efficiently24
convert the energy in solar radiation in to useful heat. Several designs have been proposed during the last25
decades, but the best compromise between heat transfer and pressure drop is observed in the so-called open26
volumetric solar receiver (VSR) (´
Avila-Mar´ın, 2011). Typically, these receivers are comprised of hundreds27
of absorber modules, each of them with a cross-section on the order of 150cm2(´
Avila-Mar´ın, 2011). A single28
VSR module consists of a permeable porous ceramic that allows concentrated solar radiation to penetrate29
within its structure, thus enhancing convective heat exchange with a suction-driven flow of atmospheric air,30
as shown in Fig. 1.31
suction-driven flow
m
solar irradiance
G
atmospheric conditions
p0,T0
pL
porous absorber
Figure 1: Schematic representation of a volumetric solar receiver module.
The operation of CRS requires the availability of smart control systems, that allows dealing with the fast32
transient events, which induce significant stress in the receiver (Augsburger and Favrat, 2013). The solar33
receiver is one of the critical components in CRS, representing a significant share of the capital investment34
(D. Gielen, 2012). Therefore, cloud passing over the heliostat field strongly affects the operation of the35
system. Transient variations on the incident radiation flux distribution increase the thermal stresses in the36
receiver, leading to severe damages or to reduce the cyclic life of the device (Crespi et al., 2018). Since37
the transient variation on the solar beam radiation are unavoidable, several authors have proposed different38
approaches for handling that issue, such as manipulate the heliostats aiming to the receiver (Garc´ıa et al.,39
2018; Tehrani and Taylor, 2016), or controlling the working fluid flow rate, which is particularly interesting40
for open volumetric receiver systems (Li et al., 2016).41
42
Hence, the availability of accurate estimation models for predicting the state and the actual operating43
temperature of the solid matrix in the volumetric absorber, would allow a safe and efficient operation of the44
plant. Indeed, such estimation models would also allow feedback control of the fluid outlet temperature,45
which is crucial for avoiding dysfunctions with the processes downstream: electricity generation, industrial46
process, etc. In addition to that, an appropriate monitoring of the receiver temperature cycling also allows47
2
to estimate the remaining life of the components and preventing dramatic failures (Li et al., 2016). Several48
works have pointed out that the solar flux distribution on the receiver surface or inside the absorber is49
highly non-uniform (´
Avila-Mar´ın, 2011; Augsburger and Favrat, 2013; Tehrani and Taylor, 2016), therefore50
it is crucial to determine the actual incident radiation flux for a safe operation. However, since measur-51
ing that flux is highly complex, developing models capable of online estimation seems as an efficient approach.52
53
Previous studies have proposed estimation models for saturated steam and molten salts receivers. Augs-54
burger and Favrat (2013) developed a simulation model for evaluating the cloud passages on a heliostat55
field, but considering an analytic function (based on the convolution of different Gaussian distributions) for56
estimating the radiation flux distribution on the receiver. The results of the model allowed to assess the flux57
evolution over short periods in terms of the geometrical spread on the receiver surface, the flux peaks and58
gradients, among other parameters. Later, Samanes and Garcia-Barberena (2014) developed a transient59
cavity receiver model, in which the main heat transfer mechanisms were modeled in fast approach, allowing60
to analyze the impact of introducing an adaptive control able to predict the dynamic response of the receiver61
for different operating conditions.62
63
Recently, Li et al. (2016) modeled an open-loop air receiver, coupled to a thermal energy storage. The64
model was validated against experimental data, allowing to adjust the heat transfer equations by using ap-65
propriate Nusselt correlations. In that context different control strategies were analyzed, aiming to keep the66
receiver outlet air temperature stable and control the integration to the thermal energy storage. However,67
that model was not applied to capture the impact of fast transient events such as cloud passing, which68
demands much more complex control schemes. Finally, Crespi et al. (2018) presented an analysis of the per-69
formance of a central receiver plant using molten salts as working fluid. The authors analyzed the dynamic70
behavior of the receiver when subjected to transient conditions due to the passage of clouds, considering71
different scenarios and assessing the receiver response. However, the control measures were applied to the72
heliostat field aiming control and not to the fluid system, as expected for systems with larger thermal inertia.73
74
In this study, we propose the implementation of a physics based approach, which accounts for the75
conjugate heat transfer phenomena in a volumetric receiver module, and implement a linear quadratic76
Gaussian (LQG) controller. Feedback control would not only allow the stabilization of the temperature77
outlet of the receiver (even under fast transient conditions), but also the estimation of the flux distribution.78
The subsequent sections of this article are organized as follows. Section 2 describes the analytical formulation79
of the proposed model, and a brief review on the tools borrowed from optimal control theory is presented80
for completeness. Section 3 presents the methodology for implementing the model and the control scheme81
proposed, which is later evaluated in terms of the results of Section 4. The results are presented under82
different scenarios of transient behavior, allowing to capture the impact and the advantages of the new83
approach. Finally, Section 5 concludes and highlights the future challenges for the proposed framework.84
2. Analytic framework85
2.1. Model formulation86
The purpose of this section is to derive a physics-based reduced-order model for the dynamics of the heat87
transfer processes in an open VSR module. In particular, we are interested in describing the behavior of the88
outlet temperature for the fluid, and of the temperature and temperature uniformity for the solid matrix89
of the porous absorber. Following the principle of Ockham’s razor, we propose a minimal representation of90
the system using only three degrees of freedom. Let Trrepresent the temperature of the solid matrix at the91
frontal section of the receiver, where the incoming radiation flux hits and penetrates into the absorber up92
to a depth Lr. Let then Tcbe the bulk temperature of the solid matrix in the rear section of the receiver of93
length Lc, which gets heated through conduction with the frontal section. The fluid enters the receiver of94
length L=Lr+Lcat a temperature T0, it exchanges heat with both sections of the solid matrix by means95
of convection and its temperature quickly saturates to the outlet value, which we will call Ta. These heat96
3
transfer processes are shown in Fig. 2, as well as the typical streamwise temperature distributions of the97
fluid and of the solid matrix.98
99
SOLID
LrLc
L
TrTc
FLUID
Ta
conduction
radiation
convection convection
inflow
T0
Ta
outflow
z
T(z)
Figure 2: Schematic of the heat transfer processes in a volumetric solar receiver and the reduced-order model. The blue (red)
curve shows a typical streamwise temperature distribution of the fluid (solid matrix).
