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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 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.

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

0−p2

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 (Ta−T0). 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(Ta−T0) + hraAra (Tr−Ta0) + hca Aca (Tc−Ta),(3a)

125

Mrcr

dTr

dt=−hraAr a (Tr−Ta0)−hrc Arc (Tr−Tc) + εG −εσ T4

r−T4

0,(3b)

126

Mccc

dTc

dt=−hcaAca (Tc−Ta) + hrc Arc (Tr−Tc),(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(t−τ)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 = (A−Kr)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(y−ˆ

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 25◦C 1.1×107m−146.68m−218.3µPa 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= 25◦C 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)G−σT4

s−T4

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)G−σT4

s−T4

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 10−5and 10−3, respectively. The simulation times for the diﬀerent transient scenarios range between274

14 and 31 hours on a 4−core 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= 700◦C 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 ±1◦C290

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= 700◦C, 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)s−1their 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= 700◦C. 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 ∼500◦C, 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= 700◦C 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 ∼700◦C 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= 25◦C, 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= 700◦C 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 70◦C 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=p0−pL, 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= 700◦C, Tr= 713.7◦C, Tc= 703.5◦C, ∆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 20◦C and 4Pa is added directly to the air outﬂow temperature and pressure sensor readings, respectively.382

Disturbances with standard deviations of 0.1◦C/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= 700◦C,404

Tr= 904.3◦C, Tc= 751◦C, ∆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 20◦C and 4Pa is added directly to the air outﬂow temperature and pressure sensor418

readings, respectively. Disturbances with standard deviations of 0.1◦C/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= 700◦C, Tr= 713.7◦C,443

Tc= 703.5◦C, ∆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 20◦C, 4Pa and 40kW/m2is added directly to the air outﬂow temperature,455

pressure, and radiation ﬂux sensor readings, respectively. Disturbances with standard deviations of 0.1◦C/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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