We start by tackling the hydrodynamics governing the problem. A blower produces a suction pressure100
pLat the receiver outlet which is lower than the atmospheric pressure p0, and therefore induces a flow of air101
through the porous structure. Due to conservation of mass, the mass flow density ˙mis constant through the102
receiver. The momentum balance determines the relation of ˙mwith the pressure difference between both103
sides of the porous structure. In a similar manner to that presented in the work of Becker et al. (2006), we104
obtain this relation using Darcy-Forcheimer’s law and the ideal gas equation of state, as follows105
p2
0p2
L
2RTaL=K1µ˙m+K2˙m2,(1)
where ˙mis expressed per unit area of receiver, Ris the ideal gas constant for air, K1and K2are the linear106
and quadratic hydraulic resistance coefficients respectively, and µis the dynamic viscosity of air. In Eq. 1107
the hydrodynamic transients have been neglected because the residence time of the fluid inside the absorber108
is much shorter than the time scales associated with the thermal response.109
110
Due to the large changes in the air temperature as it flows through the absorber, it is important to111
take into account the variations of viscosity. As in Kribus et al. (1996), the temperature dependency is112
approximated using a power-law of the form µ0(T/T0)nµ, where µ0is the viscosity at ambient temperature113
and nµ= 0.7. This power-law is evaluated at the average of fluid and solid matrix temperatures at the114
frontal and rear sections of the receiver. Then an effective viscosity is calculated using an average between115
sections weighted by their depth, as follows116
µ=µ0
LLrTr+Ta0
2T0nµ
+LcTc+Ta
2T0nµ,(2)
where Ta0is the mean fluid temperature at the frontal section of the receiver. We assume a quadratic117
temperature profile varying from T0to Taand having zero derivative at z=Lr, thus integrating we get a118
mean value of Ta0=T0+ 2/3 (TaT0). This approximation is inspired by the typical variations of the fluid119
temperature through the absorber, as shown by the blue curve in Fig. 2.120
121
4
Substituting µfrom Eq. 2 into Eq. 1, and given a value for the suction pressure pL, the mass flow density122
˙mcan be solved for as a nonlinear function of the system state (Ta, Tr, Tc)T. Thus, the dynamics of the123
system is governed by the thermal energy balances for the fluid and the two solid sections124
Maca
dTa
dt=˙mca(TaT0) + hraAra (TrTa0) + hca Aca (TcTa),(3a)
125
Mrcr
dTr
dt=hraAr a (TrTa0)hrc Arc (TrTc) + εG εσ T4
rT4
0,(3b)
126
Mccc
dTc
dt=hcaAca (TcTa) + hrc Arc (TrTc),(3c)
where Maand caare the mass of fluid inside the receiver and its heat capacity, Mr,cr,Mcand ccare the127
masses and heat capacities of the solid matrix frontal and rear sections respectively, hra and hca are the128
convective coefficients for the heat transfer through the surfaces Ara and Aca between the fluid and the129
corresponding section of the solid matrix, hrc = 2kr c/L is the thermal conductance for the heat transfer130
through the surface Arc between solid matrix sections where the material has a thermal conductivity krc,131
Gis the direct radiation flux delivered by the heliostat field, εis the absorber emissivity, σis the Stefan-132
Boltzmann constant, and tis time. The solid matrix is assumed to behave as a grey body under radiative133
equilibrium, therefore its emissivity is equal to its absorptivity (Modest, 2013). All masses and surface areas134
in Eqs. 3 are expressed per unit area of receiver cross-section.135
136
Temperature dependency of the thermal conductivity of air is important for the range of conditions137
studied, hence its effect on the heat transfer coefficients is also approximated with a power-law and evaluated138
at the respective film temperatures, as follows139
hra =h0Tr+Ta0
2T0nh
, hca =h0Tc+Ta
2T0nh
,(4)
where nh= 0.88 and h0is the heat transfer coefficient calculated using the thermal conductivity of air140
evaluated at T0. This value of nhwas obtained by a least-squares fit of a power-law to the data set for the141
thermal conductivity of air as a function of temperature in the range 300–1400K presented in the book by142
Cengel and Ghajar (2011). Note that Ta0is used for the convective heat exchange and all fluid properties at143
the front section of the receiver, as it better approximates the fluid temperature at this streamwise location.144
145
2.2. Linear quadratic Gaussian control146
This section presents a brief outline of the linear quadratic Gaussian (LQG) framework used with the147
reduced-order model. Equations 3 represent a nonlinear dynamical system and can be rewritten as ˙
x=148
f(x,u), where x= (Ta, Tr, Tc)Tis the state vector, the overdot denotes time differentiation, and uis a149
control input, such as pLfor blower actuation. A linear state-space model is obtained by linearizing around150
an equilibrium point, as follows151
˙
x=Ax +Bu,(5a)
152
y=Cx,(5b)
where the matrices Aand Barise from the linearization, xand unow represent the deviation from the153
state and actuation equilibrium points respectively, and yrepresents linear measurements of the system154
state given by the mapping C. For an arbitrary control signal u(t), the exact solution to Eq. 5a becomes155
x(t) = eAtx(0) + Zt
0
eA(tτ)Bu(τ)dτ. (6)
As described by Eq. 6, the temporal evolution of the state is governed by the matrix exponential and156
the convolution integral. The system given by Eqs. 5 is said to be controllable, if it is possible to navigate157
5
from the origin to an arbitrary state xwithin finite time using a finite control signal u(t), see, e.g., Astr¨om158
and Murray (2010). In addition, if measurements of the full state are available, then C=Iis the Nx×Nx
159
identity matrix, where Nxis the dimension of x. In this case, it is possible to design a proportional controller160
u=Krxto arbitrarily place the eigenvalues of the closed-loop system161
˙
x=Ax +Bu = (AKr)x.(7)
A typical control goal is to stabilize the system by trading off between how fast we drive the state xto162
0and how expensive the control actuation is. For this purpose, the controller gain Kris constructed so163
that it minimizes a quadratic cost function Jrthat balances the aggressive regulation of xwith the control164
expenditure, as follows165
Jr=Z
0xT(τ)Qx(τ) + uT(τ)Ru(τ)dτ, (8)
where the matrix Qis positive semi-definite and the matrix Ris positive definite. The entries in Qand R166
weight the cost of deviations of the state from zero and the cost of actuation, respectively. The full-state167
feedback controller Krthat minimizes the quadratic cost function Jris called a linear quadratic regulator168
(LQR), and can be calculated by numerically solving an algebraic Riccati equation using built-in routines169
from many computational packages.170
171
The optimal LQR controller described above relies on access to full-state measurements of the system,172
which is hardly the case in practical applications. In the context of this work, yis usually given by outflow173
temperature readings Ta, as sensors in the solid matrix are often unavailable. The system given by Eqs. 5 is174
said to be observable if any state xcan be estimated from the time-history of sensor measurements y, see,175
e.g., Astr¨om and Murray (2010). A full-state estimator is a dynamical system that produces an estimate176
ˆ
xof the full state x, given our knowledge of the process dynamics, the control input u, and the sensor177
measurements y. If the system is observable, a full-state estimator can be constructed using a filter gain178
Kf, as follows179
ˆ
˙
x=Aˆ
x+Bu +Kf(yˆ
y),(9a)
180
ˆ
y=Cˆ
x,(9b)
A typical estimator goal is for the estimated state ˆ
xto converge quickly to the true state x, while181
considering how much the model is trusted, which may have disturbances and missing dynamics, and how182
much the sensor measurements are trusted, which may have noise. Analogously to the LQR procedure, the183
estimator gain Kfis constructed so that it minimizes a quadratic cost function Jfthat balances aggressive184
estimation with noise attenuation, as follows185
Jf= lim
t→∞
E(x(t)ˆ
x(t))T(x(t)ˆ
x(t)),(10)
where Eis the expected value operator. The real system has state disturbances dand sensor noise n,186
which are assumed to be zero-mean Gaussian white-noise processes with known covariances Vdand Vn,187
respectively. These covariances appear implicitly in the calculation of the cost function Jf. The entries in188
the matrices Vdand Vnweight the uncertainty level of the model and that of the sensor measurements,189
respectively. The full-state estimator Kfthat minimizes the quadratic cost function Jfis called a linear190
quadratic estimator (LQE) or Kalman filter, and, as for the LQR, can be calculated by numerically solving191
another algebraic Riccati equation using built-in routines from many computational packages.192
193
A linear quadratic Gaussian (LQG) controller is the optimal sensor-based feedback control law that194
minimizes the cost function in Eq. 8 including sensor noise and state disturbances. Remarkably, the opti-195
mal LQG solution can be obtained by combining the the LQR controller gain Krwith the estimated state196
ˆ
xobtained by the Kalman filter gain Kf, where Krand Kfare each optimal in their corresponding cost197
functions. Thus, it is possible to calculate Krand Kfseparately and then combine them to form an optimal198
6
System
LQELQR
LQG
Figure 3: Block diagram for a linear quadratic Gaussian (LQG) controller.
LQG controller, as shown in Fig. 3.199
200
For more details on linear systems and optimal control theory, the readers are referred to textbooks such201
as Astr¨om and Murray (2010); Stengel (1994); Doyle et al. (2013).202
3. Methodology203
3.1. Model parameters204
Parameters appearing in the equations presented in Section 2.1 are selected to model a silicon carbide205
honeycomb absorber of prorosity φ= 0.64 and square channels of height l= 2mm. The cross-sectional area206
of the solid matrix per unit area of receiver cross-section is calculated as Arc = (1 φ). The areas for the207
convective heat exchange at the front and rear parts of the receiver are also normalized by the cross-sectional208
area of the absorber, and are therefore calculated as Ara = 4Lrφ/l and Aca = 4Lcφ/l, respectively, where Lr
209
and Lcare the corresponding streamwise lengths. The parameter values that remain constant throughout210
this study are listed in Table 1.211
Table 1: Constant parameters for the reduced-order model.
L LrLcT0K1K2µ0h0krc
40mm 10mm 30mm 25C 1.1×107m146.68m218.3µPa 38.89W/m2K 80W/mK
l φ MrMccacrccAra Aca Arc
2mm 0.64 11.52kg/m234.56kg/m21008J/kgK 750J/kgK 750J/kgK 12.8 38.4 0.36
The length of the frontal receiver section Lris based on the penetration depth of radiation flux into the212
absorber, which is calculated from the view factor expression for a honeycomb monolith presented in Worth213
et al. (1996). The linear and quadratic hydraulic resistance coefficients K1and K2are calculated using214
standard correlations for the friction factor for fully developed and developing laminar flow in a square duct,215
respectively. The convective heat transfer coefficient h0is calculated from the Nusselt number for thermally216
developed square duct flow at constant wall temperature, also using standard correlations, such as those217
presented in, e.g., Kandlikar et al. (2005).218
7
3.2. CFD simulations219
In addition to simulations using the model presented above, full three-dimensional computational fluid220
dynamics (CFD) simulations using ANSYS CFX are also carried out for comparison purposes. Using CFD221
for analyzing the performance of volumetric receivers have been considered by several authors Abuseada222
et al. (2019); Stadler et al. (2019), showing good agreement with experimental data. The model imple-223
mented herein follows the recommendations from the literature and considers the mass, momentum and224
energy balances within the porous absorber module of a VSR. Details of the solver setting and the numeri-225
cal methodology are described below.226
227
The porous media is treated as a continuum, hence there is only one computational domain which is228
shared between fluid and solid phases, as presented in the work by Fend et al. (2013); G´omez et al. (2013).229
The governing equations are continuity, unsteady and compressible Navier-Stokes with an ideal gas law, and230
unsteady energy balances under the non-thermal equilibrium assumption between fluid and solid phases.231
The continuum approach considers a volume-average of physical quantities at the pore level, therefore poros-232
ity of the absorber, φ= 0.64, and interactions at the fluid-solid interface enter the governing equations as233
source terms (Kaviany, 2012).234
235
The computational domain consists of a single absorber module with a square cross-section of 100mm×100mm236
and a depth L= 40mm. Hydrodynamic boundary conditions are a no-slip condition at the side-walls, zero237
(relative) total pressure at the inlet surface, and a prescribed suction pressure at the outlet pL. The par-238
ticular value used for pLdepends on the scenario being simulated and will be indicated accordingly. An239
adiabatic boundary condition is used for the fluid and the solid at the side-walls and outlet surfaces. At the240
front surface (flow inlet), a fixed temperature of Ta0= 25C is set for the air, and for the solid fraction, a241
temperature-dependent heat flux qs(Ts) is used, as follows242
qs(Ts) = ε(1 φ)GσT4
sT4
0,(11)
where Tscorresponds to the local value of the temperature in the solid phase. Equation 11 corresponds243
to the surface balance between emission and the direct radiation flux hitting the front of the solid matrix.244
The value for Gused depends on the scenario being simulated and is indicated accordingly. A similar245
radiative balance is considered at the interior of the computational domain using a volumetric heat source246
term qv(T, z) in the energy equation for the solid phase, as follows247
qv(Ts, z) = 4εφ
lF(z)GσT4
sT4
0,(12)
where zis the streamwise coordinate, lis the channel height as presented in Tab. 1, and F(z) is the view248
factor expression for a honeycomb monolith presented in Ref. (Worth et al., 1996). Equation 12 represents249
the radiative balance between the internal walls of the absorber, the heliostats and the outside air. In the250
same manner as for our proposed model, in Eqs. (11) and (12) we consider that the receiver material behaves251
as a grey body under radiative equilibrium, therefore its emissivity and absorptivity are equal (Modest, 2013).252
253
Other interactions at the pore level that are modeled as source terms in the continuum approach are the254
Darcy-Forcheimer momentum losses and the convective heat exchange between fluid and solid phases. In255
the same way as for the reduced-order model, the respective linear and quadratic loss coefficients, and the256
convective heat transfer coefficient are computed using standard correlations for friction factors and Nusselt257
number in square duct laminar flow (Kandlikar et al., 2005). A streamwise coefficient multiplier of 100 is258
used for the transverse directional loss coefficients to model the anisotropy of the hydraulic losses through259
a honeycomb absorber, as opposed to those in an open-cell foam absorber. The viscosity and thermal con-260
ductivity of air have a power-law temperature dependency, as presented in the model formulation section,261
with the difference that the properties are now evaluated at the local temperature average between fluid262
and solid phases. All other geometrical parameters or fixed thermophysical properties are the same as those263
8
presented in Tab. 1.264
265
The domain is discretized using hexahedral elements generated by a sweep method, resulting in a mesh266
with quality metrics within the recommended values (Ansys Inc., 2012). Steady-state simulations are carried267
out for G= 0.7MW/m2and pL=65.04Pa (relative pressure) using different grids and increasing the268
number of elements until mesh independence is achieved. Finally, a mesh with 1.62 ×105control volumes269
is selected. A high resolution advection scheme is selected, which includes a variable blend factor for the270
purpose of accuracy and robustness of the solution (Ansys Inc., 2011). It is also important to set a proper271
convergence criterion for the solver. The local imbalance can be evaluated by the root mean square (RMS)272
type of residual, while the global imbalance is checked by the conservation target, these are required to be273
below 105and 103, respectively. The simulation times for the different transient scenarios range between274
14 and 31 hours on a 4core Intel R
Xeon R
E3-1240 v5 processor and 16GB of RAM.275
4. Results and discussion276
4.1. Comparison with CFD: three scenarios277
The purpose of this section is to compare steady and transient solutions obtained with the model to those278
obtained using full three-dimensional CFD. Moreover, the comparison of transient results is carried out for279
three simulated scenarios that are designed to emphasize the need for control and estimation. To solve for280
Eqs. (2), we first obtain the expression of the positive root of Eq. (1) for the mass flow density ˙mas a281
function of the temperatures and suction pressure with µsubstituted in from Eq. (2). Subsequently, steady282
and transient solutions for Eqs. (2) are computed in Matlab using a nonlinear equations solver based on the283
Trust-Region-Dogleg algorithm (Conn et al., 2000), and a variable-step and variable-order stiff differential284
equations solver (Shampine and Reichelt, 1997), respectively.285
286
To start, we compute the steady solutions using a suction pressure pLsuch that the air outflow tem-287
perature is Ta= 700C for different radiation fluxes ranging from G= 0.4–1.0MW/m2with increments of288
0.1MW/m2. For the CFD simulations, this is achieved using an outlet pressure boundary condition and289
iterating the value of pLuntil the solution for the mass flow averaged outlet temperature reaches 700 ±1C290
for each value of G. The solid matrix temperature in the front section of the receiver Tris approximated291
from CFD results as the area average of the solid-phase temperature in the inlet surface of the compu-292
tational domain. The solid matrix temperature in the rear section of the receiver Tcis calculated as the293
volume average of the solid-phase temperature in the whole computational domain. For the model results, a294
nonlinear equations solver from Matlab is used to find pL,Trand Tcsuch that the steady solution to Eqs. 3295
satisfies Ta= 700C, for every value of G. Afterwards, the mass flow rate per unit area of absorber ˙mis296
calculated from Eq. 1.297
(a)(b)
Figure 4: Model and CFD steady state solutions for different values of the radiation flux. (a) Air outflow temperature and
solid matrix temperatures in the front and rear sections of the receiver. (b) Mass flow rate per unit area of absorber.
9
Steady state solutions for the mass flow rate and temperatures in the simulated VSR are shown in298
Figure 4 for different radiation fluxes, where a satisfactory agreement is observed between model and CFD299
calculations. As the value of Gincreases, a higher mass flow rate is required to maintain the desired air300
outflow temperature, and larger temperature differences occur within the solid matrix.301
4.1.1. Clear-sky daily operation302
During clear-sky operation radiation flux incident on the receiver changes slowly throughout the day303
due to the variation on the incident solar radiation and because of different losses, such as cosine losses,304
alignment problems, astigmatism, shading and obstruction among others. In order to assess the operation305
under representative conditions, a simple analysis using a ray-tracing software was implemented. A cosine306
approximation was identified to be representative for describing that daily variations. During the operation307
of the system under such conditions, the mass flow going through the absorber has to be adjusted in order to308
maintain the desired outflow air temperature. To achieve this, the blower needs to be controlled to modify309
the suction pressure that induces the flow of air. Thus, the simulations of the daily operation during a310
clear-sky scenario are carried out using a constant mass flow rate to emphasize the need for control. As311
previously mentioned, the simulated scenario considers a time-dependent radiation flux that is periodic over312
an 8h window, as follows313
G(t) = Gav GAcos (ωt),(13)
where Gav = 0.7MW/m2is the daily average, GA= 0.3MW/m2is the amplitude of the variations and314
ω= 2π/(8 ×3600)s1their angular frequency. The imposed radiation flux as function of time is shown315
in Fig. 5(a). The outlet boundary condition for the CFD simulation is set to a fixed mass flow rate316
˙m= 0.812kg/s m2(per unit area of absorber), selected such that a radiation flux of Gav results in a desired317
steady state outflow air temperature of Ta= 700C. The daily behavior of air outflow temperature and solid318
matrix temperatures from the front and rear sections of the receiver are presented with a good agreement319
between model and CFD in Fig. 5(b).320
(a)(b)
Figure 5: (a) Time-dependent radiation flux used for the clear-sky daily operation scenario. (b) CFD and model results for
air outflow temperature and solid matrix temperatures in the front and rear sections of the receiver during clear-sky daily
operation.
Without blower actuation to control the mass flow rate during daily operation, air outflow and solid321
matrix temperatures manifest variations of about 500C, as shown in Fig. 5(b). The need for control is322
evident, even if the solid matrix admitted such fluctuations (and peak values), changes of this magnitude323
on the air temperature are certainly not suitable for the downstream processes.324
4.1.2. Cloud passing325
The passing of a cloud may induce abrupt changes in the radiation flux hitting a VSR, causing large tem-326
perature variations in the solid matrix and working fluid. This scenario is simulated using a time-dependent327
radiation flux that starts at a value of 1MW/m2and after 5s drops linearly over the next 5s to 0MW/m2,328
where it stays for 30s before increasing linearly over 5s back the starting value, as shown in Fig. 6(a).The329
10
proposed scenario represents the worst case for a small heliostat field as featured by the currently operational330
CSP air systems. The approach considered herein is based on the analysis developed by Cagnoli et al. (2017)331
and Garc´ıa et al. (2018). A fixed suction pressure such that Ta= 700C when G= 1MW/m2is used for332
computations with both, model and CFD (as outlet boundary condition in the latter). The fast temperature333
transients that occur in a VSR during a cloud passing are plotted in Fig. 6(b).334
335
(a)(b)
Figure 6: (a) Time-dependent radiation flux used for the cloud passing scenario. (b) CFD and model results for air outflow
temperature and solid matrix temperatures in the front and rear sections of the receiver during the passing of a cloud.
A sharp drop of about 700C in the air outflow and solid matrix temperatures is observed when the336
blower is not being actuated to adjust the mass flow rate, as shown in Fig. 6(b). Control is clearly a necessity337
for reliability of the downstream processes, moreover, estimation of the temperatures in the solid matrix is338
required to monitor thermal expansion cycles that may occur several times per day.339
4.1.3. Cold start-up340
The cold start-up procedure carried out every day in a CSP plant involves gradually aiming the he-341
liostats at the receiver and adjusting the mass flow of working fluid to take the system from rest to its342
operating condition. In a VSR, excessive temperature differences may occur within the absorber as it heats343
up, leading to thermal shock. Therefore, the control of the heliostat field and blower must balance the344
structural integrity of the VSR with a faster start-up of the plant. The cold start-up scenario is simu-345
lated with the model and CFD using a time-dependent radiation flux and suction pressure. At t= 0, the346
system is at rest with all temperatures set to T0= 25C, no heliostats aiming at the receiver, G= 0,347
and zero pressure drop pL=p0so there is no flow of air. After 5s, the pressure drop and radiation flux348
ramp-up for 60s to reach G= 0.4MW/m2and the value of pLthat results in Ta= 700C in steady state.349
This time-dependency, shown in Fig. 7(a) for the radiation flux, is selected to simulate a typical open-loop350
control strategy used for cold-start-up procedure and emphasize the utility of a more sophisticated approach.351
352
Figure 7(b) shows the evolution of air outflow and solid matrix temperatures during a cold start-up353
procedure, while Fig. 7(c) shows the evolution of temperature differences within the absorber. The linear354
ramp-up control strategy for the radiation flux and pressure drop results in a slow thermal response, taking355
about 600s to reach the steady operation condition, and an overshoot more than 70C greater than the356
equilibrium value for the temperature differences within the absorber. In the same way as for the previous357
scenarios, the characteristic dynamics of the system are captured by the model.358
11
(a)(b)
(c)
Figure 7: (a) Time-dependent radiation flux used for the cold start-up scenario. CFD and model results during a cold start-
up procedure: (b) air outflow temperature and solid matrix temperatures in the front and rear sections of the receiver, and
(c) temperature differences within the absorber.
4.2. Clear-sky daily operation with LQG359
The formulation of the control problem involves the extension of the system to consider the pressure360
drop induced by the blower, p=p0pL, and the radiation flux reflected off the heliostat field, G, as new361
state variables. The following differential equations are included to describe the evolution of these variables362
˙
p=u, (14)
363 ˙
G= 0,(15)
where uis the only control input and represents the rate of change of the blower rotation rate. Even though364
radiation flux is changing, these variations are very slow when compared to the thermal response times of365
the system, hence Eq. 15 is a reasonable model. Furthermore, this allows for the inclusion of Gas a state366
variable, which enables its estimation in real-time using sensor-based feedback from pressure drop and air367
outflow temperature measurements. The nonlinear system described by Eqs. 3 is simulated during clear-368
sky daily operation with the radiation flux shown in Fig. 5(a) using LQG control based on the linearized369
system, which is extended to include Eqs. 14 and 15. For this scenario, the system is linearized around the370
equilibrium point Ta= 700C, Tr= 713.7C, Tc= 703.5C, ∆p= 24.76Pa, G= 0.4MW/m2, and u= 0Pa/s.371
372
Results for the simulation with LQG control are shown in Fig. 8, where the dashed curves correspond373
to the estimates of the state variables and the solid curves correspond to the actual behavior of the state374
variables of the nonlinear system. The controller stabilizes the air outflow temperature without using any375
measurements of the radiation flux. Furthermore, as shown in Fig. 8(c), the estimated radiation flux is in376
agreement with the true value, thus enabling real-time evaluation of the receiver efficiency and diagnostics377
about its state of health.378
379
12
(a)(b)
(c)
Figure 8: True and estimated state variables as a function of time during clear-sky daily operation using LQG control.
(a) Air outflow temperature and solid matrix temperatures in the front and rear sections of the receiver. (b) Pressure drop.
(c) Radiation flux.
In order to assess the robustness of the LQG controller, zero-mean Gaussian noise and disturbances with380
specified standard deviations are included in the simulations. Measurement noise with a standard deviation381
of 20C and 4Pa is added directly to the air outflow temperature and pressure sensor readings, respectively.382
Disturbances with standard deviations of 0.1C/s for the temperatures and 0.001Pa/s for the pressure are383
added to the right hand side of the corresponding equations for the nonlinear system before integrating384
every time step (process noise).385
386
Figure 9 shows the results for the simulations including noise and disturbances, where the gray thin lines387
show the sensor readings used by the LQG feedback controller. Even with large amounts of disturbances388
and noise, the system is stabilized around the desired operation point and the estimated state variables389
agree with the true behavior of the nonlinear system. Although the stabilization of air outflow temperature390
is a task that can be easily handled by a model-free control strategy, such as a proportional–integral (PI)391
controller, the proposed model-based architecture allows for the real-time estimation of the solid matrix392
temperatures and of the radiation flux.393
4.3. Cloud passing with LQG394
To formulate the control problem, again the system is extended to consider the pressure drop as a state395
using Eq. 14. However, because during this scenario the radiation flux variations are fast, its dynamics396
cannot be represented by Eq. 15, and instead G(t) is considered as an exogenous input to the system. As a397
consequence the following analysis rests on the assumption that the controller has access to the time series398
of the incoming radiation flux. In practice this can be achieved using cloud coverage prediction techniques399
that have been demonstrated in several studies (Augsburger and Favrat, 2013; Garc´ıa et al., 2018; Lopes400
et al., 2019). The nonlinear system, described by Eqs. 3 and 14, is simulated using LQG control based on401
the linearization of the extended system. The simulation considers the time-dependent radiation flux shown402
in Fig. 6(a) as an external disturbance and measurements from air temperature and pressure sensors located403
13
(a)(b)
(c)
Figure 9: True, measured and estimated state variables as a function of time during clear-sky daily operation using LQG
control. (a) Air outflow temperature and solid matrix temperatures in the front and rear sections of the receiver. (b) Pressure
drop. (c) Radiation flux.
at the receiver outlet. For this scenario, the system is linearized around the equilibrium point Ta= 700C,404
Tr= 904.3C, Tc= 751C, ∆p= 70.13Pa, G= 1MW/m2, and u= 0Pa/s.405
406
(a)(b)
Figure 10: True and estimated state variables as a function of time during the passing of a cloud using LQG control. (a) Air
outflow temperature and solid matrix temperatures in the front and rear sections of the receiver. (b) Pressure drop.
Results for the simulation with LQG control are shown in Fig. 10, where the dashed curves correspond407
to the estimates of the state variables and the solid curves correspond to the actual behavior of the state408
variables of the nonlinear system. As for the previous scenario, the controller stabilizes the air outflow409
temperature, only this time information of the radiation flux is required due to the fast transients. However,410
the controlled system still exhibits important temperature changes in the front section of the absorber, as411
shown in Fig. 10(a). Nevertheless, the size of these fluctuations decreases to less than half of that observed412
for the uncontrolled case in Fig. 6(b). Moreover, these variations can be monitored via state estimation,413
14
which may be useful to get statistics of the temperature cycling of the solid matrix for thermal fatigue studies.414
415
As in the previous section, zero-mean Gaussian noise and disturbances with specified standard deviations416
are included in the simulations to assess the robustness of the LQG controller. Measurement noise with a417
standard deviation of 20C and 4Pa is added directly to the air outflow temperature and pressure sensor418
readings, respectively. Disturbances with standard deviations of 0.1C/s for the temperatures and 0.001Pa/s419
for the pressure are added to the right hand side of the corresponding equations for the nonlinear system420
before integrating every time step (process noise).421
(a)(b)
Figure 11: True, measured and estimated state variables as a function of time during the passing of a cloud using LQG control.
(a) Air outflow temperature and solid matrix temperatures in the front and rear sections of the receiver. (b) Pressure drop.
Figure 11 shows the results for the simulations including noise and disturbances, where the gray thin lines422
show the sensor readings used by the LQG feedback controller. Even with large amounts of disturbances423
and noise, the system is stabilized around the desired operation point and the estimated state variables424
agree with the true behavior of the nonlinear system.425
(a)(b)(c)
Figure 12: Feedback control during the passing of a cloud for different actuation costs between R= 0.1–1000. (a) Air outflow
temperature. (b) Pressure drop. (c) Control input.
The effect of tuning the LQR cost function on the response of the closed-loop system during the passing426
of a cloud is explored. The air outflow temperature, pressure drop and blower control input are computed427
for different values of the actuation cost between R= 0.1–1000, as shown in Fig. 12. Tuning the LQR428
controller results in a trade-off between a cheaper actuation expenditure and a more effective controller.429
4.4. Cold start-up with LQG430
The cold start-up procedure involves actuation of the heliostat field and blower to modify the radiation431
flux and the suction pressure, respectively. Therefore, to formulate the control problem for this scenario,432
the pressure drop ∆p, the radiation flux G, and the radiation flux rate of change ˙
Gare considered as state433
variables. Their time evolution is modeled by Eq. 14 and the following differential equation434
¨
G=˙
G+v, (16)
15
which is to be interpreted as an equation of motion, where Gand ˙
Gare directly related to the position and435
velocity of the heliostats and vrepresents an actuation force that is the second input to the system along436
with u. The coefficients of the terms in Eq. 16 that represent the effects of inertia and friction are set to437
one for the sake of simplicity, but these values should be modified accordingly for the application to a real438
system. The extended nonlinear system, described by Eqs. 3, 14 and 16, is simulated using LQG control439
based on the linearization of the extended system. The simulation starts from rest, it considers the same440
target state as in the exercise presented in section 4.1.3, and feedback is obtained from measurements of the441
radiation flux (position of the heliostats) and of the temperature and pressure of the air at the outlet of the442
receiver. For this scenario, the system is linearized around the equilibrium point Ta= 700C, Tr= 713.7C,443
Tc= 703.5C, ∆p= 24.76Pa, G= 0.4MW/m2,˙
G= 0MW/(s m2), u= 0Pa/s, and v= 0MW/(s2m2).444
445
(a)
(c)
(b)
(d)
Figure 13: True and estimated state variables as a function of time during a cold start-up procedure using LQG control.
(a) Air outflow temperature and solid matrix temperatures in the front and rear sections of the receiver. (b) Pressure drop.
(c) Radiation flux. (d) Temperature difference within the absorber.
Results for the simulation with LQG control are shown in Fig. 13, where the dashed curves correspond446
to the estimates of the state variables and the solid curves correspond to the actual behavior of the state447
variables of the nonlinear system. Compared to the linear ramp-up control strategy shown in Fig. 7, the448
LQG controller is able to drive the system to the desired operating condition much faster and at the same449
time reduce the transient temperature differences within the solid matrix. This is achieved by delaying the450
blower actuation to speed up the heating of the ceramic absorber.451
452
As in the two previous scenarios, zero-mean Gaussian noise and disturbances with specified standard453
deviations are included in the simulations to assess the robustness of the LQG controller. Measurement454
noise with a standard deviation of 20C, 4Pa and 40kW/m2is added directly to the air outflow temperature,455
pressure, and radiation flux sensor readings, respectively. Disturbances with standard deviations of 0.1C/s456
for the temperatures and 0.001Pa/s for the pressure are added to the right hand side of the corresponding457
equations for the nonlinear system before integrating every time step (process noise).458
16
(a)
(c)
(b)
(d)
Figure 14: True, measured and estimated state variables as a function of time during a cold start-up procedure using LQG
control. (a) Air outflow temperature and solid matrix temperatures in the front and rear sections of the receiver. (b) Pressure
drop. (c) Radiation flux. (d) Temperature difference within the absorber.
Figure 14 shows the results for the simulations including noise and disturbances, where the gray thin459
lines show the sensor readings used by the LQG feedback controller. Again, the controller is able to reject460
disturbances noise and attenuate large amounts of noise while the estimated state variables agree with the461
true behavior of the nonlinear system.462
463
The effect of tuning the LQR cost function on the response of the closed-loop system in the cold start-up464
scenario is explored. The air outflow temperature, pressure drop, temperature differences within the solid465
matrix, and the radiation flux hitting the receiver are computed for different values of the state deviation cost466
for ˙
G, representing the velocity of the heliostats, between Q55 = 0.005–5, as shown in Fig. 15. Increasing the467
value of Q55 translates into a more aggressive heliostat actuation, that results in a faster thermal response468
at the expense of a higher overshoot in the temperature differences within the absorber. Being able to tune469
the controller is key to get a quick start-up of the receiver while avoiding thermal shock of the ceramic470
materials for a given receiver design.471
5. Conclusions472
We propose a data assimilation framework to perform simultaneous feedback control and state estima-473
tion in a volumetric solar receiver module using a physics-based model and sensors located in the outflow474
stream of air. First, we formulated a model for the conjugate heat transfer problem using only three degrees475
of freedom, thus making online calculations viable. Results are then compared to those obtained from full476
three-dimensional CFD and we show that our model is able to capture the relevant dynamics.477
478
The proposed framework was tested for three simulated transient scenarios to verify how some common479
issues in VSRs can be addressed. Firstly, during clear-sky daily operation, blower actuation is used to sta-480
bilize the outflow temperature and at the same time get an estimate of the temperature cycling in the solid481
17
(a)
(c)
(b)
(d)
Figure 15: Feedback control during a cold start-up procedure for different deviation costs for heliostat velocity between
Q55 = 0.005–5. (a) Air outflow temperature. (b) Pressure drop. (c) Temperature differences within the absorber. (d) Radiation
flux.
matrix and of the radiation flux hitting the absorber, which may be used to compute the thermal efficiency482
of the receiver and assess its state of health in real time. Secondly, during the passing of a cloud, the fluid483
temperature at the outlet is stabilized using blower actuation and assuming that information of the radiation484
flux drop is available, adding to the reliability of the downstream processes. Thirdly, for a cold start-up485
procedure, the system is guided from rest to steady state operation by actuating both, the suction pressure486
and the amount of radiation flux hitting the receiver. The controller may be tuned to trade between a faster487
start-up, increasing the energy yield, or a smaller overshoot of the temperature differences within the solid488
matrix, decreasing the risk of thermal shock. Artificial noise and disturbances are added to the system for489
all scenarios and the LQG controller proves to be robust, rejecting disturbances and attenuating noise, as490
well as compensating for model uncertainty.491
492
Interesting directions for future work consider the extension of the heat transfer model to account for493
transport processes between absorber modules, including the effect of temperature, radiation flux and mass494
flow rate distributions in the transverse plane. In addition, we are planning to test the feedback control495
and state estimation in an experimental setup, considering its interaction with the aiming control of the496
heliostat field. Online indirect measurements of the temperatures inside the ceramic absorber opens up new497
avenues for applied research, such as thermal fatigue and thermal shock studies using real operation data.498
Still, much progress is needed in these areas for the volumetric solar receiver to become a mature technology.499
Acknowledgements500
This work has been supported by Enerbosch SpA and by CORFO Chile under the grant CORFO-501
Contratos Tecnol´ogicos 18COTE-89602.502
18
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Obtaining reliable information on the thermal performance of a receiver in a solar thermal power plant during the design phase is essential for optimisation purposes and for the determination of the economic performance of the entire power plant. In open volumetric receivers, where the heat transfer circuit is open to the environment, this task is particularly challenging due to the influence of convection phenomena on the air return ratio and thus the heat losses. These losses can only be estimated by means of CFD simulations, however, since the small scale absorber structure is several orders of magnitude smaller than the receiver itself, the required computational grid becomes too large to be economically viable. In this paper a modelling approach is presented, which for the first time allows the efficient simulation of the entire receiver including the flow in front of and around the receiver while at the same time considering internal heat transfer within the absorber. A set of boundary conditions has been developed for the absorber surface of open volumetric receivers in which the characteristic behaviour of the absorber is modelled without the need of detailed resolution of the absorber structure. The model has been validated against new measurements at the solar tower Jülich. Moreover, in comparison with literature data of air return ratio measurements a very good agreement was found. A characteristic map of the receiver efficiency has been compiled showing peak efficiencies of the open volumetric receiver of the solar tower Jülich of 70.9% at a hot air temperature of 650 °C and 75.4% at 450 °C. The model can now be used for the assessment of design concepts for commercial power plants.
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Variable aperture can assist in maintaining semi-constant temperatures within a receiver's cavity under transient solar loading. An in-house code has been developed to model a receiver and effectively control its components to achieve semi-constant temperatures under transients. The code consists of a full optical analysis performed via the Monte Carlo ray tracing method in addition to a transient two-dimensional heat transfer analysis. The system studied consists of a cavity type solar receiver with 60 mm radius fixed aperture on the cavity body, a variable aperture mechanism mounted on the receiver's flange, and a 7 kW Xenon arc solar simulator. A composite shape consisting of a hemisphere attached to a cylinder is proposed to model the Xenon arc. The in-house code has been experimentally validated through experimental tests for different input currents to the solar simulator, volumetric flow rates, and aperture's radii. The optical analysis was validated based on heat flux measurements, where it had percentage errors of 0.8, 0.5, 1.1, and 3.2% for the peak power, total power, half width, and half power. For the heat transfer model, percentage errors of 3.2, 2.9, and 5.3% at the inlet, center, and outlet sections of the receiver were determined for different flow rates using maximum input current and opening radius. The aperture mechanism was capable of maintaining an exhaust temperature of 250 °C based on actual Direct Normal Irradiance data. Results showed that the variable aperture is a promising apparatus even in applications where the maximum temperatures are desired based on an observed optimum radius of 57.5 mm.
Article
The present paper explores the off-design performance of a CSP tower receiver due to the passage of clouds over the solar field. A quasi steady-state performance model is considered in the first part of the paper, accounting for the optical performance of the solar field and receiver only, yielding the expected incoming heat flux distribution on the aperture plane of the receiver and without further considerations about the thermal performance of the receiver itself. The twofold aim of this analysis is to identify the type of cloud that brings about the worst operating conditions of the receiver and to define an aiming strategy that produces the most homogeneous distribution of the incoming heat flux, this being a indirect metric of reduced thermal stress on this component. It is observed that larger clouds have a more negative impact on the heat flux distribution whereas the effect of smaller clouds is easily compensated for by other uncovered areas of the field. Large differences in performance arise when considering different aiming strategies, what leads to the selection of one of these as the optimum one given that it manages to evenly distribute the heat flux over the receiver surface. This limits the number and intensity of hot spots without affecting the overall power production. The optimum strategy is based on pointing the heliostats towards different aiming spots strategically located on the receiver surface, depending on the distance between tower and heliostat. The second part of the paper presents an in-house code developed to study the transient thermal performance of the receiver. The outcome of this second analysis confirms the results of the previous optical study: the selected optimum strategy leads to a 1% higher efficiency of the receiver. Also, the temperature distribution of the larger shadows is more harmful than when several smaller clouds pass over the solar field, both for the larger temperature gradients in the receiver and for the production of molten salts. This is also confirmed in the last part of the paper where different patterns of combined cloud passages are explored in regards to the charge/discharge process of the thermal energy storage system.
Article
An appropriate control of the heat flux distribution over the solar central receiver is essential to achieve an efficient and safe operation of solar tower systems. High solar radiation variation due to moving clouds may cause failures to the solar receiver. This paper shows a dynamic performance analysis of a solar central receiver operating when short-time cloud passages partially shade the solar field. The solar receiver incorporates an aiming methodology based on a closed loop model predictive control approach. The DNI changes are simulated using an agent-based model that closely emulates the transients in solar radiation caused by clouds. These models are coupled with a solar system model that resembles the Gemasolar solar plant. The simulations showed that the base feedback loop aiming strategy could successfully restore the solar receiver back to its steady state after transient operations caused by clouds. However, undesired overshoots in incident flux density and high heating rates in the controlled variables were found. These issues are overcome through a setpoint readjustment approach, which is temporally supported by a PI controller. The results show that the proposed aiming control strategy can provide a continuous safe operation of the solar central receiver when subject to transient flux distribution due to clouds.
Article
The performance of an open volumetric solar air receiver of honeycomb type (multiple parallel channels) for central tower CSP plants is evaluated numerically. A parametric study is conducted at the single channel level in order to investigate the influence of the main geometrical variables and of the air mass flow rate on the receiver performance. The adopted methodology consists of two steps: the first one is the optical analysis that is conducted using Tonatiuh, an open-source Monte-Carlo based ray-tracing software, providing the distribution of the absorbed heat flux on the channel inner surfaces, to be finally exploited as input data in the second step, i.e. the numerical evaluation of the thermal fluid dynamic performance of the channel. Using the commercial CFD software ANSYS Fluent, the convective heat transfer between the air flow and the absorber and the radiative heat transfer among the absorber inner walls and the channel aperture are simulated, computing the heat losses to the ambient. Different channel configurations are simulated, identifying the influence of the three key-parameters (the tilt angle with respect to the horizontal, the channel size and the air mass flow rate) on the receiver performance, in terms of solar-to-electricity efficiency.
Article
Solar irradiation is intermittent, but concentrated solar thermal (CST) plants are typically designed and analyzed solely based on their steady design point. Unlike coal power plants, however, CST plants frequently experience thermal loads well above and below their rated design point, leading to off-design operation for much of the operational year. Importantly, if a latent heat thermal energy storage (LHTES) system is employed, the receiver inlet temperature can vary under these conditions. To date, there is a clear lack of knowledge for how to handle off-design conditions in terms of developing appropriate control strategies to maximize the receiver thermal output and its operational region. In this study, a thermal model was developed and validated that is suitable for design/off-design performance analyses of molten salt cavity receivers in both steady state and transient conditions. The study investigated two control strategies – a fixed receiver flow rate (FF) and fixed receiver outlet temperature (FT) – for their off-design performance in each of two off-design operational modes (storage and non-storage). Solar field utilization (SFU) is variable in non-storage mode, but in the storage mode, it is whether variable or fixed at design point (SFU = 1). The feasible operating region in this study refers to the zone restricted by maximum allowable operational parameters defined based on design point analysis, mainly maximum receiver outlet temperature, maximum flow rate, and maximum receiver surface temperature.